In triangle ABC, the side AB is extended past B to a point D, forming an exterior angle ∠CBD = 105°. The angle at C is ∠BCA = 38°. Find the size of ∠BAC, the angle at vertex A.
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Worked Solution
Step 1: ∠CBD = 105° is the EXTERIOR angle at B (on the straight line ABD). Use angles on a straight line to find the interior angle at B.
Step 2: Interior ∠ABC = 180° − 105° = 75°.
Step 3: Apply the triangle angle sum: ∠BAC + ∠ABC + ∠BCA = 180°.
Step 4: Substitute: ∠BAC + 75° + 38° = 180°, so ∠BAC = 180° − 75° − 38°.
Step 5: Compute: 180° − 75° = 105°, then 105° − 38° = 67°.
Final answer: ∠BAC = 67°.
Answer: 67°
Correct answer: 67°
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