Question

which congruence theorem can be used to prove △bda ≅ △bdc?
options: hl, ssa, aas, sss

Explanation:

1. First, analyze the given triangles \(\triangle BDA\) and \(\triangle BDC\):

• From the diagram, we can see that \(BD\) is a common side, so \(BD = BD\) (reflexive property).
• The markings show that \(BA = BC\) (the two sides from \(B\) to \(A\) and \(B\) to \(C\) have the same tick marks) and \(DA = DC\) (the two segments from \(D\) to \(A\) and \(D\) to \(C\) have the same double - tick marks).

1. Now, recall the triangle congruence theorems:

• The SSS (Side - Side - Side) congruence theorem states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.
• For \(\triangle BDA\) and \(\triangle BDC\), we have \(BA=BC\), \(DA = DC\), and \(BD=BD\). So, by the SSS congruence theorem, \(\triangle BDA\cong\triangle BDC\).
• The HL (Hypotenuse - Leg) theorem is used for right - triangles, and we don't know if these are right - triangles from the given information. SSA is not a valid triangle congruence theorem (it does not guarantee congruence in general). AAS (Angle - Angle - Side) requires two angles and a non - included side, which is not the case here as we are given three sides.

Answer:

D. SSS