The problem provided in the image involves selecting the missing statement from a proof that shows vertical angles formed by intersecting lines are congruent. Here’s an analysis of the proof and identification of the missing step:

Diagram:

You have two intersecting lines, creating four angles, specifically:

• ∠ABD
• ∠ABC
• ∠DBE
• ∠CBE

Given Proof Outline:

Statements:

1. ∠ABD and ∠ABC form a linear pair, and ∠CBE and ∠ABC form a linear pair.

   • Reason: Definition of a linear pair.

2. ∠ABD and ∠ABC are supplementary, and ∠CBE and ∠ABC are supplementary.

   • Reason: Linear pair postulate.

3. m∠ABD + m∠ABC = 180°, and m∠CBE + m∠ABC = 180°.

   • Reason: Definition of supplementary angles.

4. (This is the missing step)

   • Reason: Substitution Property of Equality.

5. m∠ABD = m∠CBE.

   • Reason: Subtraction Property of Equality.

6. ∠ABD ≅ ∠CBE.

   • Reason: Definition of congruent angles.

Analysis:

In step 3, you establish that both pairs of angles sum to 180°:

• m∠ABD + m∠ABC = 180°
• m∠CBE + m∠ABC = 180°

The missing statement in step 4 will be obtained by equating the two expressions using the Substitution Property of Equality. That’s because both sums equal 180° and thus must be equal to each other:

• m∠ABD + m∠ABC = m∠CBE + m∠ABC

This step leads to the conclusion in step 5, where you subtract m∠ABC from both sides to show that m∠ABD = m∠CBE.

Thus, the missing statement in step 4 is:

• m∠ABD + m∠ABC = m∠CBE + m∠ABC

The correct answer is:

• m∠ABD + m∠ABC = m∠CBE + m∠ABC

Do you need any more details or have questions?

Follow-up Questions:

1. How can we generalize this proof to any intersecting lines?
2. What are the differences between supplementary and complementary angles?
3. How does the Substitution Property of Equality work in algebraic proofs?
4. What is the significance of the Linear Pair Postulate in geometry?
5. Can you explain why vertical angles are always congruent?

Tip:

In geometry, it's helpful to remember that vertical angles are always congruent due to the symmetry of intersecting lines, which creates pairs of angles that share the same measurements.