Question

The proof TriangleABC ≅ TriangleDCB that is shown.

Given: AngleA ≅ AngleD; CD||AB
Prove: TriangleABC ≅ TriangleDCB

Triangles C D B and C A B are shown. Angles C D B and C A B are congruent. Sides D C and A B are parallel.
What is the missing reason in the proof?

A 2-column table has 5 rows. Column 1 is labeled Statement with entries angle A is-congruent-to angle D, line segment C D is parallel to line segment A B, line segment C B is-congruent-to line segment B C, angle A B C is-congruent-to angle D C B, triangle A B C is-congruent-to triangle D C B. Column 2 is labeled Reason with entries given, given, reflective property, alternating interior angles are congruent, question mark.

alt. ext. Angles are ≅
ASA
AAS
corr. int. Angles are ≅

Answer 1

To complete the proof that Triangle ABC is congruent to Triangle DCB, we need to identify the correct reason for the congruence of the triangles.

Looking at your two-column table, here is the breakdown:

1. **Statement**: Angle A ≅ Angle D
**Reason**: Given

2. **Statement**: CD || AB
**Reason**: Given

3. **Statement**: CB ≅ BC (the reflective property of congruence applies here)
**Reason**: Reflective property

4. **Statement**: Angle ABC ≅ Angle DCB
**Reason**: Alternate interior angles are congruent (because CD || AB and AB is a transversal).

5. **Statement**: Triangle ABC ≅ Triangle DCB
**Reason**: **Angle-Side-Angle (ASA)**

So, the missing reason for the congruence of triangles ABC and DCB is **ASA**, as we have two angles and the included side that are congruent.

Here's the completed proof:

1. Angle A ≅ Angle D (Given)
2. CD || AB (Given)
3. CB ≅ BC (Reflexive Property)
4. Angle ABC ≅ Angle DCB (Alternate Interior Angles are Congruent)
5. Triangle ABC ≅ Triangle DCB (ASA)