Angle Theorems for Triangles 11.2

Sum of Angle Measure in a Triangle Draw a triangle and cut it out Label the angles A, B and C TeaR off each corner of the triangle What do you notice about how the angles fit together? What is the measure of a straight angle? Describe the relationship among the measure of the angles of triangle ABC

Triangle sum theorem What do you notice about the angles formed by a triangle With two parallel lines? Angles 4 and 2 are congruent Angles 5 and angles 3 are alternate interior angles (congruent) Angles 4, 1 and 5 create a straight line so they equal 180° So you can say that angle 1, 2, and 3 equal 180° 4 5 1 2 3

The sum of angles equals 180° To find the measure of a unknown angle Add the two know angles Subtract the sum from 180° The measure of the unknown angle is that measure Example : 26 + 51 = 77 180-77 = 103 Angle y is 103°

The sum of angles equals 180°

Exterior angles and remote interior angles An interior angle of a triangle is formed by two sides of the triangle. An exterior angle is formed by one side of the triangle and the extension of an adjacent side. Each exterior angle has two remote interior angles A remote interior angle is an interior angle that is not adjacent to the exterior angle 1 2 3 4 Angle 1 +2 + 3 = 180 Since angle 3 + 4 = 180 Angle 1 + angle 2 = angle 4

Using exterior angle theorem A (4y -4)° D 52° C 3y° B You can use the exterior angle theorem to find the measure of the interior angles of a triangle. Find m<a and m<b Write the exterior angle theorem Substitute the given angle measures Solve the equation for y Use the value of y to find the m<a and m<b

Find m<a and m<b Write the exterior angle theorem Find m<a and m<b Write the exterior angle theorem m<a + m<b = m<aCD Substitute the given angle measures (4y -4)° + 3y° = 52° Solve the equation for y (4y -4)° + 3y° = 52° 7y -4 = 52 +4 = +4 7y = 56 --- ---- 7 7 y = 8 ° Use the value of y to find the m<a and m<b m<a(4(8) -4)° 32- 4 = 28 ° m<b 3(8) ° = 24 ° A (4y -4)° D 52° C 3y° B