https://file.aiquickdraw.com/mathbot/file/1726454187090wpbdbis7.jpg

Let's go through the proof and complete the missing steps:

1. **Given:** \( \angle A \) and \( \angle B \) are supplements.
2. **\( m\angle A + m\angle B = 180^\circ \)** — By the definition of supplementary angles.
3. **Given:** \( \angle C \) and \( \angle D \) are supplements.
4. **\( m\angle C + m\angle D = 180^\circ \)** — By the definition of supplementary angles.
5. **\( m\angle A + m\angle B = m\angle C + m\angle D \)** — By the transitive property, since both sums equal \( 180^\circ \).
6. **\( \angle A \cong \angle C \)** — Given.
7. **\( m\angle A = m\angle C \)** — Definition of congruent angles.
8. **\( m\angle A + m\angle B = m\angle C + m\angle D \)** — Substitution property (Substitute \( m\angle A \) for \( m\angle C \)).
9. **\( m\angle B = m\angle D \)** — Subtraction property (Subtract \( m\angle A \) from both sides).
10. **\( \angle B \cong \angle D \)** — Definition of congruent angles.

Now, for statement 9 in Theorem 7, the correct choice should be:  
- \( m\angle B = m\angle D \) — This follows from the subtraction property and it is correctly listed in the options.

Therefore, the correct answer to the multiple-choice question is:
- **\( m\angle B = m\angle D \)**

Would you like a detailed explanation of any part of this proof or have any questions?

### Related Questions:
1. What is the definition of supplementary angles?
2. How does the transitive property apply in geometry proofs?
3. Why is the substitution property important in mathematical proofs?
4. What is the difference between the congruence of angles and the equality of their measures?
5. How can the subtraction property be applied to geometric equations?

#### Tip:
When working on geometric proofs, always ensure you clearly understand the properties and definitions used. This helps in logically connecting each step in the proof.

Given: ∠A and ∠B are supplements, ∠C and ∠D are supplements, and ∠A ≅ ∠C. Prove: ∠B ≅ ∠D.

This geometry proof demonstrates how to prove that ∠B ≅ ∠D using the transitive property, substitution, and subtraction properties. The problem involves supplementary and congruent angles, suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation