NAME ______________________________________________ DATE
8-1
____________ PERIOD _____
Study Guide and Intervention Angles of Polygons
Sum of Measures of Interior Angles
The segments that connect the nonconsecutive sides of a polygon are called diagonals. Drawing all of the diagonals from one vertex of an n-gon separates the polygon into n 2 triangles. The sum of the measures of the interior angles of the polygon can be found by adding the measures of the interior angles of those n 2 triangles. If a convex polygon has n sides, and S is the sum of the measures of its interior angles, then S 180(n 2).
Example 1
A convex polygon has 13 sides. Find the sum of the measures of the interior angles. S 180(n 2) 180(13 2) 180(11) 1980
Example 2
The measure of an interior angle of a regular polygon is 120. Find the number of sides. The number of sides is n, so the sum of the measures of the interior angles is 120n. S 180(n 2) 120n 180(n 2) 120n 180n 360 60n 360 n6
Exercises Find the sum of the measures of the interior angles of each convex polygon. 1. 10-gon
2. 16-gon
3. 30-gon
4. 8-gon
5. 12-gon
6. 3x-gon
The measure of an interior angle of a regular polygon is given. Find the number of sides in each polygon. 7. 150
8. 160
10. 165
9. 175
11. 168.75
13. Find x.
12. 135
D (4x 5)
E 7x (5x 5) C (6x 10)
(4x 10)
A
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Glencoe/McGraw-Hill
B
417
Glencoe Geometry
Lesson 8-1
Interior Angle Sum Theorem
NAME ______________________________________________ DATE
8-1
____________ PERIOD _____
Study Guide and Intervention
(continued)
Angles of Polygons Sum of Measures of Exterior Angles
There is a simple relationship among the
exterior angles of a convex polygon. Exterior Angle Sum Theorem
If a polygon is convex, then the sum of the measures of the exterior angles, one at each vertex, is 360.
Example 1
Find the sum of the measures of the exterior angles, one at each vertex, of a convex 27-gon. For any convex polygon, the sum of the measures of its exterior angles, one at each vertex, is 360.
Example 2
Find the measure of each exterior angle of regular hexagon ABCDEF. The sum of the measures of the exterior angles is 360 and a hexagon has 6 angles. If n is the measure of each exterior angle, then
A
B C
F
6n 360 n 60
E
D
Exercises Find the sum of the measures of the exterior angles of each convex polygon. 1. 10-gon
2. 16-gon
3. 36-gon
Find the measure of an exterior angle for each convex regular polygon. 4. 12-gon
5. 36-gon
6. 2x-gon
Find the measure of an exterior angle given the number of sides of a regular polygon. 7. 40
8. 18
10. 24
11. 180
©
Glencoe/McGraw-Hill
9. 12
12. 8
418
Glencoe Geometry
NAME ______________________________________________ DATE
8-2
____________ PERIOD _____
Study Guide and Intervention Parallelograms
Sides and Angles of Parallelograms A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Here are four important properties of parallelograms.
P
Q
S
R
If PQRS is a parallelogram, then Q P S R and P S Q R
The opposite angles of a parallelogram are congruent.
P R and S Q
The consecutive angles of a parallelogram are supplementary.
P and S are supplementary; S and R are supplementary; R and Q are supplementary; Q and P are supplementary.
If a parallelogram has one right angle, then it has four right angles.
If mP 90, then mQ 90, mR 90, and mS 90.
Example
If ABCD is a parallelogram, find a and b. AB and C D are opposite sides, so A B C D . 2a 34 a 17
2a
A 8b D
112 34
B C
A and C are opposite angles, so A C. 8b 112 b 14
Exercises Find x and y in each parallelogram. 1.
2.
3x
8y 6x
4y 88
3.
6x
3y
4. 6x
12
5.
55
5x
6.
60
Glencoe/McGraw-Hill
12x
2y 30x
2y
©
3y
150 72x
423
Glencoe Geometry
Lesson 8-2
The opposite sides of a parallelogram are congruent.
NAME ______________________________________________ DATE
8-2
____________ PERIOD _____
Study Guide and Intervention
(continued)
Parallelograms Diagonals of Parallelograms Two important properties of parallelograms deal with their diagonals.
A
B P
D
C
If ABCD is a parallelogram, then: The diagonals of a parallelogram bisect each other.
AP PC and DP PB
Each diagonal separates a parallelogram into two congruent triangles.
ACD CAB and ADB CBD
Example
Find x and y in parallelogram ABCD. The diagonals bisect each other, so AE CE and DE BE. 6x 24 4y 18 x4 y 4.5
A
B
18 6x
D
E 24 4y
C
Exercises Find x and y in each parallelogram. 1.
2.
4y 3x
12
8
4.
28
5. 30 y
10
3. 2y
4y
60 2x
4x
12 3x
6.
4
2y
x
y 17
3x
Complete each statement about ABCD. Justify your answer.
A
B E
7. BAC
D
C
8. D E 9. ADC 10. A D ||
©
Glencoe/McGraw-Hill
424
Glencoe Geometry
