Chapter 8 powerpoint 3

8.1 Angle Measures in Polygons

In a polygon, two vertices that are endpoints of the same side are called ___________________________.

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

Diagonals from one vertex form __________________.


Polygon Interior Angles Theorem –

The sum of the measures of the interior angles of a convex n-gon is ________________


Interior Angles of a 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 octagon.

The sum of the measures of the interior angles of a convex polygon is 2340°. Classify the polygon by the number of sides.


Polygon Exterior Angles Theorem –

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


A convex hexagon has exterior angles with measures 34°, 49°, 58°, 67°, and 75°. What is the measure of an exterior angle at the sixth vertex?


180( 2)n

n

8.2 Properties of Parallelograms

A _____________________ is a quadrilateral with both pairs of opposite sides

____________________.

The term “parallelogram PQRS can be written as _____________.

In _____________, ____________ and ____________ by definition.


Theorem 8.3 – If a quadrilateral is a parallelogram, then its opposite sides are _________________.

Theorem 8.4 – If a quadrilateral is a parallelogram, then its opposite angles are ________________.


Find the values of x and y.

D E

G F53°

y°

4x-1

15


Find the values of x and y.

D E

G F53°

2x

y+2

16

10


Theorem 8.5 – If a quadrilateral is a parallelogram, then its consecutive angles are _________________.

Solve for the variable.

42°

D E

G F2x°


Solve for the variable.

4(p+3)°D E

G F135°


Theorem 8.6 – If a quadrilateral is a parallelogram, then its ______________ bisect each other.

P Q

S R

T


Solve for PR, ST, and the measures of angles SRQ and PQR.

P Q

S R

T

8.3 Showing that a Quadrilateral is a Parallelogram

Theorem 8.7 – If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a _________________.

Theorem 8.8 – If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a ________________________.

D E

FG


Given:Prove: ABCD is a parallelogram D E

FG


Theorem 8.9 – If one pair of opposite sides of a quadrilateral are __________________________, then the quadrilateral is a _________________.

Theorem 8.10– If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a ________________________.

D E

FG


For what value of x is quadrilateral DEFG a parallelogram?

D E

FG

8.4 Properties of Rhombuses, Rectangles, and Squares

Rhombus –

Rectangle –

Square –


Rhombus Corollary – A quadrilateral is a rhombus if and only if it has

_____________________________.

Rectangle Corollary – A quadrilateral is a rectangle if and only if it has

_____________________________.

Square Corollary – A quadrilateral is a square if and only if it is a _____________

and a ____________________.


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

a.

b.

AB CD

AB BC


For any rhombus ABCD, decide whether the statement is always, sometimes, or never true. Draw a sketch and explain your reasoning.

a.

b.

AB BC

mA mB


Diagonal Theorems –

Theorem 8.11 – A parallelogram is a rhombus if and only if its diagonals are _____________________

Theorem 8.12 –A parallelogram is a rhombus if and only if each diagonal ___________________ a pair of opposite angles.

Theorem 8.13 –A parallelogram is a rectangle is and only if its diagonals are ____________________


For any rhombus DEFG, decide whether the statement is always, sometimes, or never true. Draw a sketch and explain your reasoning.

1.

2.

3.

DEG FEG

DEG EFD

DG GF


Classify the special quadrilateral. Explain your reasoning.


Sketch rhombus ABCD. List everything you know about it.


Sketch square ABCD. List everything you know about it.

8.5 Using Properties of Trapezoids and Kites

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

These parallel sides are called the __________________.

For each of the bases, there is a pair of _____________________________.

The nonparallel sides are called the _________________.

If the legs are congruent, then the trapezoid is an _______________________.


Show that ABCD is a trapezoid.A (1,2), B (4,5), C (7,3), D (7,-2)


Theorem 8.14 –If a trapezoid is isosceles, then each pair of base angles is __________________.

Theorem 8.15 –If a trapezoid has a pair of congruent base angles, then it is an _______________trapezoid.

Theorem 8.16 –A trapezoid is isosceles if and only if its diagonals are __________________.


Theorem 8.17 Midsegment Theorem for Trapezoids –

The midsegment of a trapezoid is ________________ to each base and its

length is ______________ the sum of the lengths of the bases.

A B

D C

P Q


In this diagram, ABCD is an isosceles trapezoid, and PQ is the midsegment.

a. Find b. Find

A B

D C

P Q

m B

PQ


A ____________ is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are __________________.


Theorem 8.18 – If a quadrilateral is a kite, then its diagonals are ______________________.

Theorem 8.19 – If a quadrilateral is a kite, then exactly one pair of opposite angles are ____________.

A

B

C

D

A

B

C

D


In the diagram, PQRS is a kite. Find

S

P

Q

R

m Q

83° 41°


In a kite, the measures of the angles are 6x°, 24°, 84°, and 126°. Find the value of x.What are the measures of the angles that are congruent?

8.6 Identifying Special Quadrilaterals


Quadrilateral ABCD has at least one pair of opposite angles congruent.What types of quadrilaterals meet this condition?


Quadrilateral WXYZ has at least one pair of opposite sides that are parallel.What types of quadrilaterals meet this condition?


What is the most specific name for quadrilateral DEFG?

D

E

F

G


Is enough information given in the diagram to show that quadrilateral ABCD is a rhombus? Explain.

A

B

C

D


Give the most specific name for the quadrilateral. Explain your reasoning.

A B

CD



A B

CD



A B

CD


You are given the following information about quadrilateral ABCD.

Is enough information given to conclude that quadrilateral ABCD is a trapezoid? Explain.

6

12

115

65

AB

CD

m A

m D

Education

Chapter 8 powerpoint 3