1. Solving for Interior & Exterior Angles of Triangles and Polygons with Equations

2. Angles
When the sides of a polygon are extended, other angles are formed.
The original angles are the interior angles.
The angles that form linear pairs with the interior angles are the exterior angles.

3. Theorem 4.1 Triangle Sum Theorem:
The sum of the measures of the interior angles of a triangle is 180o.
Theorem 4.2 Exterior Angle Theorem:
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

4. Solve for the variable using an Equation :
Demonstrate your steps mathematically :
19˚
4x˚
41˚
Check:
Solve for the missing measure:

5. Solve for the variable using an Equation :
Solve for the missing measure:
Demonstrate your steps mathematically :
6x˚
91˚
41˚
Check:

6. Solve for the variable using an Equation :
40˚
Demonstrate your steps mathematically :
2x˚
2x˚
Check:
Solve for the missing measure:

7. Ex.4: Find . Ex.5: Find the measure of in the diagram shown.

8. Find Angle Measures in Polygons
Diagonal: Connects two non- consecutive vertices Divides the shape into triangles. How many triangles in the pentagon? A E B C D

9. How many triangles in a hexagon, quadrilateral?
4, 2
What is the pattern?
The number of triangles equals the # of sides minus 2
triangles = (n – 2)
How many triangles in a 15-gon? 13

10. So how many degrees in a pentagon? Number of triangles? 3
Number of degree in each triangle?
180
Total number of degrees in the pentagon?
540 = 3 * 180
Degrees = (n – 2) * 180, where n = # of sides

11. Find the measure of the interior angles of the indicated convex polygon
octagon
octagon = 8
(n-2) *180 = degrees
n=8
(8-2) * 180
6 * 180
1080 = degrees

12. Find the measure of the interior angles of the indicated convex polygon
(n-2) * 180 = degrees
n=13
(13-2) * 180
11* 180
1980 = degrees

13. How can we find the number of sides a shape has based on the sum of the interior angles?
D = (n-2) * 180
n = (D/180) + 2
So how many sides is the figure with 1620 degrees?
n = (1620/180) + 2
n = 9 + 2
n = 11

14. The sum of the measures of the interior angles of a convex polygon is Classify the polygon by the number of sides.
n = (Interior Angles/180) + 2
n = (1440/180) + 2
n = (8) +2
n=10

15. Exterior Angles The sum of exterior angles is always equal to 360˚
Interior and Exterior
angles always add to
180 5 4 3 2 1 A E B C D

16. Why Exterior Angles equal 360
What are the measure of the interior angles?
180*4 = 720
720 / 6 = 120
What are the exterior angles?
180 – 120 = 60
60 * 6 = 360
Regular Hexagon

17. ALGEBRA Combine like terms. Subtract 288 from each side.
Find an unkown interior angle measure
ALGEBRA
Find the value of x in the diagram shown.
SOLUTION
The polygon is a quadrilateral. Use the Corollary to the Polygon Interior Angles Theorem to write an equation involving x. Then solve the equation.
x° + 108° + 121° + 59° =
360°
x =
360
Combine like terms.
x = 72
Subtract 288 from each side.
The value of x is 72.
ANSWER

18. Example
Use the diagram at the right. Find m S and m T. 3.
103°, 103°
ANSWER
The measures of three of the interior angles of a quadrilateral are 89°, 110°, and 46°. Find the measure of the fourth interior angle. 4.
115°
ANSWER

19. Polygon Exterior Angles Theorem
Standardized Test Practice
SOLUTION
Use the Polygon Exterior Angles Theorem to write and
solve an equation.
x° + 2x° + 89° + 67° =
360°
Polygon Exterior Angles Theorem
3x =
360
Combine like terms.
x = 68
Solve for x.
The correct answer is B.
ANSWER

20. What is the value of x in the diagram shown?
a = 136°, b = 35°, c = 126°
x = 360
297 + x = 360
x = 360 – 297
x = 63

21. Example
A convex hexagon has exterior angles with measures 34°, 49°, 58°, 67°, and 75°. What is the measure of an exterior angle at the sixth vertex? 5.
ANSWER
77°