1. Polygons
5.4
Pre-Algebra

2. Warm Up 1. How many sides does a hexagon have?
2. How many sides does a pentagon have?
3. How many angles does an octagon have?
4. Evaluate (n – 2)180 for n = 7. 6 5 8
900

3. Learn to classify and find angles in polygons.

4. Vocabulary polygon regular polygon trapezoid parallelogram rectangle
rhombus
square

5. The cross section of a brilliant-cut diamond is a pentagon
The cross section of a brilliant-cut diamond is a pentagon. The most beautiful and valuable diamonds have precisely cut angles that maximize the amount of light they reflect.
A polygon is a closed plane figure formed by three or more segments. A polygon is named by the number of its sides.

6. Polygon
Number of Sides 3
Triangle 4
Quadrilateral 5
Pentagon 6
Hexagon 7
Heptagon 8
Octagon n
n-gon

7. Example: Finding Sums of the Angle Measures in Polygons
A. Find the sum of the angle measures in a hexagon.
Divide the figure into triangles.
4 triangles
4 • 180° = 720°

8. Example: Finding Sums of the Angle Measures in Polygons Continued
B. Find the sum of the angle measures in a octagon.
Divide the figure into triangles.
6 triangles
6 • 180° = 1080°

9. Try This A. Find the sum of the angle measures in a hexagon.
Divide the figure into triangles.
4 triangles
4 • 180° = 720°

10. Try This B. Find the sum of the angle measures in a heptagon.
Divide the figure into triangles.
5 triangles
5 • 180° = 900°

11. The pattern is that the number of triangles is always 2 less than the number of sides. So an n-gon can be divided into n – 2 triangles. The sum of the angle measures of any n-gon is 180°(n – 2).
All the sides and angles of a regular polygon have equal measures.

12. Example: Finding the Measure of Each Angle in a Regular Polygon
Find the angle measures in the regular polygon.
6 congruent angles
6x = 180°(6 – 2)
6x = 180°(4)
6x = 720° 6x 6
720°6 =
x = 120°

13. Example: Finding the Measure of Each Angle in a Regular Polygon
Find the angle measures in the regular polygon.
4 congruent angles
4y = 180°(4 – 2)
4y = 180°(2)
4y = 360° 4y 4
360°4 =
y = 90°

14. Try This Find the angle measures in the regular polygon.
5 congruent angles
5a = 180°(5 – 2) a°
5a = 180°(3) a° a°
5a = 540° 5a 5
540° = a° a°
a = 108°

15. Try This Find the angle measures in the regular polygon.
8 congruent angles b°
8b = 180°(8 – 2)
8b = 180°(6)
8b = 1080° 8b 8
1080° =
b = 135°

17. Example: Classifying Quadrilaterals
Give all the names that apply to the figure.
quadrilateral
Four-sided polygon
parallelogram
2 pairs of parallel sides
rectangle
4 right angles
rhombus
4 congruent sides
square
4 congruent sides and 4 right angles

18. Example: Classifying Quadrilaterals Continued
Give all the names that apply to the figure.
quadrilateral
Four-sided polygon
parallelogram
2 pairs of parallel sides
rhombus
4 congruent sides

19. Try This Give all the names that apply to the figure. A. quadrilateral
Four-sided polygon
parallelogram
2 pairs of parallel sides
rectangle
4 right angles

20. Try This Give all the names that apply to the figure. B. quadrilateral
Four-sided polygon

21. Lesson Quiz: Part 1
1. Find the sum of the angle measures in a quadrilateral.
2. Find the sum of the angle measures in a hexagon.
3. Find the measure of each angle in a regular octagon.
360°
720°
135°

22. Lesson Quiz: Part 2
4. Write all of the names that apply to the figure below.
quadrilateral, rhombus, parallelogram