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Difficulty: MediumTriangle Properties and Angle Theorems

In △ABC\triangle ABC, the measure of exterior angle ∠ACD\angle ACD is 135∘135^\circ, where DD lies on the extension of side BCBC past CC. If the measure of interior angle ∠A\angle A is 25∘25^\circ greater than the measure of interior angle ∠B\angle B, what is the measure, in degrees, of ∠B\angle B?

Answer: 55 degrees

Answer

55
According to the Exterior Angle Theorem, the measure of exterior angle ∠ACD\angle ACD is equal to the sum of the measures of its remote interior angles, ∠A\angle A and ∠B\angle B. This gives the equation m∠A+m∠B=135∘\text{m}\angle A + \text{m}\angle B = 135^\circ. Using the information that m∠A=m∠B+25∘\text{m}\angle A = \text{m}\angle B + 25^\circ, we substitute this expression into the equation to get (m∠B+25∘)+m∠B=135∘(\text{m}\angle B + 25^\circ) + \text{m}\angle B = 135^\circ. Simplifying this equation gives 2m∠B+25=1352\text{m}\angle B + 25 = 135, which simplifies to 2m∠B=1102\text{m}\angle B = 110, and dividing by 2 yields m∠B=55∘\text{m}\angle B = 55^\circ.

Step-by-Step Solution

1
Apply the Exterior Angle Theorem to express the relation between the exterior angle and the two remote interior angles.
m∠A+m∠B=135∘\text{m}\angle A + \text{m}\angle B = 135^\circ
The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
2
Substitute the relationship between the interior angles into the equation.
(m∠B+25∘)+m∠B=135∘(\text{m}\angle B + 25^\circ) + \text{m}\angle B = 135^\circ
The problem states that the measure of interior angle ∠A\angle A is 25∘25^\circ greater than the measure of interior angle ∠B\angle B.
3
Solve the algebraic equation for the measure of interior angle ∠B\angle B.
m∠B=55∘\text{m}\angle B = 55^\circ
Combining like terms gives 2m∠B+25=1352\text{m}\angle B + 25 = 135. Subtracting 25 from both sides gives 2m∠B=1102\text{m}\angle B = 110. Dividing by 2 yields m∠B=55∘\text{m}\angle B = 55^\circ.

Key Concept

Exterior Angle Theorem and remote interior angles relation
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Topics

Triangle Properties and Angle Theorems

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