To find the measures of ∠ABC and ∠CBD with the given answer choices, we need to use the fact that the angles in a triangle add up to 180°.
Let's start by assigning values to ∠ABC and ∠CBD based on the answer choices:
- If ∠ABC = 20°, then ∠CBD = 180° - ∠ABD - ∠ABC = 180° - 112° - 20° = 48°.
- If ∠ABC = 45°, then ∠CBD = 180° - ∠ABD - ∠ABC = 180° - 112° - 45° = 23°.
- If ∠ABC = 67°, then ∠CBD = 180° - ∠ ABD - ∠ABC = 180° - 112° - 67° = 1°.
- If ∠ABC = 82°, then ∠CBD = 180° - ∠ABD - ∠ABC = 180° - 112° - 82° = -14°.
Since it is not possible for angles in a triangle to have negative measures, we can conclude that the only valid values for ∠ABC and ∠CBD are:
- ∠ABC = 45° and ∠CBD = 23°.
Therefore, ∠ABC= 45° and ∠CBD = 23° are the correct measures.