Special Topics Eleanor Roosevelt High School Chin-Sung Lin.

Special Topics

Eleanor Roosevelt High School

Chin-Sung Lin

Similar Triangles

Mr. Chin-Sung Lin

ERHS Math Geometry

Definition of Similar Triangles

Mr. Chin-Sung Lin

Triangles are similar if their corresponding angles are equal and their corresponding sides are proportional

(The number represented by the ratio of similitude is called the constant of proportionality)

ERHS Math Geometry

Example of Similar Triangles

Mr. Chin-Sung Lin

A X, B Y, C Z

AB = 6, BC = 8, and CA = 10

XY = 3, YZ = 4 and ZX = 5

Show that ABC~XYZX

Y Z

3

4

5

A

B C

6

8

10

ERHS Math Geometry


Mr. Chin-Sung Lin

A X, B Y, C Z

AB BC CA 2

XY YZ ZX 1

Therefore ABC~XYZ

= = =

ERHS Math Geometry

X

Y Z

3

4

5

A

B C

6

8

10


Mr. Chin-Sung Lin

The sides of a triangle have lengths 4, 6, and 8.

Find the sides of a larger similar triangle if the constant of proportionality is 5/2

4

6

8 ?

?

?

ERHS Math Geometry


Mr. Chin-Sung Lin

Assume x, y, and z are the sides of the larger triangle, then

x 5 y 5 z 5

4 2 8 2 6 2

4

6

8 x = 10

z = 15

y = 20

ERHS Math Geometry

= = =


Mr. Chin-Sung Lin

In ABC, AB = 9, BC = 15, AC = 18.

If ABC~XYZ, and XZ = 12, find XY and YZ

ERHS Math Geometry

X

Y Z

?

?

129

15

18

A

B C


Mr. Chin-Sung Lin

Since ABC~XYZ, and XZ = 12, then

XY YZ 12

9 15 18

X

Y Z

6

10

129

15

18

A

B C

ERHS Math Geometry

= =


Mr. Chin-Sung Lin

In ABC, AB = 4y – 1, BC = 8x + 2, AC = 8.

If ABC~XYZ, and XZ = 6, find XY and YZ

ERHS Math Geometry

X

Y Z

?

?

64y – 1

8x + 2

8

A

B C


Mr. Chin-Sung Lin

Since ABC~XYZ, and XZ = 6, then

XY YZ 6

4y–1 8x+2 8

X

Y Z

3y–¾

6x+ 3/2

64y – 1

8x + 2

8

A

B C

ERHS Math Geometry

= =

Prove Similarity

Mr. Chin-Sung Lin

ERHS Math Geometry

Angle-Angle Similarity Theorem (AA~)

Mr. Chin-Sung Lin

If two angles of one triangle are congruent to two corresponding angles of another triangle, then the triangles are similar

Given: ABC and XYZ with A X, and C Z

Prove: ABC~XYZ

X

Y Z

A

B C

ERHS Math Geometry

Example of AA Similarity Theorem

Mr. Chin-Sung Lin

Given: mA = 45 and mD = 45

Prove: ABC~DEC

45o

A

B

C 45o

D

E

ERHS Math Geometry

Example of AA Similarity Theorem

Mr. Chin-Sung Lin

Statements Reasons

1. mA = 45 and mD = 45 1. Given

2. A D 2. Substitution property

3. ACB DCE 3. Vertical angles

4. ABC~DEC 4. AA similarity theorem

45oA

B

C45o

D

E

ERHS Math Geometry

Side-Side-Side Similarity Theorem (SSS~)

Mr. Chin-Sung Lin

Two triangles are similar if the three ratios of corresponding sides are equal

Given: AB/XY = AC/XZ = BC/YZ

Prove: ABC~XYZ

ERHS Math Geometry

X

Y Z

A

B C

Side-Angle-Side Similarity Theorem (SAS~)

Mr. Chin-Sung Lin

Two triangles are similar if the ratios of two pairs of corresponding sides are equal and the corresponding angles included between these sides are congruent

Given: A X, AB/XY = AC/XZ

Prove: ABC~XYZ

X

Y Z

A

B C

ERHS Math Geometry

Example of SAS Similarity Theorem

Mr. Chin-Sung Lin

Prove: ABC~DEC

Calculate: DE

16A

B

CD

E

1012

8

6 ?

ERHS Math Geometry

Example of SAS Similarity Theorem

Mr. Chin-Sung Lin

Prove: ABC~DEC

Calculate: DE

16A

B

CD

E

1012

8

6 5

ERHS Math Geometry

Triangle Proportionality Theorem

Mr. Chin-Sung Lin

ERHS Math Geometry


Mr. Chin-Sung Lin

If a line parallel to one side of a triangle intersects the other two sides, then it divides them proportionally

Given: DE || BC

Prove: AD AE

DB EC =

D E

A

B C

ERHS Math Geometry


Mr. Chin-Sung Lin

If a line parallel to one side of a triangle intersects the other two sides, then it divides them proportionally

DE || BC

AD AE

DB EC

AD AE DE

AB AC BCD E

A

B C

= =

ERHS Math Geometry

=

Converse of Triangle Proportionality Theorem

Mr. Chin-Sung Lin

If the points at which a line intersects two sides of a triangle divide those sides proportionally, then the line is parallel to the third side

Given: AD AE

DB EC

Prove: DE || BC

=

D E

A

B C

ERHS Math Geometry

Example of Triangle Proportionality Theorem

Mr. Chin-Sung Lin

Given: DE || BC, AD = 4, BD = 3, AE = 6

Calculate: CE and BC

8

3

4 6

?D E

A

BC

?

