Angles of Triangles

Objectives Find angle measures in triangles.

Measures of Angles of a Triangle The word “triangle” means “three angles” When the sides of a triangles are extended, however, other angles are formed When the sides of a triangles are extended, however, other angles are formed The original 3 angles of the triangle are the interior angles The original 3 angles of the triangle are the interior angles The angles that are adjacent to interior angles are the exterior angles The angles that are adjacent to interior angles are the exterior angles Each vertex has a pair of exterior angles Each vertex has a pair of exterior angles Original Triangle Extend sides Interior Angle Exterior Angle

Triangle Interior and Exterior Angles Smiley faces are interior angles and hearts represent the exterior angles Each vertex has a pair of congruent exterior angles; however it is common to show only one exterior angle at each vertex.

Triangle Interior and Exterior Angles )) ))) ( A B C ( )) (( D E F  Interior Angles  Exterior Angles (formed by extending the sides)

Triangle Sum Theorem The Triangle Angle-Sum Theorem gives the relationship among the interior angle measures of any triangle.

Triangle Sum Theorem If you tear off two corners of a triangle and place them next to the third corner, the three angles seem to form a straight line. You can also show this in a drawing.

Draw a triangle and extend one side. Then draw a line parallel to the extended side, as shown. The three angles in the triangle can be arranged to form a straight line or 180°. Two sides of the triangle are transversals to the parallel lines. Triangle Sum Theorem

Theorem 4.1 – Triangle Sum Theorem The sum of the measures of the angles of a triangle is 180°. m  X + m  Y + m  Z = 180° m  X + m  Y + m  Z = 180° X Y Z

Triangle Sum Theorem

Given m  A = 43° and m  B = 85°, find m  C. ANSWER  C has a measure of 52°. CHECK Check your solution by substituting 52° for m  C. 43° + 85° + 52° = 180° SOLUTION m  A + m  B + m  C = 180° Triangle Sum Theorem 43° + 85° + m  C = 180° Substitute 43° for m  A and 85° for m  B. 128° + m  C = 180° Simplify. m  C = 52° Simplify. 128° + m  C – 128° = 180° – 128° Subtract 128° from each side. Example 1

A. Find p in the acute triangle. 73° + 44° + p° = 180° p = 180 p = 63 –117 Triangle Sum Theorem Subtract 117 from both sides. Example 2a

B. Find m in the obtuse triangle. 23° + 62° + m° = 180° 85 + m = 180 m = 95 –85 Triangle Sum Theorem Subtract 85 from both sides. 23  62  m Example 2b

A. Find a in the acute triangle. 88° + 38° + a° = 180° a = 180 a = 54 –126 88° 38° a°a° Triangle Sum Theorem Subtract 126 from both sides. Your Turn:

B. Find c in the obtuse triangle. 24° + 38° + c° = 180° 62 + c = 180 c = 118 –62 –62 c°c° 24° 38° Triangle Sum Theorem. Subtract 62 from both sides. Your Turn:

2x° + 3x° + 5x° = 180° 10x = 180 x = Find the angle measures in the scalene triangle. Triangle Sum Theorem Simplify. Divide both sides by 10. The angle labeled 2x° measures 2(18°) = 36°, the angle labeled 3x° measures 3(18°) = 54°, and the angle labeled 5x° measures 5(18°) = 90°. Example 3

3x° + 7x° + 10x° = 180° 20x = 180 x = Find the angle measures in the scalene triangle. Triangle Sum Theorem Simplify. Divide both sides by 20. 3x°3x°7x°7x° 10x° The angle labeled 3x° measures 3(9°) = 27°, the angle labeled 7x° measures 7(9°) = 63°, and the angle labeled 10x° measures 10(9°) = 90°. Your Turn:

ANSWER 65° ANSWER 75° ANSWER 50° Find m  A. 1. Find m  B. 2. Find m  C. 3. Your Turn:

Substitution Subtract 20 from each side. Answer: GARDENING The flower bed shown is in the shape of a right triangle. Find if is 20. Example 6:

Answer: The piece of quilt fabric is in the shape of a right triangle. Find if is 62. Your Turn:

Investigating Exterior Angles of a Triangles B A A B C You can put the two torn angles together to exactly cover one of the exterior angles

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 remote interior angles. m  1 = m  2 + m  3 The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. m  1 = m  2 + m 

ANSWER  1 has a measure of 130°. SOLUTION m  1 = m  A + m  C Exterior Angle Theorem Given m  A = 58° and m  C = 72°, find m  1. Substitute 58° for m  A and 72° for m  C. = 58° + 72° Simplify. = 130° Example 7

ANSWER 120° ANSWER 155° ANSWER 113° Find m  Find m  Find m  Your Turn: