8.1 Find Angle Measures in Polygons - RJS SOLUTIONS p Find angle measures in polygons. Your Notes...

VOCABULARY

Diagonal

Find Angle Measures in PolygonsGoal p Find angle measures in polygons.

Your Notes

THEOREM 8.1: POLYGON INTERIOR ANGLES THEOREM

The sum of the measures of the 1

2

456

3

n 5 6

interior angles of a convex n-gon is (n 2 ) p .

m∠1 1 m∠2 1 . . . 1 m∠n 5 (n 2 ) p

COROLLARY TO THEOREM 8.1: INTERIOR ANGLES OFA QUADRILATERAL

The sum of the measures of the interior angles of a quadrilateral is .

Find the sum of the measures of the interior angles of a convex hexagon.

Solution

A hexagon has sides. Use the Polygon Interior Angles Theorem.

(n 2 ) p 5 ( 2 ) p Substitute for n.

5 p Subtract.

5 Multiply.

The sum of the measures of the interior angles of a hexagon is .

Example 1 Find the sum of angle measures in a polygon

206 Lesson 8.1 • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved.

8.1

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VOCABULARY

Diagonal A diagonal of a polygon is a segment that joins two nonconsecutive vertices.

Find Angle Measures in PolygonsGoal p Find angle measures in polygons.

Your Notes

THEOREM 8.1: POLYGON INTERIOR ANGLES THEOREM

The sum of the measures of the 1

2

456

3

n 5 6

interior angles of a convex n-gon is (n 2 2 ) p 1808 .

m∠1 1 m∠2 1 . . . 1 m∠n 5 (n 2 2 ) p 1808

COROLLARY TO THEOREM 8.1: INTERIOR ANGLES OFA QUADRILATERAL

The sum of the measures of the interior angles of a quadrilateral is 3608 .

Find the sum of the measures of the interior angles of a convex hexagon.

Solution

A hexagon has 6 sides. Use the Polygon Interior Angles Theorem.

(n 2 2 ) p 1808 5 ( 6 2 2 ) p 1808 Substitute 6 for n.

5 4 p 1808 Subtract.

5 7208 Multiply.

The sum of the measures of the interior angles of a hexagon is 7208 .

Example 1 Find the sum of angle measures in a polygon


8.1


The sum of the measures of the interior angles of a convex polygon is 12608. Classify the polygon by the number of sides.

SolutionUse the Polygon Interior Angles Theorem to write an equation involving the number of sides n. Then solve the equation to find the number of sides.

(n 2 ) p 5 Polygon Interior Angles Theorem

n 2 5 Divide each side by .

n 5 Add to each side.

The polygon has sides. It is a .

Example 2 Find the number of sides of a polygon

1. Find the sum of the measures of the interior angles of the convex decagon.

Checkpoint Complete the following exercise.

Find the value of x in the diagram shown.

Solution x8 718

1358 1128The polygon is a quadrilateral. Use the Corollary to the Polygon Interior Angles Theorem to write an equation involving x. Then solve the equation.

x8 1 1 1 5 Corollary to Theorem 8.1

x 1 5 Combine like terms.

x 5 Subtract from each side.

Example 3 Find an unknown interior angle measure

Copyright © Holt McDougal. All rights reserved. Lesson 8.1 • Geometry Notetaking Guide 207

Your Notes


The sum of the measures of the interior angles of a convex polygon is 12608. Classify the polygon by the number of sides.

SolutionUse the Polygon Interior Angles Theorem to write an equation involving the number of sides n. Then solve the equation to find the number of sides.

(n 2 2 ) p 1808 5 12608 Polygon Interior Angles Theorem

n 2 2 5 7 Divide each side by 1808 .

n 5 9 Add 2 to each side.

The polygon has 9 sides. It is a nonagon .

Example 2 Find the number of sides of a polygon

1. Find the sum of the measures of the interior angles of the convex decagon.

14408



Solution x8 718

1358 1128The polygon is a quadrilateral. Use the Corollary to the Polygon Interior Angles Theorem to write an equation involving x. Then solve the equation.

x8 1 1358 1 1128 1 718 5 3608 Corollary to Theorem 8.1

x 1 318 5 360 Combine like terms.

x 5 42 Subtract 318 from each side.

Example 3 Find an unknown interior angle measure


Your Notes


2. The sum of the measures of the interior angles of a convex polygon is 16208. Classify the polygon by the number of sides.

3. Use the diagram at the right.

688

1058

1098J

N

LK

M

Find m∠K and m∠L.

Checkpoint Complete the following exercises.

THEOREM 8.2: POLYGON EXTERIOR ANGLES THEOREM

1

2 3

4

5

n 5 5

The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is .

m∠1 1 m∠2 1 . . . 1 m∠n 5


Solution

858

898

2x8

x8

Use the Polygon Exterior Angles Theorem to write and solve an equation.

x8 1 1 1 5 Polygon Exterior Angles Theorem

x 1 5 Combine like terms.

x 5 Solve for x.

Example 4 Find unknown exterior angle measures


Your Notes


2. The sum of the measures of the interior angles of a convex polygon is 16208. Classify the polygon by the number of sides.

11-gon

3. Use the diagram at the right.

688

1058

1098J

N

LK

M

Find m∠K and m∠L.

m∠K 5 m∠L 5 1298


THEOREM 8.2: POLYGON EXTERIOR ANGLES THEOREM

1

2 3

4

5

n 5 5

The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 3608 .

m∠1 1 m∠2 1 . . . 1 m∠n 5 3608


Solution

858

898

2x8

x8

Use the Polygon Exterior Angles Theorem to write and solve an equation.

x8 1 2x8 1 858 1 898 5 3608 Polygon Exterior Angles Theorem

3 x 1 174 5 360 Combine like terms.

x 5 62 Solve for x.

Example 4 Find unknown exterior angle measures


Your Notes


Lamps The base of a lamp is in the shape of a regular 15-gon. Find (a) the measure of each interior angle and (b) the measure of each exterior angle.

Solutiona. Use the Polygon Interior Angles Theorem to find the

sum of the measures of the interior angles.

(n 2 ) p 5 ( 2 ) p

5

Then find the measure of one interior angle. A regular 15-gon has congruent interior angles. Divide by : 4 5 .

The measure of each interior angle in the 15-gon is .

b. By the Polygon Exterior Angles Theorem, the sum of the measures of the exterior angles, one angle at each vertex, is . Divide by :

4 5 .

The measure of each exterior angle in the 15-gon is .

Example 5 Find angle measures in regular polygons

4. A convex pentagon has exterior angles with measures 668, 778, 828, and 628. What is the measure of an exterior angle at the fifth vertex?

5. Find the measure of (a) each interior angle and (b) each exterior angle of a regular nonagon.



Homework

Your Notes


Lamps The base of a lamp is in the shape of a regular 15-gon. Find (a) the measure of each interior angle and (b) the measure of each exterior angle.