ERHS Math Geometry


Mr. Chin-Sung Lin

Given: DE || BC, AD = 4, BD = 3, AE = 6

Calculate: CE and BC

8

3

4 6

4.5D E

A

BC

14

ERHS Math Geometry


Mr. Chin-Sung Lin

Given: DE || BC, AE = 6, BD = 4, DE = 8, and BC = 12

Calculate: EC and AD

8

4

? 6

?D E

A

BC

12

ERHS Math Geometry


Mr. Chin-Sung Lin

Given: DE || BC, AE = 6, BD = 4, DE = 8, and BC = 12

Calculate: EC and AD

8

4

8 6

3D E

A

BC

12

ERHS Math Geometry


Mr. Chin-Sung Lin

Given: DE || BC, BD = 5, AC = 10, DE = 8, and BC = 12

Calculate: AE and AB

8

5

?

10?

D E

A

BC

12

ERHS Math Geometry


Mr. Chin-Sung Lin

Given: DE || BC, BD = 5, AC = 10, DE = 8, and BC = 12

Calculate: AE and AB

8

5

15

1020/3

D E

A

BC

12

ERHS Math Geometry

Pythagorean Theorem

Mr. Chin-Sung Lin

ERHS Math Geometry

Pythagorean Theorem

Mr. Chin-Sung Lin

A triangle is a right triangle if and only if the square of the length of the longest side is equal to the sum of the squares of the lengths of the other two sides

ABC, mC = 90

if and only if

a2 + b2 = c2

ERHS Math Geometry

A

C Ba

bc

Pythagorean Example - Distance

Mr. Chin-Sung Lin

Find the distance between A and B.

ERHS Math Geometry

A (5, 3)

B(2, 1)

?


Mr. Chin-Sung Lin


ERHS Math Geometry

A (5, 3)

B(2, 1)C (5, 1)

| 3 – 1 | = 2

| 5 – 2 | = 3

?


Mr. Chin-Sung Lin


ERHS Math Geometry

A (5, 3)

B(2, 1)

√13

C (5, 1)

| 3 – 1 | = 2

| 5 – 2 | = 3

Parallelograms

Mr. Chin-Sung Lin

ERHS Math Geometry

Theorems of Parallelogram

Mr. Chin-Sung Lin

Theorem of Dividing Diagonals

Theorem of Opposite Sides

Theorem of Opposite Angles

Theorem of Bisecting Diagonals

Theorem of Consecutive Angles

ERHS Math Geometry

Criteria for Proving Parallelograms

Mr. Chin-Sung Lin

Parallel opposite sides

Congruent opposite sides

Congruent & parallel opposite sides

Congruent opposite angles

Supplementary consecutive angles

Bisecting diagonals

ERHS Math Geometry

Rectangles

Mr. Chin-Sung Lin

ERHS Math Geometry

Rectangles

Mr. Chin-Sung Lin

A rectangle is a parallelogram containing one right angle

A B

CD

ERHS Math Geometry

Properties of Rectangle

Mr. Chin-Sung Lin

The properties of a rectangle

All the properties of a parallelogram

Four right angles (equiangular)

Congruent diagonals A B

CD

ERHS Math Geometry

Proving Rectangles

Mr. Chin-Sung Lin

To show that a quadrilateral is a rectangle, by showing that the quadrilateral is equiangular or a parallelogram

that contains a right angle, or with congruent diagonals

If a parallelogram does not contain a right angle, or doesn’t have congruent diagonals, then it is not a rectangle

ERHS Math Geometry

Rhombuses

Mr. Chin-Sung Lin

ERHS Math Geometry

Rhombus

Mr. Chin-Sung Lin

A rhombus is a parallelogram that has two congruent consecutive sides

A

B

C

D

ERHS Math Geometry

Properties of Rhombus

Mr. Chin-Sung Lin

The properties of a rhombus


Four congruent sides (equilateral)

Perpendicular diagonals

Diagonals that bisect opposite pairs of angles

A

B

C

D

ERHS Math Geometry

Proving Rhombus

Mr. Chin-Sung Lin

To show that a quadrilateral is a rhombus, by showing that the quadrilateral is equilateral or a parallelogram

that contains two congruent consecutive sides with perpendicular diagonals, or with diagonals bisecting opposite angles

If a parallelogram does not contain two congruent consecutive sides, or doesn’t have perpendicular diagonals, then it is not a rectangle

