In isosceles triangle ∆ABC, AC is the base and AD is the angle bisector of ∠A. What are the measures of the angles of this triangle if m∠ADB = 110°?

Accepted Solution

Angle values in fraction formA = 220/3B = 100/3C = 200/3Angle values in decimal formA = 73.333B = 33.333C = 73.333For each decimal value, the '3's go on forever. =============================================Work Shown:Check out the diagram below. Since AD bisects angle BAC, this means the two smaller pieces (angle DAB and angle DAC) are equal to one another. Let's call those smaller pieces x for now. They combine to 2x which is the measure of angle BAC. Angle BCA is 2x since this is the other base angle, and the two base angles are congruent.Let y be the measure of angle ADC. It's supplementary to the 110 degree angle.y+110 = 180y = 180-110y = 70Now focus on triangle ACD. All three angles of a triangle add to 180A+D+C = 180x+y+2x = 1803x+y = 1803x+70 = 1803x = 180-703x = 110x = 110/3This means each base angle is 2x = 2*(110/3) = 220/3So we know that A = 220/3 and C = 220/3 are the two base angles of triangle ABC.The last step is to find the vertex angle BA+B+C = 180220/3 + B + 220/3 = 180B+440/3 = 180B = 180 - 440/3B = 180*(3/3) - 440/3B = 540/3 - 440/3B = 100/3------------------------------------So in summary we have A = 220/3B = 100/3C = 220/3as the three angles for triangle ABC. Those are the exact fractional values. The approximate decimal values areA = 73.333B = 33.333C = 73.333