Solutiona. Use the Polygon Interior Angles Theorem to find the

sum of the measures of the interior angles.

(n 2 2 ) p 1808 5 ( 15 2 2 ) p 1808

5 23408

Then find the measure of one interior angle. A regular 15-gon has 15 congruent interior angles. Divide 23408 by 15 : 23408 4 15 5 1568 .

The measure of each interior angle in the 15-gon is 1568 .

b. By the Polygon Exterior Angles Theorem, the sum of the measures of the exterior angles, one angle at each vertex, is 3608 . Divide 3608 by 15 : 3608 4 15 5 248 .

The measure of each exterior angle in the 15-gon is 248 .

Example 5 Find angle measures in regular polygons

4. A convex pentagon has exterior angles with measures 668, 778, 828, and 628. What is the measure of an exterior angle at the fifth vertex?

738

5. Find the measure of (a) each interior angle and (b) each exterior angle of a regular nonagon.

a. 1408

b. 408



Homework

Your Notes


Use Properties of Parallelograms

8.2

VOCABULARY

Parallelogram

THEOREM 8.3

If a quadrilateral is a parallelogram, then its opposite sides are congruent.

If PQRS is a parallelogram, then

R

SP > } RS and } QR > .

THEOREM 8.4

If a quadrilateral is a parallelogram, then its opposite angles are congruent.


R

SP∠P > and > ∠S.

Find the values of x and y. F

J H

G688

x 1 6

13y8

SolutionFGHJ is a parallelogram by the definition of a parallelogram. Use Theorem 8.3 to find the value of x.

FG 5 Opposite sides of a ~ are >.

x 1 6 5 Substitute x 1 6 for FG and for .

x 5 Subtract 6 from each side.

By Theorem 8.4, ∠F > , or m∠F 5 . So, y8 5 .

In ~FGHJ, x 5 and y 5 .

Example 1 Use properties of parallelograms

Goal p Find angle and side measures in parallelograms.


Your Notes


Use Properties of Parallelograms

8.2

VOCABULARY

Parallelogram A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

THEOREM 8.3

If a quadrilateral is a parallelogram, then its opposite sides are congruent.


R

SP }

PQ > } RS and } QR > }

PS .

THEOREM 8.4

If a quadrilateral is a parallelogram, then its opposite angles are congruent.


R

SP∠P > ∠R and ∠Q > ∠S.

Find the values of x and y. F

J H

G688

x 1 6

13y8

SolutionFGHJ is a parallelogram by the definition of a parallelogram. Use Theorem 8.3 to find the value of x.

FG 5 HJ Opposite sides of a ~ are >.

x 1 6 5 13 Substitute x 1 6 for FG and 13 for HJ .

x 5 7 Subtract 6 from each side.

By Theorem 8.4, ∠F > ∠H , or m∠F 5 m∠H . So, y8 5 688 .

In ~FGHJ, x 5 7 and y 5 68 .

Example 1 Use properties of parallelograms

Goal p Find angle and side measures in parallelograms.


Your Notes



THEOREM 8.5

If a quadrilateral is a parallelogram, then its consecutive angles are

.

R

SPy8

y8x8

x8

If PQRS is a parallelogram, then x8 1 y8 5 .

Gates As shown, a gate contains A

B

C

D

several parallelograms. Find m∠ADC when m∠DAB 5 658.

SolutionBy Theorem 8.5, the consecutive angle pairs in ~ABCD are . So, m∠ADC 1 m∠DAB 5 . Because m∠DAB 5 658, m∠ADC 5 2 5 .

Example 2 Use properties of a parallelogram

1. x 2. y

3. z

Checkpoint Find the indicated measure in ~KLMN shown at the right.

L M

NK

1238

2x 2 3

37

y8z8

Your Notes



THEOREM 8.5

If a quadrilateral is a parallelogram, then its consecutive angles are supplementary .

R

SPy8

y8x8

x8

If PQRS is a parallelogram, then x8 1 y8 5 1808 .

Gates As shown, a gate contains A

B

C

D

several parallelograms. Find m∠ADC when m∠DAB 5 658.

SolutionBy Theorem 8.5, the consecutive angle pairs in ~ABCD are supplementary . So, m∠ADC 1 m∠DAB 5 1808 . Because m∠DAB 5 658, m∠ADC 5 1808 2 658 5 1158 .


1. x 2. y

x 5 20 y 5 123

3. z

z 5 57

Checkpoint Find the indicated measure in ~KLMN shown at the right.

L M

NK

1238

2x 2 3

37

y8z8

Your Notes


THEOREM 8.6

If a quadrilateral is a parallelogram, then its diagonals each other.

R

M

SP

} QM > and

} PM >

The diagonals of ~STUV

x

y

1

1

T U

VS

W

intersect at point W. Find the coordinates of W.

SolutionBy Theorem 8.6, the diagonals of a parallelogram each other. So, W is the of the diagonals } TV and } SU . Use the .

Coordinates of midpoint W of

} SU 5 1 2 5 1 2


4. The diagonals of ~VWXY

x

y

1

1

W X

YV

Z

intersect at point Z. Find the coordinates of Z.

5. Given that ~FGHJ is GF

M

HJ

5a parallelogram, find MH and FH.


In Example 3, you can use either diagonal to find the coordinates of W. Using } SU simplifies calculations because one endpoint is (0, 0).


Homework

Your Notes


THEOREM 8.6

If a quadrilateral is a parallelogram, then its diagonals bisect each other.

R

M

SP

} QM > }

SM and

} PM > }

RM

The diagonals of ~STUV

x

y

1

1

T U

VS

W

intersect at point W. Find the coordinates of W.

SolutionBy Theorem 8.6, the diagonals of a parallelogram bisect each other. So, W is the midpoint of the diagonals } TV and } SU . Use the Midpoint Formula .

Coordinates of midpoint W of

} SU 5 1 6 1 0 } 2 , 5 1 0

} 2 2 5 1 3, 5 } 2 2


4. The diagonals of ~VWXY

x

y

1

1

W X

YV

Z

intersect at point Z. Find the coordinates of Z.

Z 1 7 } 2 , 3 2

5. Given that ~FGHJ is GF

M

HJ

5a parallelogram, find MH and FH.

MH 5 5, FH 5 10


In Example 3, you can use either diagonal to find the coordinates of W. Using } SU simplifies calculations because one endpoint is (0, 0).


Homework

Your Notes


8.3


Show that a Quadrilateral is a Parallelogram

Basketball In the diagram at the right,

D B

A

C

18 in.

18 in.36 in.

36 in.

} AB and } DC represent adjustable supports of a basketball hoop. Explain why } AD is always parallel to } BC .

SolutionThe shape of quadrilateral ABCD changes as the adjustable supports move, but its

do not change. Both pairs of opposite are congruent, so ABCD is a parallelogram by .

By the definition of a parallelogram, } AD i .

Example 1 Solve a real-world problem

Goal p Use properties to identify parallelograms.

