Geometry Fall Semester

Name: ______________

Chapter 8: Quadrilaterals

Guided Notes

CH. 8 Guided Notes, page 2

8.1 Find Angle Measures in Polygons

Term Definition Example

consecutive vertices

nonconsecutive

vertices

diagonal

Theorem 8.1 Polygon Interior Angles Theorem

The sum of the measures of the interior

angles of a convex n-gon is S = 180(n – 2)º.

Corollary to Theorem 8.1

Interior Angles of a

Quadrilateral

The sum of the measures of the interior

angles of a quadrilateral is 360°.

Theorem 8.2 Polygon Exterior Angles Theorem

The sum of the measures of the exterior

angles of a convex polygon, one angle at

each vertex, is 360°.

CH. 8 Guided Notes, page 3

Convex Polygon Number of Sides Number of Triangles

Sum of Angle Measures

Triangle 3 1 (1 • 180) = 180º Quadrilateral 4 2 (2 • 180) = 360º

Pentagon 5 3 (3 • 180) = 540º Hexagon 6 4 (4 • 180) = 720º Heptagon 7 5 (5 • 180) = 900º Octagon 8 6 (6 • 180) = 1080º n-gon n n – 2 180(n – 2)º

Examples:

1. The sum of the measures of the interior angles of a convex polygon is

1260° . Classify the polygon by the number of sides.

2. Find the value of x in each of the diagrams . a).

b).

CH. 8 Guided Notes, page 4 3. The base of a lamp is in the shape of a regular 15-gon. Find the measure of each interior angle and the measure of each exterior angle.

CH. 8 Guided Notes, page 5

8.2 Use Properties of Parallelograms

Term Definition Example

parallelogram

Theorem 8.3

If a quadrilateral is a parallelogram, then its

opposite sides are congruent.

Theorem 8.4

If a quadrilateral is a parallelogram, then its

opposite angles are congruent.

Theorem 8.5

If a quadrilateral is a parallelogram, then its

consecutive angles are supplementary.

Theorem 8.6

If a quadrilateral is a parallelogram, then its

diagonals bisect each other.

Example:

1. Find the values of x and y.

CH. 8 Guided Notes, page 6 2. As shown, a gate contains several parallelograms. Find

m!ADC when

m!DAB = 65°. 3. The diagonals of parallelogram

STUV intersect at point

W . Find the coordinates of

W.

CH. 8 Guided Notes, page 7

8.3 Show that a Quadrilateral is a Parallelogram

Term Definition Example

Theorem 8.7 (Converse of

Thm 8.3)

If both pairs of opposite sides of a

quadrilateral are congruent, then the

quadrilateral is a parallelogram.

Theorem 8.8 (Converse of

Thm 8.4)

If both pairs of opposite angles of a

quadrilateral are congruent, then the

quadrilateral is a parallelogram.

Theorem 8.9

If one pair of opposite sides of a

quadrilateral is congruent and parallel, then

the quadrilateral is a parallelogram.

Theorem 8.10

(Converse of

Thm 8.6)

If the diagonals of a quadrilateral bisect

each other, then the quadrilateral is a

parallelogram.

A quadrilateral is a parallelogram if any one of the following is true.

1. Both pairs of opposite sides are parallel. Definition 2. Both pairs of opposite sides are congruent. Theorem 8.7 3. Both pairs of opposite angles are congruent. Theorem 8.8 4. A pair of opposite sides is both congruent and parallel. Theorem 8.9 5. Diagonals bisect each other. Theorem 8.10

CH. 8 Guided Notes, page 8 Examples: 1. In the diagram at the right,

AB and

DC represent adjustable supports of a basketball hoop. Explain why

AD is always parallel to

BC . 2. The headlights of a car have the shape shown at the right. Explain how you know that

!B "!D .

3. For what value of x is quadrilateral PQRS a paralleogram?

CH. 8 Guided Notes, page 9 4. Show that quadrilateral KLMN is a parallelogram.

CH. 8 Guided Notes, page 10

8.4 Properties of Rhombuses, Rectangles, and Squares

Term Definition Example

rhombus

rectangle

square

Rhombus Corollary

A quadrilateral is a rhombus if and only if it

has four congruent sides.

Rectangle Corollary

A quadrilateral is a rectangle if and only if it

has four right angles.

Square Corollary

A quadrilateral is a square if and only if it is

a rhombus and a rectangle.

Theorem 8.11

A parallelogram is a rhombus if and only if

its diagonals are perpendicular.

Theorem 8.12

A parallelogram is a rhombus if and only if each diagonal

bisects a pair of opposite angles.

Theorem 8.13

A parallelogram is a rectangle if and only if its

diagonals are congruent.

CH. 8 Guided Notes, page 11

If a quadrilateral is a rectangle, then the following properties hold true.

1. Opposite sides are congruent and parallel. 2. Opposite angles are congruent. 3. Consecutive angles are supplementary. 4. Diagonals are congruent and bisect each other. 5. All four angles are right angles.

Examples: 1. 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 2. Classify this special quadrilateral. Explain your reasoning.

3. You are building a frame for 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? 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?

CH. 8 Guided Notes, page 12

8.5 Use Properties of Trapezoids and Kites

Term Definition Example

trapezoid

parts of

trapezoids

1. Bases—

2. Legs—

3. Base Angles—

isosceles trapezoid

Theorem 8.14

If a trapezoid is isosceles, then each pair of

base angles is congruent.

Theorem 8.15

If a trapezoid has a pair of congruent base

angles, then it is an isosceles trapezoid.

Theorem 8.16

A trapezoid is isosceles if and only if its

diagonals are congruent.

CH. 8 Guided Notes, page 13 midsegment of a

trapezoid (median)

Theorem 8.17 Midsegment Theorem for Trapezoids

The midsegment of a trapezoid is parallel to

each base and its length is one half the sum

of the lengths of the bases.

kite

Theorem 8.18

If a quadrilateral is a kite, then its

diagonals are perpendicular.

Theorem 8.19

If a quadrilateral is a kite, then exactly one

pair of opposite angles is congruent.

Examples: 1. Show that

CDEF is a trapezoid.

CH. 8 Guided Notes, page 14 2. A shelf fitting into a cupboard in the corner of a kitchen is an isosceles trapezoid. Find

m!N,m!L, and

m!M.

3. In the diagram,

MN is the midsegment of trapezoid

PQRS. Find

MN.

4. Find

m!T in the kite shown at the right.

CH. 8 Guided Notes, page 15

8.6 Identify Special Quadrilaterals

Quadrilateral Hierarchy Diagram

Properties of Quadrilaterals Property Parallelogram Rectangle Rhombus Square Kite Trapezoid

All sides are ! . Exactly 1 pair of opposite sides are ! .

Both pairs of opposite sides are ! .

Both pairs of opposite sides are //.

Exactly one pair of opposite sides are //.

All angles are ! . Exactly one pair of opposite angles are ! .

Both pairs of opposite angles are ! .

Diagonals are ! . Diagonals are ! . Diagonals bisect each other.