find angle measures in polygons -...

FIND ANGLE MEASURES IN

POLYGONS

“A life not lived for others is not a life worth living.” –Albert Einstein

Diagonals

A diagonal is a segment that connects any two

nonconsecutive vertices in a polygon.

Side

Diagonal

Concept 1: Polygon Interior Angles

Theorem

Polygon Interior Angles Theorem

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

is 180°(n-2).

n=3

180°

n=5

540°

n=6

720°

Concept 2: Interior Angles of a

Quadrilateral

The sum of the measures of the interior angles

of a quadrilateral is 360°.

n=4

360°

n=4

360° n=4

360°

Example 1

Find the sum of the measures of the interior angles

of a convex octagon.

Example 2

The sum of the measures of the interior angles of a

convex polygon is 2160°. Classify the polygon by

the number of sides.

Example 3

Find the value of x in the diagram.

55°

100°

75° 166°

x°

Concept 3: Polygon Exterior Angles

Theorem

The sum of the measures of the exterior angles of a

convex polygon, one angle at each vertex, is 360°.

1

2

3

4

5

Example 4

Find the value of x

100°

35°

60°

75°

x°

Example 5

Find the measure of each interior angle of a

pentagon and the measure of each exterior angle

of a pentagon.

USE PROPERTIES OF

PARALLELOGRAMS “Bad is never good until worse happens.” –Danish Proverb

Parallelogram

A parallelogram is a quadrilateral with opposite

sides that are parallel.

Concept 4: Theorem 8.3 and 8.4

Theorem 8.3

If a quadrilateral is a parallelogram, then its

opposite sides are congruent.

Theorem 8.4


opposite angles are congruent.

Example 1

Find the values of x, y, and z.

55°

x° (4y-35)°

125°

3z+5

17


Theorem 8.5


consecutive angles are supplementary.

x°

x° y°

y°

Example 2

Find the measure of each angle (exclude straight

angles).

35°

75°

50°

56°


Theorem 8.6

If a quadrilateral is a parallelogram, then its diagonals

bisect each other.

Example 3

Find the value of the variables.

2y-5

5x

20 6

3v 12 15

z

SHOW THAT A

QUADRILATERAL IS A

PARALLELOGRAM

“Things could be worse. Suppose your errors were counted and published every day, like those of a baseball player.” –Anon.


Theorem 8.7

If both pairs of opposite sides of a quadrilateral are

congruent, then the quadrilateral is a parallelogram.

Theorem 8.8

If both pairs of opposite angles of a quadrilateral are

congruent, then the quadrilateral is a parallelogram.

, then If

Example 1

Determine whether the quadrilateral is a

parallelogram. Explain.

5

5

6 6

120°

120°

60°

60°

10

10


Theorem 8.9

If one pair of opposite sides of a quadrilateral are

congruent and parallel, then the quadrilateral is a

parallelogram

, then If


Theorem 8.10

If the diagonals of a quadrilateral bisect each other,

then the quadrilateral is a parallelogram.

, then If

Example 2

Determine if the shape is a parallelogram. Explain.

10

10

5 5

6

6

8

8

Example 3

For what value of x is the quadrilateral a

parallelogram?

2x

6

8

8 5x-16

2x-4

6 6

120°

(8x-8)°

60°

60°

5x-4 11

Example 4

The vertices of quadrilateral ABCD are given. Draw

ABCD in a coordinate plane and show that it is a

parallelogram.

A(0, 1), B(4, 4), C(12, 4), D(8, 1)

PROPERTIES OF RHOMBUSES,

RECTANGLES, AND SQUARES

“The superior man(/woman) blames himself(/herself), the inferior man(/woman) blames others.” –Don Shula

Concept 8: Rhombus, Rectangle, and

Square

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.

Concept 8: Rhombus, Rectangle, and

Square

Rhombus

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

sides.

Rectangle

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

angles.

Square

A quadrilateral is a square if and only if it is a rhombus and a

rectangle.

Example 1

For any square ABCD, decide whether the statement

is always, sometimes, or never true.

AB=BC

BC=AD

m∠A=m∠C

Example 2

For any rectangle ABCD, decide whether the

statement is always, sometimes, or never true.

AB=BC

BC=AD

m∠B=m∠C

Example 3

Classify the special quadrilateral. Explain your

reasoning.

4y+5

2y+35

5x-9 x+31

(5y-5)°

(4y+5)°


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.

Concept 10: Theorem 8.13

Theorem 8.13

A parallelogram is a rectangle if and only if its

diagonals are congruent.

Example 4

Name each quadrilateral (parallelogram, rhombus,

rectangle, and square) for which the statement is

true.

Diagonals bisect each other.

Diagonals are congruent.

Diagonals intersect at a right angle.

Diagonals bisect the angles.

Example 5

The diagonals of rhombus ABCD intersect at E.

Given that m∠DCA=72°, find the indicated

measures.

m∠BCA

m∠BAC

m∠BEA

m∠ABC

m∠ABD

Example 6

The diagonals of rectangle ABCD intersect at E.

Given that m∠DCA=72°, find the indicated

measures.

m∠BCA

m∠BAC

m∠BEA

m∠ABC

m∠ABD

A B

C D

E

USE PROPERTIES OF

TRAPEZOIDS AND KITES “I think we consider too much the good luck of the early bird, and not enough the bad luck of the early worm.”

–Franklin D. Roosevelt

Trapezoids

Trapezoids are quadrilaterals with only one pair of

parallel sides called bases. Angles on the same

base are called base angles. The other two sides

are called the legs. An isosceles trapezoid is one

with congruent legs.

base

base

leg leg

base angle pair

base angle pair

Example 1

Points A, B, C, and D are the vertices of a

quadrilateral. Determine whether ABCD is a trapezoid.

A(0, 4), B(4, 4), C(8, -2), D(2, 1)

Concept 11: Theorem 8.14, 8.15, and 8.16

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.

Example 2

EFGH is an isosceles trapezoid. The measure of

angle E is 72°. Find the other 3 angle measures.

Example 3

If AC=BD is the trapezoid isosceles? Explain.

A B

C D

Midsegment of a trapezoid

The midsegment of a trapezoid is the segment that

connects the midpoints of the legs of a trapezoid.

midsegment

Concept 12: Theorem 8.17

The midsegment of a trapezoid is parallel to each

base and its length is one half the sum of the

lengths of the bases.

1

2(𝑏1 + 𝑏2)

𝑏1

𝑏2

Example 4

Find the value of x.

𝑥

25

17

Example 5


18.7

12𝑥 − 1.7

5𝑥

Kite

A kite is a quadrilateral that has two pairs of

consecutive congruent sides, but opposite sides are

not congruent.


Theorem 8.18

If a quadrilateral is a kite, then its diagonals are

perpendicular.

Theorem 8.19

If a quadrilateral is a kite, the exactly one pair of

opposite angles are congruent.

Example 6


50° 100°

x°

40°

120°

x°

IDENTIFY SPECIAL

QUADRILATERALS “Doing what’s right isn’t the problem. It’s knowing what’s right.” –Lyndon B. Johnson

Concept 14: Determining Shapes

Example 1

What types of quadrilaterals meet this condition.

Quadrilateral ABCD has at least one pair of congruent

opposite angles.

Quadrilateral ABCD has diagonals being congruent.

Quadrilateral ABCD has diagonals intersecting at a

right angle.

Example 2

What is the name of the quadrilateral ABCD?

A B

C D

A B

C D

Example 3

Is enough information given in the diagram to show

that quadrilateral ABCD is an isosceles trapezoid?

Explain.

A B

C D 68° 68°

112°

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