THEOREM 8.7

If both pairs of opposite of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

CB

DA

If } AB > and } BC > , then ABCD is a parallelogram.

THEOREM 8.8

If both pairs of opposite of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

CB

DA

If ∠A > and ∠B > , then ABCD is a parallelogram.

Your Notes


8.3


Show that a Quadrilateral is a Parallelogram

Basketball In the diagram at the right,

D B

A

C

18 in.

18 in.36 in.

36 in.

} AB and } DC represent adjustable supports of a basketball hoop. Explain why } AD is always parallel to } BC .

SolutionThe shape of quadrilateral ABCD changes as the adjustable supports move, but its side lengths do not change. Both pairs of opposite sides are congruent, so ABCD is a parallelogram by Theorem 8.7 .

By the definition of a parallelogram, } AD i }

BC .


Goal p Use properties to identify parallelograms.

THEOREM 8.7

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

CB

DA

If } AB > }

CD and } BC > } AD , then ABCD is a parallelogram.

THEOREM 8.8

If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

CB

DA

If ∠A > ∠C and ∠B > ∠D , then ABCD is a parallelogram.

Your Notes


THEOREM 8.9

If one pair of opposite sides of a quadrilateral are and , then the quadrilateral is a parallelogram.

CB

DAIf } BC } AD and } BC } AD , then ABCD is a parallelogram.

THEOREM 8.10

If the diagonals of a quadrilateral each other, then the

quadrilateral is a parallelogram.

CB

DA

If } BD and } AC each other, then ABCD is a parallelogram.

Lights The headlights of a car have the CB

DA

shape shown at the right. Explain how you know that ∠B > ∠D.

Solution

In the diagram, } BC i and } BC > . By , quadrilateral ABCD is a parallelogram.

By , you know that opposite angles of a parallelogram are congruent. So, ∠B > .

Example 2 Identify a parallelogram

1. In quadrilateral GHJK, m∠G 5 558, m∠H 5 1258, and m∠J 5 558. Find m∠K. What theorem can you use to show that GHJK is a parallelogram?

2. What theorem can you use 7

7

4

4to show that the quadrilateral is a parallelogram?



Your Notes


THEOREM 8.9

If one pair of opposite sides of a quadrilateral are congruent and parallel , then the quadrilateral is a parallelogram.

CB

DAIf } BC i } AD and } BC > } AD , then ABCD is a parallelogram.

THEOREM 8.10

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

CB

DA

If } BD and } AC bisect each other, then ABCD is a parallelogram.

Lights The headlights of a car have the CB

DA

shape shown at the right. Explain how you know that ∠B > ∠D.

Solution

In the diagram, } BC i } AD and } BC > } AD . By Theorem 8.9 , quadrilateral ABCD is a parallelogram. By Theorem 8.4 , you know that opposite angles of a parallelogram are congruent. So, ∠B > ∠D .

Example 2 Identify a parallelogram

1. In quadrilateral GHJK, m∠G 5 558, m∠H 5 1258, and m∠J 5 558. Find m∠K. What theorem can you use to show that GHJK is a parallelogram?

m∠K 5 1258; Theorem 8.8

2. What theorem can you use 7

7

4

4to show that the quadrilateral is a parallelogram?

Theorem 8.10



Your Notes



For what value of x is quadrilateral

P RT

S

5x 2x 1 9

PQRS a parallelogram?

SolutionBy Theorem 8.10, if the diagonals of PQRS each other, then it is a parallelogram. You are given that } QT > . Find x so that } PT > .

PT 5 Set the segment lengths equal.

5x 5 Substitute 5x for PT and for .

x 5 Subtract from each side.

x 5 Divide each side by .

When x 5 , PT 5 5( ) 5 and RT 5 2( ) 1 9 5 .

Quadrilateral PQRS is a parallelogram when x 5 .

Example 3 Use algebra with parallelograms

CONCEPT SUMMARY: WAYS TO PROVE A QUADRILATERAL IS A PARALLELOGRAM

1. Show both pairs of opposite sides are parallel. (Definition)

2. Show both pairs of opposite sides are congruent. (Theorem 8.7)

3. Show both pairs of opposite angles are congruent. (Theorem 8.8)

4. Show one pair of opposite sides are congruent and parallel. (Theorem 8.9)

5. Show the diagonals bisect each other. (Theorem 8.10)

Your Notes



For what value of x is quadrilateral

P RT

S

5x 2x 1 9

PQRS a parallelogram?

SolutionBy Theorem 8.10, if the diagonals of PQRS bisect each other, then it is a parallelogram. You are given that } QT >

} ST . Find x so that } PT > } RT .

PT 5 RT Set the segment lengths equal.

5x 5 2x 1 9 Substitute 5x for PT and 2x 1 9 for RT .

3 x 5 9 Subtract 2x from each side.

x 5 3 Divide each side by 3 .

When x 5 3 , PT 5 5( 3 ) 5 15 and RT 5 2( 3 ) 1 9 5 15 .

Quadrilateral PQRS is a parallelogram when x 5 3 .

Example 3 Use algebra with parallelograms

CONCEPT SUMMARY: WAYS TO PROVE A QUADRILATERAL IS A PARALLELOGRAM

1. Show both pairs of opposite sides are parallel. (Definition)

2. Show both pairs of opposite sides are congruent. (Theorem 8.7)

3. Show both pairs of opposite angles are congruent. (Theorem 8.8)

4. Show one pair of opposite sides are congruent and parallel. (Theorem 8.9)

5. Show the diagonals bisect each other. (Theorem 8.10)

Your Notes


Show that quadrilateral KLMN

x

y

1

1

K(2, 2)

L(4, 4)

M(6, 0)

N(4, 22)

is a parallelogram.

SolutionOne way is to show that a pair of sides are congruent and parallel. Then apply .

First use the Distance Formula to show that } KL and } MN are .

KL 5 Ï}}

5 Ï}

MN 5 Ï}}}

5 Ï}

Because KL 5 MN 5 Ï}

, } KL } MN .

Then use the slope formula to show that } KL } MN .

Slope of } KL 5 5

Slope of } MN 5 5

} KL and } MN have the same slope, so they are .

} KL and } MN are congruent and parallel. So, KLMN is a parallelogram by .

Example 4 Use coordinate geometry

3. For what value of x is quadrilateralDFGH a parallelogram?

F G

HD

2x

4x 2 7

4. Explain another method that can be used to show that quadrilateral KLMN in Example 4 is a parallelogram.



Homework

Your Notes


Show that quadrilateral KLMN

x

y

1

1

K(2, 2)

L(4, 4)

M(6, 0)

N(4, 22)

is a parallelogram.

SolutionOne way is to show that a pair of sides are congruent and parallel. Then apply Theorem 8.9 .

First use the Distance Formula to show that } KL and } MN are congruent .

