The proof TriangleABC ≅ TriangleDCB that is shown.

Question

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)