ERHS Math Geometry

Application Example

Mr. Chin-Sung Lin

ABCD is a parallelogram. AB = 2x + 1, DC = 3x - 11, AD = x + 13

Prove: ABCD is a rhombusA B

D C

2x+1

3x-11

x+13

ERHS Math Geometry

Application Example

Mr. Chin-Sung Lin

ABCD is a parallelogram. AB = 2x + 1, DC = 3x - 11, AD = x + 13

Prove: ABCD is a rhombus

x = 12AB = AD = 25ABCD is a rhombus

A B

D C

2x+1

3x-11

x+13

ERHS Math Geometry

Application Example

ABCD is a parallelogram, AB = 3x - 2, BC = 2x + 2, and CD = x + 6. Show that ABCD is a rhombus A

B

C

D

ERHS Math Geometry

Mr. Chin-Sung Lin

Application Example

ABCD is a parallelogram, AB = 3x - 2, BC = 2x + 2, and CD = x + 6. Show that ABCD is a rhombus

x = 4

AB = BC = 10

ABCD is a rhombus

A

B

C

D

ERHS Math Geometry

Mr. Chin-Sung Lin

Squares

Mr. Chin-Sung Lin

ERHS Math Geometry

Squares

Mr. Chin-Sung Lin

A square is a rectangle that has two congruent consecutive sides

A B

CD

ERHS Math Geometry

Squares

Mr. Chin-Sung Lin

A square is a rectangle with four congruent sides (an equilateral rectangle)

ERHS Math Geometry

A B

CD

Squares

Mr. Chin-Sung Lin

A square is a rhombus with four right angles (an equiangular rhombus)

ERHS Math Geometry

A B

CD

Squares

Mr. Chin-Sung Lin

A square is an equilateral quadrilateral

A square is an equiangular quadrilateral

ERHS Math Geometry

A B

CD

Squares

Mr. Chin-Sung Lin

A square is a rhombus

A square is a rectangle

ERHS Math Geometry

A B

CD

Properties of Square

Mr. Chin-Sung Lin

The properties of a square


All the properties of a rectangle

All the properties of a rhombus

A B

CD

ERHS Math Geometry

Proving Squares

Mr. Chin-Sung Lin

To show that a quadrilateral is a square, by showing that the quadrilateral is a

rectangle with a pair of congruent consecutive sides, or

a rhombus that contains a right angle

ERHS Math Geometry

Application Example

ABCD is a square, mA = 4x - 30, AB = 3x + 10 and BC = 4y. Solve x and y

A B

CD

ERHS Math Geometry

Mr. Chin-Sung Lin

Application Example

ABCD is a square, mA = 4x - 30, AB = 3x + 10 and BC = 4y. Solve x and y

4x – 30 = 90

x = 30

y = 25

A B

CD

ERHS Math Geometry

Mr. Chin-Sung Lin

Trapezoids

Mr. Chin-Sung Lin

ERHS Math Geometry

Trapezoids

Mr. Chin-Sung Lin

A trapezoid is a quadrilateral that has exactly one pair of parallel sides

The parallel sides of a trapezoid are called bases. The nonparallel sides of a trapezoid are the legs

A B

CD

Upper base

Lower base

LegLeg

ERHS Math Geometry

Isosceles Trapezoids

Mr. Chin-Sung Lin

A trapezoid whose nonparallel sides are congruent is called an isosceles trapezoid

ERHS Math Geometry

A B

CD

Upper base

Lower base

LegLeg

Properties of Isosceles Trapezoids

Mr. Chin-Sung Lin

The properties of a isosceles trapezoid

Base angles are congruent

Diagonals are congruent

The property of a trapezoid

Median is parallel to and average of the bases

ERHS Math Geometry

Proving Trapezoids

Mr. Chin-Sung Lin

To prove that a quadrilateral is a trapezoid, show that two sides are parallel and the other two sides are not parallel

To prove that a quadrilateral is not a trapezoid, show that both pairs of opposite sides are parallel or that both pairs of opposite sides are not parallel

ERHS Math Geometry

Proving Isosceles Trapezoids

Mr. Chin-Sung Lin

To prove that a trapezoid is an isosceles trapezoid, show that one of the following statements is true:

The legs are congruent

The lower/upper base angles are congruent

The diagonals are congruent

ERHS Math Geometry

Numeric Example of Trapezoids

Mr. Chin-Sung Lin

Isosceles Trapezoid ABCD, AB || CD and AD BC

Solve for x and yA B

CD

2xo

xo 3yo

ERHS Math Geometry

Numeric Example of Trapezoids

Mr. Chin-Sung Lin

Isosceles Trapezoid ABCD, AB || CD and AD BC

Solve for x and y

x = 60

y = 20

A B

CD

2xo

xo 3yo

ERHS Math Geometry

Q & A

Mr. Chin-Sung Lin

ERHS Math Geometry

The End

Mr. Chin-Sung Lin

ERHS Math Geometry

Special Topics Eleanor Roosevelt High School Chin-Sung Lin.

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