KL 5 Ï}}

(4 2 2)2 1 (4 2 2)2 5 Ï}

8

MN 5 Ï}}}

(6 2 4)2 1 [0 2(22)]2 5 Ï}

8

Because KL 5 MN 5 Ï}

8 , } KL > } MN .

Then use the slope formula to show that } KL i } MN .

Slope of } KL 5 4 2 2

4 2 2 5 1

Slope of } MN 5 0 2 (22)

6 2 4 5 1

} KL and } MN have the same slope, so they are parallel .

} KL and } MN are congruent and parallel. So, KLMN is a parallelogram by Theorem 8.9 .

Example 4 Use coordinate geometry

3. For what value of x is quadrilateralDFGH a parallelogram?

F G

HD

2x

4x 2 7 x 5 3.5

4. Explain another method that can be used to show that quadrilateral KLMN in Example 4 is a parallelogram.

Sample Answer: Draw the diagonals and find the point of intersection. Show the diagonals bisect each other and apply Theorem 8.10.



Homework

Your Notes



RHOMBUS COROLLARY

A quadrilateral is a rhombus if and A

D C

B

only if it has four congruent .

ABCD is a rhombus if and only if } AB > } BC > } CD > } AD .

RECTANGLE COROLLARY

A quadrilateral is a rectangle if and A

D C

B

only if it has four .

ABCD is a rectangle if and only if ∠A, ∠B, ∠C, and ∠D are right angles.

SQUARE COROLLARY

A quadrilateral is a square if and only if A

D C

B

it is a and a .

ABCD is a square if and only if } AB > } BC > } CD > } AD and ∠A, ∠B, ∠C, and ∠D are right angles.

8.4 Properties of Rhombuses, Rectangles, and SquaresGoal p Use properties of rhombuses, rectangles, and

squares.

Your Notes VOCABULARY

Rhombus

Rectangle

Square



RHOMBUS COROLLARY

A quadrilateral is a rhombus if and A

D C

B

only if it has four congruent sides .

ABCD is a rhombus if and only if } AB > } BC > } CD > } AD .

RECTANGLE COROLLARY

A quadrilateral is a rectangle if and A

D C

B

only if it has four right angles .

ABCD is a rectangle if and only if ∠A, ∠B, ∠C, and ∠D are right angles.

SQUARE COROLLARY

A quadrilateral is a square if and only if A

D C

B

it is a rhombus and a rectangle .

ABCD is a square if and only if } AB > } BC > } CD > } AD and ∠A, ∠B, ∠C, and ∠D are right angles.

8.4 Properties of Rhombuses, Rectangles, and SquaresGoal p Use properties of rhombuses, rectangles, and

squares.

Your Notes VOCABULARY

Rhombus A rhombus is a parallelogram with four congruent sides.

Rectangle A rectangle is a parallelogram with four right angles.

Square A square is a parallelogram with four congruent sides and four right angles.


1. For any square CDEF, is it always or sometimes true that } CD > } DE ? Explain your reasoning.

2. A quadrilateral has four congruent sides and four congruent angles. Classify the quadrilateral.


Classify the special quadrilateral. 1278

Explain your reasoning.

The quadrilateral has four congruent . One of the angles is not a , so the rhombus is not also a . By the Rhombus Corollary, the quadrilateral is a .

Example 2 Classify special quadrilaterals

For any rhombus RSTV, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning.

a. ∠S > ∠V b. ∠T > ∠V

Solutiona. By definition, a rhombus is a

V T

SR

parallelogram with four congruent . By Theorem 8.4, opposite

angles of a parallelogram are . So, ∠S > ∠V. The

statement is true.

b. If rhombus RSTV is a , then

V T

SR

all four angles are congruent right angles. So ∠T > ∠V if RSTV is a

. Because not all rhombuses are also , the statement is

true.

Example 1 Use properties of special quadrilaterals


Your Notes


1. For any square CDEF, is it always or sometimes true that } CD > } DE ? Explain your reasoning.

Always; a square has four congruent sides.

2. A quadrilateral has four congruent sides and four congruent angles. Classify the quadrilateral.

square


Classify the special quadrilateral. 1278

Explain your reasoning.

The quadrilateral has four congruent sides . One of the angles is not a right angle , so the rhombus is not also a square . By the Rhombus Corollary, the quadrilateral is a rhombus .

Example 2 Classify special quadrilaterals

For any rhombus RSTV, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning.

a. ∠S > ∠V b. ∠T > ∠V

Solutiona. By definition, a rhombus is a

V T

SR

parallelogram with four congruent sides . By Theorem 8.4, opposite angles of a parallelogram are congruent . So, ∠S > ∠V. The statement is always true.

b. If rhombus RSTV is a square , then

V T

SR

all four angles are congruent right angles. So ∠T > ∠V if RSTV is a square . Because not all rhombuses are also squares , the statement is sometimes true.

Example 1 Use properties of special quadrilaterals


Your Notes



Sketch rhombus FGHJ. List everything you know about it.

SolutionBy definition, you need to draw a

J H

GF

figure with the following properties:

• The figure is a .• The figure has four congruent .

Because FGHJ is a parallelogram, it has these properties:

• Opposite sides are and .• Opposite angles are . Consecutive angles

are .• Diagonals each other.

By Theorem 8.11, the diagonals of FGHJ are . By Theorem 8.12, each diagonal

bisects a pair of .

Example 3 List properties of special parallelograms

THEOREM 8.11

A parallelogram is a rhombus if and A

D C

B

only if its diagonals are .

~ABCD is a rhombus if and only if ⊥ .

THEOREM 8.12


D C

B

only if each diagonal bisects a pair of opposite angles.

~ABCD is a rhombus if and only if } AC bisects ∠ and ∠ and } BD bisects ∠ and ∠ .

THEOREM 8.13

A parallelogram is a rectangle if and A

D C

B

only if its diagonals are .

~ABCD is a rectangle if and only if > .

Your Notes



Sketch rhombus FGHJ. List everything you know about it.

SolutionBy definition, you need to draw a

J H

GF

figure with the following properties:

• The figure is a parallelogram .• The figure has four congruent sides .

Because FGHJ is a parallelogram, it has these properties:

• Opposite sides are parallel and congruent .• Opposite angles are congruent . Consecutive angles

are supplementary .• Diagonals bisect each other.

By Theorem 8.11, the diagonals of FGHJ are perpendicular . By Theorem 8.12, each diagonal bisects a pair of opposite angles .

Example 3 List properties of special parallelograms

THEOREM 8.11


D C

B

only if its diagonals are perpendicular .

~ABCD is a rhombus if and only if }

AC ⊥ } BD .

THEOREM 8.12


D C

B

only if each diagonal bisects a pair of opposite angles.

~ABCD is a rhombus if and only if } AC bisects ∠ BCD and ∠ BAD and } BD bisects ∠ ABC and ∠ ADC .

THEOREM 8.13

A parallelogram is a rectangle if and A

D C

B

only if its diagonals are congruent .

