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Chapter 8: Quadrilaterals

Guided Notes

Geometry Fall Semester

8.1 Find Angle Measures in Polygons Term

Definition

consecutive vertices

nonconsecutive vertices

diagonal Theorem 8.1 Polygon Interior

The sum of the measures of the interior angles of a convex n-gon is S = 180(n – 2)º.

Angles Theorem Corollary to Theorem 8.1

The sum of the measures of the interior angles of a quadrilateral is 360°.

Interior Angles of a Quadrilateral Theorem 8.2

The sum of the measures of the exterior

Polygon Exterior

angles of a convex polygon, one angle at

Angles Theorem

each vertex, is 360°.

CH. 8 Guided Notes, page 2

Example

CH. 8 Guided Notes, page 3 Convex Polygon

Number of Sides

Number of Triangles

Sum of Angle Measures

Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon n-gon

3 4 5 6 7 8 n

1 2 3 4 5 6 n – 2

(1 • 180) = 180º (2 • 180) = 360º (3 • 180) = 540º (4 • 180) = 720º (5 • 180) = 900º (6 • 180) = 1080º 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.

8.2 Use Properties of Parallelograms Term

Definition

parallelogram If a quadrilateral is a parallelogram, then its

Theorem 8.3

opposite sides are congruent.

If a quadrilateral is a parallelogram, then its

Theorem 8.4

opposite angles are congruent.

If a quadrilateral is a parallelogram, then its

Theorem 8.5

consecutive angles are supplementary.

If a quadrilateral is a parallelogram, then its

Theorem 8.6

diagonals bisect each other.

Example: 1. Find the values of x and y.

CH. 8 Guided Notes, page 5

Example

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

If both pairs of opposite sides of a

Theorem 8.7 (Converse of

quadrilateral are congruent, then the quadrilateral is a parallelogram.

Thm 8.3) If both pairs of opposite angles of a

Theorem 8.8 (Converse of

quadrilateral are congruent, then the quadrilateral is a parallelogram.

Thm 8.4) If one pair of opposite sides of a

Theorem 8.9

quadrilateral is congruent and parallel, then the quadrilateral is a parallelogram.

If the diagonals of a quadrilateral bisect

Theorem 8.10 (Converse of

each other, then the quadrilateral is a parallelogram.

Thm 8.6)

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

rhombus

rectangle

square Rhombus Corollary Rectangle Corollary

A quadrilateral is a rhombus if and only if it has four congruent sides.

A quadrilateral is a rectangle if and only if it has four right angles.

A quadrilateral is a square if and only if it is

Square Corollary

a rhombus and a rectangle.

A parallelogram is a rhombus if and only if

Theorem 8.11

its diagonals are perpendicular.

A parallelogram is a rhombus if and only if each diagonal

Theorem 8.12

bisects a pair of opposite angles.

A parallelogram is a rectangle if and only if its

Theorem 8.13

diagonals are congruent.

Example

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

trapezoid

1. Bases—

parts of

2. Legs—

trapezoids 3. Base Angles—

isosceles trapezoid If a trapezoid is isosceles, then each pair of

Theorem 8.14

base angles is congruent.

If a trapezoid has a pair of congruent base

Theorem 8.15

angles, then it is an isosceles trapezoid.

A trapezoid is isosceles if and only if its

Theorem 8.16

diagonals are congruent.

Example

CH. 8 Guided Notes, page 13 midsegment of a trapezoid (median) Theorem 8.17

The midsegment of a trapezoid is parallel to

Midsegment

each base and its length is one half the sum

Theorem for

of the lengths of the bases.

Trapezoids kite

If a quadrilateral is a kite, then its

Theorem 8.18

diagonals are perpendicular.

If a quadrilateral is a kite, then exactly one

Theorem 8.19

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.

8.6 Identify Special Quadrilaterals

CH. 8 Guided Notes, page 15

Quadrilateral Hierarchy Diagram Properties of Quadrilaterals Property 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.

Parallelogram Rectangle Rhombus Square Kite

Trapezoid