They have two pairs of congruent sides where the sides included angle are congruent: the Side-Angle-Side (SAS) Congruence Theorem.From the exercise we know that AC is a straight segment, which makes ∠ADB and ∠CDB a linear pair. Since m∠CDB=90^(∘) it must be that m∠ADB=90^(∘) as well.
From the diagram, we also see that the two triangles share DB as a side, which means it is congruent according to the Reflexive Property of Congruence.
From the diagram, we see that the triangles have two pairs of congruent angles and one pair of congruent corresponding sides. This means we can claim congruence by the ASA Congruence Theorem.
△ ABD ≅ △ CBD
b
Examining the diagram, we notice that the triangles have a pair of congruent angles and a pair of congruent sides. From the diagram, we also notice that the triangles share BC as a side which means it is congruent according to the Reflexive Property of Congruence.
Since the congruent angles is the included angle to the two pairs of congruent sides, we can claim congruence by the SAS Congruence Theorem.
△ ABC ≅ △ DCB
c
Examining the diagram, we can identify a pair of vertical angles, ∠DCE and ∠ACB. According to the Vertical Angles Theorem, these angles are congruent.
Also, from the diagram we see that DE∥ BA. According to the Alternate Interior Angles Theorem, we know that ∠E≅ ∠B and ∠D ≅ ∠A.
As we can see, the two triangles have three pairs of congruent angles, which means the triangles are similar. We also see that they have a pair of congruent sides. However, these sides are not corresponding because they are not between the same two pairs of angles. Therefore, we cannot claim congruence between the triangles.
d
Looking at the diagram, we see that the triangles have two pairs of congruent angles. We also see that they share AB as a side.
Notice that they have two pairs of congruent angles, which means they are similar. If we separate the triangles, we also see that AB and B'A' is the included side to the same two pairs of angles.
Therefore, we can claim congruence by the ASA Congruence Theorem.
△ ABD ≅ △ BAC
e
Examining the diagram, we notice that the two triangles have three pairs of congruent angles. Therefore, we can claim that the triangles are similar. However, there is nothing that helps us identify the length of any of the sides, and therefore we cannot claim congruence.
f
From the diagram, we notice that the triangles have three pairs of congruent sides.
Since the two triangles have three pairs of congruent sides, we can claim congruence by the SSS Congruence Theorem.
△ RQS ≅ △ KGH