~ABCD is a rectangle if and only if }

AC > } BD .

Your Notes


3. Sketch rectangle WXYZ. List everything that you know about it.

4. Suppose the diagonals of the frame in Example 4 are not congruent.

Could the frame still be a rectangle? Explain.


Framing You are building a frame for

16 in.16 in.

20 in.

20 in.

a painting. The measurements of the frame are shown at the right.

a. The frame must be a rectangle. Given the measurements in the diagram, can you assume that it is? Explain.

b. You measure the diagonals of the frame. The diagonals are about 25.6 inches. What can you conclude about the shape of the frame?

Solutiona. No, you cannot. The boards on opposite sides are the

same length, so they form a . But you do not know whether the angles are .

b. By Theorem 8.13, the diagonals of a rectangle are . The diagonals of the frame are , so the frame forms a .



Homework

Your Notes


3. Sketch rectangle WXYZ. List everything that you know about it.

W X

Z Y

WXYZ is a parallelogram with four right angles. Opposite sides are parallel and congruent. Opposite angles are congruent and consecutive angles are supplementary. The diagonals are congruent and bisect each other.

4. Suppose the diagonals of the frame in Example 4 are not congruent.

Could the frame still be a rectangle? Explain.

No; by Theorem 8.13, a rectangle must have congruent diagonals.


Framing You are building a frame for

16 in.16 in.

20 in.

20 in.

a painting. The measurements of the frame are shown at the right.

a. The frame must be a rectangle. Given the measurements in the diagram, can you assume that it is? Explain.

b. You measure the diagonals of the frame. The diagonals are about 25.6 inches. What can you conclude about the shape of the frame?

Solutiona. No, you cannot. The boards on opposite sides are the

same length, so they form a parallelogram . But you do not know whether the angles are right angles .

b. By Theorem 8.13, the diagonals of a rectangle are congruent . The diagonals of the frame are congruent , so the frame forms a rectangle .



Homework

Your Notes



8.5 Use Properties of Trapezoids and KitesGoal p Use properties of trapezoids and kites.

VOCABULARY

Trapezoid

Bases of a trapezoid

Base angles of a trapezoid

Legs of a trapezoid

Isosceles trapezoid

Midsegment of a trapezoid

Kite

Your Notes



8.5 Use Properties of Trapezoids and KitesGoal p Use properties of trapezoids and kites.

VOCABULARY

Trapezoid A trapezoid is a quadrilateral with exactly one pair of parallel sides.

Bases of a trapezoid The parallel sides of a trapezoid are the bases.

Base angles of a trapezoid A trapezoid has two pairs of base angles. Each pair shares a base as a side.

Legs of a trapezoid The nonparallel sides of a trapezoid are the legs.

Isosceles trapezoid An isosceles trapezoid is a trapezoid in which the legs are congruent.

Midsegment of a trapezoid The midsegment of a trapezoid is the segment that connects the midpoints of its legs.

Kite A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

Your Notes


Show that CDEF is a trapezoid.

x

y

1

4C(0, 0)

D(1, 3) E(4, 4)

F(6, 2)SolutionCompare the slopes of opposite sides.

Slope of } DE 5 5

Slope of } CF 5 5 5

The slopes of } DE and } CF are the same, so } DE } CF .

Slope of } EF 5 5 5

Slope of } CD 5 5 5

The slopes of } EF and } CD are not the same, so } EF is to } CD .

Because quadrilateral CDEF has exactly one pair of , it is a trapezoid.

Example 1 Use a coordinate plane

THEOREM 8.14

If a trapezoid is isosceles, then each

A D

B C

pair of base angles is .

If trapezoid ABCD is isosceles, then ∠A > ∠ and ∠ > ∠C.

THEOREM 8.15

If a trapezoid has a pair of congruent

A D

B C

, then it is an isosceles trapezoid.

If ∠A > ∠D (or if ∠B > ∠C), then trapezoid ABCD is isosceles.

THEOREM 8.16

A trapezoid is isosceles if and only A D

B C

if its diagonals are .

Trapezoid ABCD is isosceles if and only if > .


Your Notes


Show that CDEF is a trapezoid.

x

y

1

4C(0, 0)

D(1, 3) E(4, 4)

F(6, 2)SolutionCompare the slopes of opposite sides.

Slope of } DE 5 4 2 3 } 4 2 1 5 1 } 3

Slope of } CF 5 2 2 0 } 6 2 0 5 2 } 6 5 1 } 3

The slopes of } DE and } CF are the same, so } DE i } CF .

Slope of } EF 5 2 2 4 } 6 2 4 5 22

} 2 5 21

Slope of } CD 5 3 2 0 } 1 2 0 5 3 } 1 5 3

The slopes of } EF and } CD are not the same, so } EF is not parallel to } CD .

Because quadrilateral CDEF has exactly one pair of parallel sides , it is a trapezoid.

Example 1 Use a coordinate plane

THEOREM 8.14

If a trapezoid is isosceles, then each

A D

B C

pair of base angles is congruent .

If trapezoid ABCD is isosceles, then ∠A > ∠ D and ∠ B > ∠C.

THEOREM 8.15

If a trapezoid has a pair of congruent

A D

B C

base angles , then it is an isosceles trapezoid.

If ∠A > ∠D (or if ∠B > ∠C), then trapezoid ABCD is isosceles.

THEOREM 8.16

A trapezoid is isosceles if and only A D

B C

if its diagonals are congruent .

Trapezoid ABCD is isosceles if and only if }

AC > } BD .


Your Notes


Kitchen A shelf fitting into a L

K N

M

508

cupboard in the corner of a kitchen is an isosceles trapezoid. Find m∠N, m∠L, and m∠M.

Solution

Step 1 Find m∠N. KLMN is an , so ∠N and ∠ are congruent base angles, and m∠N 5 m∠ 5 .

Step 2 Find m∠L. Because ∠K and ∠L are consecutive interior angles formed by @##$ KL intersecting two parallel lines, they are . So, m∠L 5 2 5 .

Step 3 Find m∠M. Because ∠M and ∠ are a pair of base angles, they are congruent, and m∠M 5 m∠ 5 .

So, m∠N 5 , m∠L 5 , and m∠M 5 .

Example 2 Use properties of isosceles trapezoids

1. In Example 1, suppose the coordinates of point E are (7, 5). What type of quadrilateral is CDEF? Explain.

2. Find m∠C, m∠A, and m∠D A

B C

D

1358in the trapezoid shown.



Your Notes


Kitchen A shelf fitting into a L

K N

M

508

cupboard in the corner of a kitchen is an isosceles trapezoid. Find m∠N, m∠L, and m∠M.

Solution

Step 1 Find m∠N. KLMN is an isosceles trapezoid , so ∠N and ∠ K are congruent base angles, and m∠N 5 m∠ K 5 508 .

Step 2 Find m∠L. Because ∠K and ∠L are consecutive interior angles formed by @##$ KL intersecting two parallel lines, they are supplementary . So, m∠L 5 1808 2 508 5 1308 .

