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°?

Angle values in fraction form

Angle values in decimal form

For each decimal value, the '3's go on forever.

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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 = 180

y = 180-110

y = 70

Now focus on triangle ACD. All three angles of a triangle add to 180

A+D+C = 180

x+y+2x = 180

3x+y = 180

3x+70 = 180

3x = 180-70

3x = 110

x = 110/3

This means each base angle is

2x = 2*(110/3) = 220/3

So 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 B

A+B+C = 180

220/3 + B + 220/3 = 180

B+440/3 = 180

B = 180 - 440/3

B = 180*(3/3) - 440/3

B = 540/3 - 440/3

B = 100/3

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So in summary we have

as the three angles for triangle ABC. Those are the exact fractional values.

The approximate decimal values are