Chapter 5

Quadrilaterals

Mid-Term Exam Review Project

Chapter 5: Break Down

• 5-1: Properties of Parallelograms

• 5-2: Ways to Prove that Quadrilaterals are Parallelograms

• 5-3: Theorems Involving Parallelograms

• 5-4 : Special Quadrilaterals

• 5-5: Trapezoids

Chapter 5: Section 1

Properties of Parallelograms

Parallelograms

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

But there are more characteristics of parallelograms than just that…

These additional characteristics are applied in several theorems

throughout the section.

Theorem 5-1

“Opposite sides of a parallelogram are congruent.”

This theorem states that the sides in a parallelogram that are opposite of each other are congruent. This means that in ABCD, side AB is congruent to side CD and side AC to side BD.

A B

C D

Theorem 5-2

“Opposite Angles of a parallelogram are congruent”

This theorem states that in a parallelogram, the opposite angles of a

parallelogram are congruent.

Theorem 5-3

“Diagonals of a parallelogram bisect each other”

This theorem states that in any parallelogram the diagonals bisect

each other.

Section 5-2

Ways to Prove that Quadrilaterals are Parallelograms

5 Ways to Prove that a Quadrilateral is a Parallelogram

• Show that both pairs of opposite sides are parallel

• Show that both pairs of opposite sides are congruent

• Show that one pair of opposite sides are both congruent and parallel

• Show that both pairs of opposite angles are congruent

• Show that the diagonals bisect each other

Theorem 5-4

“If both pairs of opposite sides of a quadrilateral are congruent, then the

quadrilateral is a parallelogram.”

Theorem 5-4T S

Q R

This theorem states that if side TS is congruent to side QR and side QT is congruent to side SR, then the quadrilateral is a parallelogram.

Theorem 5-5

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

parallelogram.”

Theorem 5-5

• Theorem 5-5 states that if side VL and side AD are both parallel and congruent, then the quadrilateral is a parallelogram.

V L

DA

Theorem 5-6

“If both pairs of opposite angles of a quadrilateral are congruent, then the

quadrilateral is a parallelogram.”

Theorem 5-6

• This theorem states that if angle M is congruent to angle E, and angle I is congruent to angle K, then quadrilateral MIKE is a parallelogram.

M

EK

I

Theorem 5-7

If the diagonals of a quadrilateral bisect each other, then the

quadrilateral is a parallelogram.

Theorem 5-7

• This theorem states that if segment HA is congruent to segment AT, and segment MA is congruent to segment AD, that quadrilateral HDMT is a parallelogram.

D

A

M T

H

Chapter 5: Section 3

• Theorems Involving Parallel Lines

Theorem 5-8

• “If two lines are parallel, then all points on one line are equidistant from the other line.”

All of the points on a line are equidistant from points on

the perpendicular bisector of a parallel line.

Theorem 5-9

• “If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.”

Theorem 5-10

• “A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side.”

Theorem 5-11

• “The segment that joins the midpoints of two sides of a triangle.”

1) is parallel to the third side

2) is half the length of the third side

Section 5-4

Special Parallelograms

Special Parallelograms

• There are four distinct special parallelograms that have unique characteristics.

• These special parallelograms include:

• Rectangles

• Rhombi

• Squares

Rectangles

• A rectangle is a quadrilateral with four right angles. Every rectangle, therefore, is a parallelogram.

Why?

• Every rectangle is a parallelogram because all angles in a rectangle are right, and all right angles are congruent.

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

Additionally:

• If a rectangle is a parallelogram, then it retains all of the characteristics of a parallelogram. If this is true then name all congruencies in the following diagram.

S A

ED

R

Solution:

S A

R

D E

Rhombi

• A rhombus is a quadrilateral with four congruent sides. Therefore, every rhombus is a parallelogram.

Why?

• Every rhombus is a parallelogram because all four sides are congruent, and therefore both pairs of opposite sides are congruent.

Squares

• A square is a quadrilateral with four right angles and four congruent sides. Therefore, every square is a rectangle, a rhombus, and a parallelogram.