Step 3 Find m∠M. Because ∠M and ∠ L are a pair of base angles, they are congruent, and m∠M 5 m∠ L 5 1308 .

So, m∠N 5 508 , m∠L 5 1308 , and m∠M 5 1308 .

Example 2 Use properties of isosceles trapezoids

1. In Example 1, suppose the coordinates of point E are (7, 5). What type of quadrilateral is CDEF? Explain.

Parallelogram; opposite pairs of sides are parallel.

2. Find m∠C, m∠A, and m∠D A

B C

D

1358in the trapezoid shown.

m∠C 5 1358, m∠A 5 458, m∠D 5 458



Your Notes


3. Find MN in the trapezoid at P

S

Q

R

M

N

30 ft

12 ftthe right.


THEOREM 8.17: MIDSEGMENT THEOREM FOR TRAPEZOIDS

The midsegment of a trapezoid is A B

C

M N

D

parallel to each base and its length is one half the sum of the lengths of the bases.

If } MN is the midsegment of trapezoid ABCD, then

} MN i , } MN i , and MN 5 ( 1 ).

In the diagram, } MN is the midsegment P Q

NM

S R

16 in.

9 in.

of trapezoid PQRS. Find MN.

SolutionUse Theorem 8.17 to find MN.

MN 5 ( 1 ) Apply Theorem 8.17.

5 ( 1 ) Substitute for PQ and for SR.

5 Simplify.

The length MN is inches.

Example 3 Use the midsegment of a trapezoid


Your Notes

LAH_GE_11_FL_NTG_Ch08_206-232.indd 224LAH_GE_11_FL_NTG_Ch08_206-232.indd 224 12/17/09 1:45:38 AM12/17/09 1:45:38 AM

3. Find MN in the trapezoid at P

S

Q

R

M

N

30 ft

12 ftthe right.

MN 5 21 ft


THEOREM 8.17: MIDSEGMENT THEOREM FOR TRAPEZOIDS

The midsegment of a trapezoid is A B

C

M N

D

parallel to each base and its length is one half the sum of the lengths of the bases.

If } MN is the midsegment of trapezoid ABCD, then

} MN i } AB , } MN i }

DC , and MN 5 1 } 2 ( AB 1 CD ).

In the diagram, } MN is the midsegment P Q

NM

S R

16 in.

9 in.

of trapezoid PQRS. Find MN.

SolutionUse Theorem 8.17 to find MN.

MN 5 1 } 2 ( PQ 1 SR ) Apply Theorem 8.17.

5 1 } 2 ( 16 1 9 ) Substitute 16 for PQ and 9 for SR.

5 12.5 Simplify.

The length MN is 12.5 inches.

Example 3 Use the midsegment of a trapezoid


Your Notes



THEOREM 8.18

If a quadrilateral is a kite, then its diagonals are . B

C

A

D

If quadrilateral ABCD is a kite, then ⊥ .

THEOREM 8.19

If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

B

C

A

D

If quadrilateral ABCD is a kite and } BC > } BA , then ∠A ∠C and ∠B ∠D.

Find m∠T in the kite shown at the right.

Solution

TS

RQ708

888By Theorem 8.19, QRST has exactly one pair of opposite angles. Because ∠Q À ∠S, ∠ and ∠T must be congruent. So, m∠ 5 m∠T. Write and solve an equation to find m∠T.

m∠T 1 m∠R 1 1 5 Corollary to Theorem 8.1

m∠T 1 m∠T 1 1 5 Substitute m∠T for m∠R.

(m∠T ) 1 5 Combine like terms.

m∠T 5 Solve for m∠T.

Example 4 Apply Theorem 8.19

4. Find m∠G in the kite shown G H

IJ

858

758

at the right.

Checkpoint Complete the following exercise.Homework

Your Notes



THEOREM 8.18

If a quadrilateral is a kite, then its diagonals are perpendicular . B

C

A

D

If quadrilateral ABCD is a kite, then

} AC ⊥ } BD .

THEOREM 8.19

If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

B

C

A

D

If quadrilateral ABCD is a kite and } BC > } BA , then ∠A > ∠C and ∠B À ∠D.

Find m∠T in the kite shown at the right.

Solution

TS

RQ708

888By Theorem 8.19, QRST has exactly one pair of congruent opposite angles. Because ∠Q À ∠S, ∠ R and ∠T must be congruent. So, m∠ R 5 m∠T. Write and solve an equation to find m∠T.

m∠T 1 m∠R 1 708 1 888 5 3608 Corollary to Theorem 8.1

m∠T 1 m∠T 1 708 1 888 5 3608 Substitute m∠T for m∠R.

2 (m∠T ) 1 1588 5 3608 Combine like terms.

m∠T 5 1018 Solve for m∠T.

Example 4 Apply Theorem 8.19

4. Find m∠G in the kite shown G H

IJ

858

758

at the right.

m∠G 5 1008

Checkpoint Complete the following exercise.Homework

Your Notes


8.6 Identify Special Quadrilaterals

Quadrilateral ABCD has both pairs of opposite sides congruent. What types of quadrilaterals meet this condition?

SolutionThere are many possibilities.

Opposite sides are congruent. All sides are congruent.

Example 1 Identify quadrilaterals

1. Quadrilateral JKLM has both pairs of opposite angles congruent. What types of quadrilaterals meet this condition?


What is the most specific name A B

D C

for quadrilateral ABCD?

SolutionThe diagram shows that both pairs of opposite sides are congruent. By Theorem 8.7, ABCD is a . All sides are congruent, so ABCD is a by definition.

are also rhombuses. However, there is no information given about the angle measures of ABCD. So, you cannot determine whether it is a .

Example 2 Identify a quadrilateralIn Example 2, ABCD is shaped like a square. But you must rely only on marked information when you interpret a diagram.

Goal p Identify special quadrilaterals.


Your Notes


8.6 Identify Special Quadrilaterals

Quadrilateral ABCD has both pairs of opposite sides congruent. What types of quadrilaterals meet this condition?

SolutionThere are many possibilities.

Parallelogram Rectangle Rhombus Square

Opposite sides are congruent. All sides are congruent.

Example 1 Identify quadrilaterals

1. Quadrilateral JKLM has both pairs of opposite angles congruent. What types of quadrilaterals meet this condition?

parallelogram, rectangle, square, rhombus


What is the most specific name A B

D C

for quadrilateral ABCD?

SolutionThe diagram shows that both pairs of opposite sides are congruent. By Theorem 8.7, ABCD is a parallelogram . All sides are congruent, so ABCD is a rhombus by definition.

Squares are also rhombuses. However, there is no information given about the angle measures of ABCD. So, you cannot determine whether it is a square .

Example 2 Identify a quadrilateralIn Example 2, ABCD is shaped like a square. But you must rely only on marked information when you interpret a diagram.

Goal p Identify special quadrilaterals.


