The question is asking about the congruence of triangles â–³ ABC and â–³ ABD. Let's highlight these triangles on the diagram.
Let's also indicate that the segment AB bisects angles ∠CBD and ∠CAD.
Angle bisectors cut an angle into two congruent angles. This means that triangles â–³ ABC and â–³ ABD have two congruent angle pairs.
∠CAB&≅ ∠DAB
∠ABC&≅ ∠ABD
Since side AB is included in both triangles and is therefore congruent, we have two congruent angles and a congruent side. The Angle-Side-Angle (ASA) Congruence Postulate implies that the two triangles are congruent.
△ ABC≅△ ABD
b
We now want to prove that triangles â–³ CAF and â–³ DAE are congruent. Let's focus on the congruent triangles mentioned in the question, â–³ ABC and â–³ ABD, and mark the given angle congruence and the congruent corresponding sides.
Notice that at vertex A two intersecting straight lines form a vertical angle pair. These angles are also congruent, so let's indicate this on the diagram.
We can see that triangles â–³ CAF and â–³ DAE have two pairs of congruent angles. It is also marked on the diagram that the included side is also congruent.
Hence, the Angle-Side-Angle (ASA) Congruence Postulate implies that the two triangles are congruent.
△ CAF≅△ DAE
c
Last, we want to prove the congruence of triangles â–³ BHG and â–³ BEA. Let's focus on the triangles mentioned in the question and mark the given congruences on the diagram.
Let's summarize what we know about triangles â–³ BHG and â–³ BEA.
It is given that HB≅ EB.
It is given that ∠BHG≅ ∠BEA.
We can also see that angles ∠BGH and ∠BAE are put together from congruent angles. This means that they have the same shape and size, so they are congruent.
∠BGH≅ ∠BAE
We can see that triangles â–³ BHG and â–³ BEA have two pairs of congruent angles and a pair of congruent sides that are not included between the congruent angles.
Hence, the Angle-Angle-Side (AAS) Congruence Theorem implies that the two triangles are congruent.
△ BHG≅△ BEA