Why?

• A square is a rectangle a rhombus, and a parallelogram because all four sides are congruent and all four angles are right angles.

Theorems

• Theorem’s 5-12 through 5-13 are very simple and self explanatory. They read as follows:

• 5-12: The diagonals of a rectangle are congruent

• 5-13: The diagonals of a rhombus are perpendicular.

• 5-14: Each diagonal of a rhombus bisects two angles of the rhombus.

Theorem 5-12

• According to theorem 5-12, segment SE is congruent to segment DA

S A

D E

R

Theorem 5-13

• According to theorem 5-13, segment AD is congruent to segment CB.

D

B C

A

Theorem 5-14

• According to theorem 5-14, segment AD bisects angle BAC and angle BCD, and segment BC bisects angle ABD and angle ACD.

D

B C

A

Theorem 5-15

“The Midpoint of the hypotenuse of a right triangle is equidistant from the

three vertices.”

Theorem 5-15

M

According to this theorem, M is equidistant from points A, B, and C.

A

BC

Theorem 5-16

If an angle of a parallelogram is a right angle, then the parallelogram is

a rectangle.

Theorem 5-16

• In MATH, we know that angle H is a right angle. This means that all of the angles are right angles, and that MATH is a rectangle. (According to Theorem 5-16)

TH

AM

Theorem 5-17

If two consecutive sides of a parallelogram are congruent, then the

parallelogram is a rhombus.

Theorem 5-17

• In MATH, if we know that segment MA is congruent to segment AT, then we know that MATH is a rhombus, because segment MA is congruent to segment TH and segment AT is congruent to segment MH. Therefore, all sides are congruent to each other.

M

H

T

A

Chart of Special QuadrilateralsProperty Parallelogram Rectangle Rhombus Square

Opp. Sides Parallel ♫ ♫ ♫ ♫

Opp. Sides Congruent ♫ ♫ ♫ ♫

Opp. Angles Congruent ♫ ♫ ♫ ♫

A diag. forms two congruent ∆s ♫ ♫ ♫ ♫

Diags. Bisect each other ♫ ♫ ♫ ♫

Diags are congruent ♫ ♫

Diags. are perpendicular ♫ ♫

A diag. bisects two angles ♫ ♫

All angles are Right Angles ♫ ♫

All sides are congruent ♫ ♫

Section 5-5

Trapezoids

Trapezoid

• A trapezoid is defined as a quadrilateral with exactly one pair of parallel sides.

• The parallel sides are called the bases.

• The other sides are the legs

Isosceles Trapezoid

• A trapezoid with congruent legs is called an Isosceles Trapezoid.

Theorems

• The Following Theorems Concern Trapezoids and their dimensions.

Theorem 5-18

Base angles of an isosceles trapezoid are congruent.

• This theorem states that trapezoid HAIR is isosceles, then angle R is congruent to angle I, and angle H is congruent to angle A.

H

R I

A

Theorem 5-19

The Median of a trapezoid:

(1) Is parallel to the bases;

(2) Has a length equal to the average of the bases.

The Median of a Trapezoid

• The median of a trapezoid is the segment that joins the midpoints of the legs.

• In Trapezoid ANDR, EW is the median because it joins the midpoints, E and W, of the legs.

A

W E

N

R D

Theorem 5-19

• According to this theorem, Segment EW (a) is parallel to AN and DR, and a length equal to the average of the lengths of AN and DR.

A

W E

N

R D

Use the Theorems to Complete:

• Given: AN = 10; DR = 20; AR is congruent to ND

• WE =?• Angle R is congruent to ?• Angle A is congruent to ?

A

W E

N

R D

Solution:

• WE = 15

• Angle R is congruent to Angle D

• Angle A is congruent to Angle N

A

W E

N

R D

The End

Brought to you by:•ReuvenSmells™ Productions and

•GiveUsAnA Publishing

This presentation by:

• Chris O’Connell

• Brent Sneider

• Jason Fernandez

• Mike

• Russell Waldman

FIN