Your Notes


Is enough information given in the F

J H

G1028 1028

788

diagram to show that quadrilateral FGHJ is an isosceles trapezoid? Explain.

Solution

Step 1 Show that FGHJ is a . ∠G and ∠H are but ∠F and ∠G are not.

So, i , but } FJ is not to } GH . By definition, FGHJ is a .

Step 2 Show that trapezoid FGHJ is . ∠F and ∠G are a pair of congruent . So, FGHJ is an by Theorem 8.15.

Yes, the diagram is sufficient to show that FGHJ is an isosceles trapezoid.

Example 3 Identify a quadrilateral

2. What is the most specific name R

ST

7

7

8

8

for quadrilateral QRST? Explain your reasoning.

3. Is enough information given in the B C

E D

diagram to show that quadrilateral BCDE is a rectangle? Explain.



Homework

Your Notes


Is enough information given in the F

J H

G1028 1028

788

diagram to show that quadrilateral FGHJ is an isosceles trapezoid? Explain.

Solution

Step 1 Show that FGHJ is a trapezoid . ∠G and ∠H are supplementary but ∠F and ∠G are not.

So, }

FG i }

HJ , but } FJ is not parallel to } GH . By definition, FGHJ is a trapezoid .

Step 2 Show that trapezoid FGHJ is isosceles . ∠F and ∠G are a pair of congruent base angles . So, FGHJ is an isosceles trapezoid by Theorem 8.15.

Yes, the diagram is sufficient to show that FGHJ is an isosceles trapezoid.

Example 3 Identify a quadrilateral

2. What is the most specific name R

ST

7

7

8

8

for quadrilateral QRST? Explain your reasoning.

Kite; there are two pairs of consecutive congruent sides.

3. Is enough information given in the B C

E D

diagram to show that quadrilateral BCDE is a rectangle? Explain.

Yes; you know that m∠D 5 908 by the Corollary to Theorem 8.1. Both pairs of opposite angles are congruent, so BCDE is a parallelogram by Theorem 8.8. By definition, BCDE is a rectangle.



Homework

Your Notes


8.7 Coordinate Proof with QuadrilateralsGoal p Use coordinate geometry to prove properties of

quadrilaterals.

Your Notes

Determine if the quadrilaterals with the given vertices are congruent.O(0, 0), B(2, 4), C(4, 4), D(2, 0)

E(24, 0), F(22, 0), G(0, 4) H(22, 4)

SolutionGraph the quadrilaterals. Show that corresponding and

are congruent.

Use the Distance Formula.

EH 5 FG 5 OB 5 CD 5 Ï}

5

EF 5 HG 5 OD 5 BC 5

Since both pairs of opposite sides in each quadrilateral are , and are parallelograms.

angles in a parallelogram are congruent,

so and . Lines @##$ OB and @##$ EH are , because both have slope 2, and they are cut

by transversal @##$ EO so ∠E and ∠O are , and ∠E ≅ ∠O. By substitution, .

Similar reasoning can be used to show that and .

Because all corresponding sides and angles congruent, OBCD congruent to EHGF.

Example 1 Determine if quadrilaterals are congruent

1. O(0, 0), B(24, 22), C(24, 6), D(22, 6)R(6, 22), S(2, 24), T(2, 4) U(4, 6)

Checkpoint Find the side lengths of the quadrilaterals with the given vertices. Then determine if the quadrilaterals are congruent.


y

x1

1


8.7 Coordinate Proof with QuadrilateralsGoal p Use coordinate geometry to prove properties of

quadrilaterals.

Your Notes

Determine if the quadrilaterals with the given vertices are congruent.O(0, 0), B(2, 4), C(4, 4), D(2, 0)

E(24, 0), F(22, 0), G(0, 4) H(22, 4)

SolutionGraph the quadrilaterals. Show that corresponding sides and angles are congruent.

Use the Distance Formula.

EH 5 FG 5 OB 5 CD 5 Ï}

20 5 2 Ï}

5

EF 5 HG 5 OD 5 BC 5 2

Since both pairs of opposite sides in each quadrilateral are congruent, OBCD and EHGF are parallelograms.

Opposite angles in a parallelogram are congruent,

so ∠E ≅ ∠G and ∠O ≅ ∠C. Lines @##$ OB and @##$ EH are parallel, because both have slope 2, and they are cut by transversal @##$ EO so ∠E and ∠O are corresponding angles, and ∠E ≅ ∠O. By substitution, ∠G ≅ ∠C .Similar reasoning can be used to show that ∠H ≅ ∠B and ∠F ≅ ∠D .Because all corresponding sides and angles are congruent, OBCD is congruent to EHGF.

Example 1 Determine if quadrilaterals are congruent

1. O(0, 0), B(24, 22), C(24, 6), D(22, 6)R(6, 22), S(2, 24), T(2, 4) U(4, 6)

OB 5 RS 5 Ï}

20 ; BC 5 ST 5 8; CD 5 2; TU 5 Ï}

8 ; DO 5 Ï

}

40 ; RU 5 Ï}

68 ; not congruent

Checkpoint Find the side lengths of the quadrilaterals with the given vertices. Then determine if the quadrilaterals are congruent.


y

x1

1

HG

F O

B C

DE


Example 2 Determine if quadrilaterals are similar

Determine if the quadrilaterals with the given vertices are similar.O(0, 0), B(2, 2), C(6, 2), D(4, 0)

O(0, 0), F(3, 3), G(9, 3) H(6, 0)

SolutionGraph the quadrilaterals. Find the ratios of corresponding side lengths.

OF } OB 5 3 Ï

}

2 }

2 Ï}

2 5 } FG

} BC 5 6 } 4 5 }

CD } HG 5 3 Ï

}

2 }

2 Ï}

2 5 } OH

} OD 5 6 } 4 5 }

Because OB 5 CD and BC 5 OD, is a .

Because OF 5 HG and FG 5 OH, is a .

angles in a parallelogram are , so ∠O ≅ ∠C and ∠O ≅ ∠G. Therefore ∠C ≅ ∠G. Parallel lines @##$ FG and @##$ BC are cut by @##$ FB , so ∠F and ∠CBO are and ∠F ≅ ∠CBO.

Likewise @##$ GH and @##$ DC are because both have slope 1, and they are cut by transversal @##$ DH , so ∠H and ∠ODC are , and .

Because corresponding side lengths are proportional and corresponding angles congruent, OBCD similar to OFGH.


2. O(0, 0), B(4, 22), C(4, 24), D(0, 24)O(0, 0), T(2, 22), U(1, 23) V(0, 23)

Checkpoint Determine if the quadrilaterals with the given vertices are similar.

y

x1

1

Your Notes


Example 2 Determine if quadrilaterals are similar

Determine if the quadrilaterals with the given vertices are similar.O(0, 0), B(2, 2), C(6, 2), D(4, 0)

O(0, 0), F(3, 3), G(9, 3) H(6, 0)

SolutionGraph the quadrilaterals. Find the ratios of corresponding side lengths.

