1. Warm Up
True or false?
1. Some trapezoids are parallelograms.
2. Some figures with 4 right angles are squares.
3. Some quadrilaterals have only one right angle.
false
true
true

2. Learn to identify regular and not regular polygons and to find the angle measures of regular polygons.

3. Vocabulary
polygon
regular polygon
diagonal

4. A polygon is a closed plane figure formed by three or more line segments. A regular polygon is a polygon in which all sides are congruent and all angles are congruent.
Polygons are named by the number of their sides and angles.

6. Additional Example 1A: Identifying Polygons
Tell whether each shape is a polygon. If so, give its name and tell whether it appears to be regular or not regular.
The shape is a closed plane figure formed by 3 or more line segments.
polygon
There are 5 sides and 5 angles.
pentagon
All 5 sides do not appear to be congruent.
not regular

7. Additional Example 1B: Identifying Polygons
Tell whether each shape is a polygon. If so, give its name and tell whether it appears to be regular or not regular.
The shape is a closed plane figure formed by 3 or more line segments.
polygon
There are 8 sides and 8 angles.
octagon
The sides and angles appear to be congruent.
regular

8. Check It Out: Example 1A
Tell whether each shape is a polygon. If so, give its name and tell whether it appears to be regular or not regular.
There are 4 sides and 4 angles.
quadrilateral
The sides and angles appear to be congruent.
regular

9. Check It Out: Example 1B
Tell whether each shape is a polygon. If so, give its name and tell whether it appears to be regular or not regular.
There are 4 sides and 4 angles.
quadrilateral
All 4 sides do not appear to be congruent.
not regular

10. You can divide any quadrilateral into two triangles by drawing a diagonal, a line segment that connects two non-adjacent vertices. The sum of the interior angle measures in a triangle is 180°, so the sum of the interior angle measures in a quadrilateral is 360°.

11. Understand the Problem
Additional Example 2: Problem Solving Application
Malcolm designed a wall hanging that was a regular 9-sided polygon (called a nonagon). What is the measure of each angle of the nonagon? 1
Understand the Problem
The answer will be the measure of each angle in a nonagon.
List the important information:
A regular nonagon has 9 congruent sides and t 9 congruent angles.

12. Additional Example 2 Continued
Make a Plan
Make a table to look for a pattern using regular polygons.
Solve 3
Draw some regular polygons and divide each into triangles.

13. Additional Example 2 Continued
720°
The prefixes in the names of the polygons tell you how many sides and angles there are.
tri- = three quad- = four
penta- = five hexa- = six octa- = eight
Reading Math

14. Additional Example 2 Continued
Solve Cont. 3
The number of triangles is always 2 fewer than the number of sides.
A nonagon can be divided into 9 – 2 = 7 triangles.
The sum of the interior angle measures in a nonagon is 7  180° = 1,260°.
So the measure of each angle is 1,260° ÷ 9 = 140°.

15. Additional Example 2 Continued4
Look Back
Each angle in a nonagon is obtuse. 140° is a reasonable answer, because an obtuse angle is between 90° and 180°.

16. Understand the Problem
Check It Out: Example 2
Sara designed a picture that was a regular 6-sided polygon (called a hexagon). What is the measure of each angle of the hexagon? 1
Understand the Problem
The answer will be the measure of each angle in a hexagon.
List the important information:
A regular hexagon has 6 congruent sides and 6 congruent angles.

17. Check It Out: Example 2 Continued
Make a Plan
Make a table to look for a pattern using regular polygons.
Solve 3
Draw some regular polygons and divide each into triangles.

18. Check It Out: Example 2 Continued
Solve Cont. 3
The number of triangles is always 2 fewer than the number of sides A hexagon can be divided into 6 – 2 = 4 triangles.
The sum of the interior angles in a octagon is  180° = 720°.
So the measure of each angle is 720° ÷ 6 = 120°.

19. Check It Out: Example 2 Continued4
Look Back
Each angle in a hexagon is obtuse. 120° is a reasonable answer, because an obtuse angle is between 90° and 180°.

20. Lesson Quiz
1. Name each polygon and tell whether it appears to be regular or not regular.
2. What is the measure of each angle in a regular dodecagon (12-sided figure)?
nonagon, regular; octagon, not regular
150°