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Difficulty: MediumTriangle Properties and Angle Theorems

In the figure, line L1L_1 is parallel to line L2L_2. Vertex AA of △ABC\triangle ABC lies on L1L_1, and vertices BB and CC lie on L2L_2. Side ABAB is perpendicular to L2L_2. Point DD lies on L2L_2 such that CC is between BB and DD. If the measure of the exterior angle ∠ACD\angle ACD is 132∘132^\circ, what is the measure, in degrees, of ∠BAC\angle BAC?

Answer: 42

Answer

42
The correct answer is 4242. Because ABAB is perpendicular to L2L_2, ∠ABC\angle ABC is 90∘90^\circ. The exterior angle ∠ACD\angle ACD is given as 132∘132^\circ, which means the adjacent interior angle ∠ACB\angle ACB must be supplementary to it: 180∘−132∘=48∘180^\circ - 132^\circ = 48^\circ. Since the interior angles of a triangle always sum to 180∘180^\circ, the remaining angle ∠BAC\angle BAC is 180∘−(90∘+48∘)=42∘180^\circ - (90^\circ + 48^\circ) = 42^\circ. Alternatively, applying the Exterior Angle Theorem, the exterior angle is equal to the sum of the two remote interior angles: ∠ACD=∠ABC+∠BAC\angle ACD = \angle ABC + \angle BAC, so 132∘=90∘+∠BAC132^\circ = 90^\circ + \angle BAC, which simplifies to ∠BAC=42∘\angle BAC = 42^\circ.

Step-by-Step Solution

1
Determine the measure of interior angle ∠ABC\angle ABC.
∠ABC=90∘\angle ABC = 90^\circ
Since side ABAB is perpendicular to L2L_2, the angle it makes with L2L_2 at vertex BB is 90∘90^\circ.
2
Find the measure of interior angle ∠ACB\angle ACB.
∠ACB=48∘\angle ACB = 48^\circ
The interior angle ∠ACB\angle ACB and the exterior angle ∠ACD\angle ACD form a linear pair along line L2L_2, making them supplementary: ∠ACB=180∘−132∘=48∘\angle ACB = 180^\circ - 132^\circ = 48^\circ.
3
Calculate the measure of ∠BAC\angle BAC using the angle sum of a triangle.
42∘42^\circ
The interior angles of △ABC\triangle ABC sum to 180∘180^\circ. Subtracting the known angles gives ∠BAC=180∘−(90∘+48∘)=42∘\angle BAC = 180^\circ - (90^\circ + 48^\circ) = 42^\circ.

Key Concept

Triangle Angle Sum Theorem and Supplementary Angle Relationships
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Topics

Triangle Properties and Angle Theorems

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