OF } OB 5 3 Ï

}

2 }

2 Ï}

2 5

3 }

2 FG

} BC 5 6 } 4 5 3

} 2

CD } HG 5 3 Ï

}

2 }

2 Ï}

2 5

3 }

2 OH

} OD 5 6 } 4 5 3

} 2

Because OB 5 CD and BC 5 OD, OBCD is a parallelogram.

Because OF 5 HG and FG 5 OH, OFGH is a parallelogram.

Opposite angles in a parallelogram are congruent, so ∠O ≅ ∠C and ∠O ≅ ∠G. Therefore ∠C ≅ ∠G. Parallel lines @##$ FG and @##$ BC are cut by transversal @##$ FB , so ∠F and ∠CBO are corresponding angles and ∠F ≅ ∠CBO.

Likewise @##$ GH and @##$ DC are parallel lines because both have slope 1, and they are cut by transversal @##$ DH , so ∠H and ∠ODC are corresponding angles, and ∠H ≅ ∠ODC .

Because corresponding side lengths are proportional and corresponding angles are congruent, OBCD is similar to OFGH.


2. O(0, 0), B(4, 22), C(4, 24), D(0, 24)O(0, 0), T(2, 22), U(1, 23) V(0, 23)

OBCD is not similar to OTUV.

Checkpoint Determine if the quadrilaterals with the given vertices are similar.

y

x

B C

D H

GF

1

1

O

Your Notes


Show that the parallelogram is not an isosceles trapezoid.

Solution

Use the Distance Formula. BD 5 and AC 5 Ï}

. Since the measures of the are

, the trapezoid an isosceles trapezoid.

Example 3 Demonstrate properties of quadrilaterals

3. What other ways could you use to show that ABCD is not an isosceles triangle?

Checkpoint Use properties of trapezoids.

Without introducing any new variables, supply the missing coordinates for P so that OPQR is a parallelogram.

SolutionChoose coordinates so that sides of the quadrilateral are .

} PQ must be to be to } OR , so the y-coordinate of P is .

To find the x-coordinate of P, write expressions for the slopes of } OP and } RQ . Use x for the of P.

slope of } RQ 5 c 2 }

2 a 5 } slope of } OP 5 c 2 0

} x 2 0 5 }

The slopes are , so 5 . Therefore 5 .

The point P has coordinates .

Example 4 Determine coordinates for a vertex


y

x1

1

C

DA

B

y

x

P(?, ?)

O(0, 0)

Q(b, c)

R(a, 0)

Your Notes


Show that the parallelogram is not an isosceles trapezoid.

Solution

Use the Distance Formula. BD 5 3 Ï}

5 and AC 5 Ï}

34 . Since the measures of the diagonals are not congruent, the trapezoid is not an isosceles trapezoid.

Example 3 Demonstrate properties of quadrilaterals

3. What other ways could you use to show that ABCD is not an isosceles triangle?

Show that the base angles are not congruent.

Checkpoint Use properties of trapezoids.

Without introducing any new variables, supply the missing coordinates for P so that OPQR is a parallelogram.

SolutionChoose coordinates so that opposite sides of the quadrilateral are parallel.

} PQ must be horizontal to be parallel to } OR , so the y-coordinate of P is c.

To find the x-coordinate of P, write expressions for the slopes of } OP and } RQ . Use x for the x-coordinate of P.

slope of } RQ 5 c 2 0 }

b 2 a 5

c }

b 2 a slope of } OP 5 c 2 0

} x 2 0 5 c }

x

The slopes are equal, so c } b 2 a 5 c } x . Therefore b − a 5 x.

The point P has coordinates 1 b 2 a, c 2 .

Example 4 Determine coordinates for a vertex


y

x1

1

C

DA

B

y

x

P(?, ?)

O(0, 0)

Q(b, c)

R(a, 0)

Your Notes



Prove that the diagonals of a square are perpendicular.

Solution

Step 1 Place a square with side length a on the axes. Draw one side of the square from the origin to a point B(0, a) on the positive y-axis. Draw the second side from B to C(a, ) within the first quadrant. Draw the third side from C to D( , ) on the positive x-axis. Draw the fourth side from D back to the origin. Draw the diagonals.

Step 2 Find the of each diagonal.

Slope of } OC 5

} 5 } 5

Slope of } BD 5 } 5 } 5

Since the of their slopes is , the diagonals are .

Example 5 Write a coordinate proof

4. O(0, 0), S(a, 0), T(a, b), U(0, b)O(0, 0), B(2a, 0), C(2a, 2b), D(0, 2b)

Checkpoint Verify that two quadrilaterals are congruent.

y

xO(0, 0)

Homework

Your Notes



Prove that the diagonals of a square are perpendicular.

Solution

Step 1 Place a square with side length a on the axes. Draw one side of the square from the origin to a point B(0, a) on the positive y-axis. Draw the second side from B to C(a, a) within the first quadrant. Draw the third side from C to D(a, 0) on the positive x-axis. Draw the fourth side from D back to the origin. Draw the diagonals.

Step 2 Find the slope of each diagonal.

Slope of } OC 5 a 2 0

} a 2 0

5 a }

a 5 1

Slope of } BD 5 0 2 a

} a 2 0

5 2a }

a 5 21

Since the product of their slopes is –1 , the diagonals are perpendicular.

Example 5 Write a coordinate proof

4. O(0, 0), S(a, 0), T(a, b), U(0, b)O(0, 0), B(2a, 0), C(2a, 2b), D(0, 2b)

Quadrilateral OSTU and OBCD are both rectangles with horizontal and vertical sides. All angles in a rectangle are right angles, so the corresponding angles in the quadrilaterals are congruent. From the distance formula, the length of the corresponding sides are also equal: OS 5 OB 5 a, ST 5 BC 5 b, TU 5 CD 5 a, and UO 5 DO 5 b. Because corresponding angles and sides are congruent, OSTU is congruent to OBCD.

Checkpoint Verify that two quadrilaterals are congruent.

y

xD (a, 0)

C (a, a)B(0, a)

O(0, 0)

Homework

Your Notes


Words to ReviewGive an example of the vocabulary word.

Diagonal

Rhombus

Square

Bases, Legs, and Base angles of a trapezoid


Parallelogram

Rectangle

Trapezoid

Isosceles trapezoid

Kite

Review your notes and Chapter 8 by using the Chapter Review on pages 585–588 of your textbook.

232 Words to Review • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved.


Words to ReviewGive an example of the vocabulary word.

Diagonal

diagonal

Rhombus

Square

Bases, Legs, and Base angles of a trapezoid

bases legs

pair of base angles


midsegment

Parallelogram

Rectangle

Trapezoid

Isosceles trapezoid

Kite

Review your notes and Chapter 8 by using the Chapter Review on pages 585–588 of your textbook.

232 Words to Review • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved.


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