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

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.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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 + 288 = 360 Combine like terms. x = 72 Subtract 288 from each side. The value of x is 72. ANSWER

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

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 + 156 = 360 Combine like terms. x = 68 Solve for x. The correct answer is B. ANSWER

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

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°