The problem states that ∠A and ∠B are complementary, and ∠A and ∠C are supplementary. It also provides the value of the measure of ∠A as 42° and asks for the measures of ∠B and ∠C.

Here’s how to approach it:

Step 1: Complementary Angles (∠A and ∠B)

• Complementary angles add up to 90°. Since ∠A and ∠B are complementary: $m∠A + m∠B = 90°$ Given $m∠A = 42°$, we can solve for $m∠B$: $42° + m∠B = 90°$ $m∠B = 90° - 42° = 48°$

Step 2: Supplementary Angles (∠A and ∠C)

• Supplementary angles add up to 180°. Since ∠A and ∠C are supplementary: $m∠A + m∠C = 180°$ Given $m∠A = 42°$, we can solve for $m∠C$: $42° + m∠C = 180°$ $m∠C = 180° - 42° = 138°$

Final Answer:

• $m∠B = 48°$
• $m∠C = 138°$

Would you like more details or further clarifications?

Here are five related questions to explore:

1. What is the definition of complementary angles?
2. How do you determine the value of an unknown angle when given a complementary angle pair?
3. What is the difference between complementary and supplementary angles?
4. If $m∠A$ were 60°, what would $m∠B$ and $m∠C$ be?
5. How can these angle relationships be applied in geometry problems involving polygons?

Tip: Complementary angles always add up to 90°, while supplementary angles sum to 180°.