find m<4 in degrees

Question

Consider the following isosceles triangle. Note the use of dashes on congruent sides of the triangle.

1. 
   

If m∠3 = 58°, find m∠4 (in degrees), the angle formed by the bisectors of ∠3 and ∠2. Use arithmetic or algebra, as needed, to find the measure.

Answer 1

m<3 = 58, so m<2 = 58 as well, since isosceles triangle. Since <2 and <3 are bisected, each angle is divided into two angles of 29 degrees each. In the bottom triangle we have those two half angles of <2 and <3 which will total 58 degrees. Since the sum of the interior angles of a triangle are 180 degrees, m<4 = 180 - 58 = 122 degrees.

Answer 2

The answer is 122°. Angle 2 and 3 are both 58 degrees because this is an isosceles triangle. The bottom triangle is made up of angle 4 and half of angles 2 and 3 (since the lines are bisectors). 180 = angle4 - 0.5*angle3 + 0.5*angle2, and we find that angle 4 is 122°