6.1 The Polygon Angle-Sum Theorems

Polygon: Closed figure with at least 3 sides

Convex: Concave:

Ex: What is the sum of the interior angles of a heptagon?

(7 – 2) 180

(9) 180 = 900

Ex: What is the measure of

n = 5

(5 – 2) 180 = 542

m

m

Ex: What is the measure of each interior angle of a regular nonagon?

Ex: What is the value of x, y, and z?

2x + 20 + 112 + 96 = 360

x = 66

y = 84

z = 94

Ex: What is the value of the <1 in the regular pentagon?

*All interior angles are congruent, so all exterior angles are congruent

*Exterior angles of regular polygon =

Ex: The measure of the exterior angle of a polygon is 40. Find the measure of the exterior angle and the number of sides.

Interior angle: 180-40 = 140Number of sides:

360 = 40n

n = 9

6.2 Properties of Parallelograms

Parallelogram: A quadrilateral (4 sides) with both pairs of opposite sides parallel. ll and ll

-Can name as

-Opposite sides are congruent.

and

-Opposite angles are congruent.

∠Q ∠S and ∠P ∠R

-Consecutive angles (angles that share a side) are supplementary

m

m

-Diagonals bisect each other.

and

Find the values of x and y.

2y + 9 = 273x + 6 = 12

2y = 183x = 6

y = 9 x = 2

Find the values of x and y.

Write the 2 equations.Use substitution.

2x = y + 42x = x + 2 +4 y = 8

x + 2 = yx = 6

with diagonals and

Prove:

StatementsReasons

1. ABCD is a parallelogram with

diagonals and 1. Given

2. and 2. Definition of bisect

3. 3. Opposite sides of parallelograms are

4. 4. SSS

EH = 6.75

6.3 Proving that a Quadrilateral is a Parallelogram

Ex: For what values of x and y make PQRS a parallelogram?

3x – 5 = 2x + 1

X = 6

Y = 8

Ex: For what values of x and y make EFGH a parallelogram?

3y – 2 + y + 10 =1804x + 13 + 12x + 7 = 180

4y + 8 = 18016x + 20 = 180

4y = 17616x = 160

Y = 43 x = 10

Ex: For what values of x and y make ACBD a parallelogram?

2x = 4y -1 = 2y -7

X = 2y = 6

Ex: Can you prove that the quadrilateral is a parallelogram based on the given information?

No, not enough infoYes, alternate interior angles are congruent, so both sets of lines are Parallel.

6.4 Properties of Rhombuses, Rectangles, and Squares

Rhombus: a parallelogram with four congruent sides.

Rectangle: a parallelogram with four right angles.

Square: a parallelogram with four congruent sides and four right angles. (A square is a rhombus and a rectangle.)

1. If a parallelogram is a rhombus, then its diagonals are perpendicular.

2. If a parallelogram is a rhombus, then its diagonals bisect a pair of opposite angles.

3. A parallelogram is a rectangle if and only if its diagonals are congruent.

Rhombus ParallelogramRhombusRectangle

Ex. Find the measures of the angles.

m<1 = 26 m< 1 = 32

m<2 = 128 m<2 = 90

m< 3 = 128 m< 3 = 58

m < 4 = 32

Ex: LMNP is a rectangle. Find the value of x and the length of each diagonal.

LN = 3x + 1 and MP = 8x - 4

3x + 1 = 8x – 4

5 = 5x

1 = x

LN and MP = 4

Ex: Find the variables and the side lengths.

3y = 155x = 15

Y = 5 x =3

Sides = 15

6.5 Conditions for Rhombuses, Rectangles, and Squares

6.6 Trapezoids and Kites

Trapezoid: quadrilateral with exactly ONE pair of parallel sides.

Bases: Parallel sides (BC and AD)

Base angles: ∠A and ∠D and ∠B and ∠C

Legs: Nonparallel sides (AB and CD)

Isosceles Trapezoid: A trapezoid that has congruent legs

-Base angles are congruent

∠A ∠D and ∠B ∠C

-Diagonals are congruent

Ex: Find the measures of the numbered angles.

m< 1 = 49

m<2 = 131

m<3 = 131

Ex: Find EF.

EF = (AD + BC)

3x = (x +3 + 12)3x + 5 = (4 + 7x + 4)

6x = x + 156x + 10 = 7x + 8

5x = 15 2 = x

X = 3 EF = 11

EF = 9

Kite: A quadrilateral with two pairs of consecutive congruent sides and opposite sides are not congruent.

-Diagonals are perpendicular

-One pair of congruent opposite angles

Ex: Find the measures of the numbered angles.

m<1 = 108

m<2 = 108

m<1= 90

m<2= 52

m<3= 38

m<4= 37

m<5= 53

6.7 Polygons in the Coordinate Plane

Ex: Classify the triangle as scalene, isosceles, or equilateral.

A (1,3) B(3,1) C(-2, -2)

AB = = = = 2

BC = = =

AC = = =

Isosceles Triangle

How to classify a parallelogram as just a parallelogram, a rhombus, a rectangle, or a square:

Ex:

Step 1. Sketch graph.

Step 2. Find the slope of 2 consecutive sides.

SP = = = PA = = = 3

Not opposite reciprocals so either a parallelogram or rhombus

Step 3. Find the lengths of the 2 consecutive sides.

SP = PA =

= =

= =

Same side lengths so either a square or rhombus

Answer: Rhombus

Ex:

Step 1. Sketch graph.

Step 2. Find the slope of 2 consecutive sides.

HI = = = IJ = = = -2

Opposite reciprocals so either a square or rectangle

Step 3. Find the lengths.

HI = IJ =

= =

= =

Different side lengths so either a rectangle or parallelogram

Answer: Rectangle

so