Question

g.gsr.3.4 (mc)
we are given $\angle bad \cong \angle dcb$, and $\angle abd \cong \angle cdb$. what else are we able to prove in order to prove these triangles are congruent using aas?
image of a quadrilateral abcd with diagonal bd, angles at a, b, c, d marked
\bigcirc using the reflexive property.
\bigcirc $\angle abd \cong \angle cbc$ because they are corresponding angles.
\bigcirc because they are opposite sides.
\bigcirc using the reflexive property.

Explanation:

Step1: Recall AAS Congruence

AAS (Angle - Angle - Side) congruence criterion states that if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, then the triangles are congruent.

Step2: Analyze the Given Triangles

We have two triangles, $\triangle ABD$ and $\triangle CDB$. We know that $\angle BAD\cong\angle DCB$ and $\angle ABD\cong\angle CDB$. For AAS, we need a pair of congruent non - included sides. The side $BD$ is common to both $\triangle ABD$ and $\triangle CDB$. By the Reflexive Property of congruence, a segment is congruent to itself, so $BD\cong BD$.

Step3: Evaluate the Options

• The first option (assuming it is about $BD\cong BD$ using Reflexive Property) is correct.
• The option about $\angle ABD\cong\angle CBC$ is incorrect as there is no such correspondence and they are not corresponding angles.
• The option about opposite sides is not relevant here as we need a side for AAS and the common side is what we need.
• The last option (if it is also about $BD\cong BD$) is the same as the first correct idea.

Answer:

The option stating that $BD\cong BD$ using the Reflexive Property (the first option among the given options, assuming the first option is about $BD\cong BD$).