Question

triangle congruence: asa and aas
the proof that $\triangle abc \cong \triangle dcb$ is shown.
given: $\angle a \cong \angle d$, $\overline{cd} \parallel \overline{ab}$
prove: $\triangle abc \cong \triangle dcb$

statement	reason
2. $\overline{cd} \parallel \overline{ab}$	2. given
3. $\overline{cb} \cong \overline{bc}$	3. refl. prop.
4. $\angle abc \cong \angle dcb$	4. alt. int. $\angle$s are $\cong$
5. $\triangle abc \cong \triangle dcb$	5.?

what is the missing reason in the proof?
(options: con. int. $\angle$s are $\cong$, aas, alt. ext. $\angle$s are $\cong$, asa)

Explanation:

Step1: Analyze Given Information

We have two triangles, $\triangle ABC$ and $\triangle DCB$. The given information includes $\angle A \cong \angle D$, $\overline{CD} \parallel \overline{AB}$, and $\overline{CB} \cong \overline{BC}$ (reflexive property). We also found that $\angle ABC \cong \angle DCB$ using the alternate interior angles theorem (since $CD \parallel AB$ and $BC$ is a transversal).

Step2: Identify Triangle Congruence Criterion

Now, we need to prove $\triangle ABC \cong \triangle DCB$. Let's list the corresponding parts:

• $\angle A \cong \angle D$ (given)
• $\angle ABC \cong \angle DCB$ (alternate interior angles)
• $\overline{CB} \cong \overline{BC}$ (reflexive property)

This matches the AAS (Angle - Angle - Side) congruence criterion, where two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle.

Answer:

AAS (The option labeled "AAS" among the given choices)