We want to write a two-column proof of the following statement.
&Given:&& BDHAis a parallelogram.
& && CG≅CA
&Prove:&& ∠ BDH≅ ∠ G
Let's start by drawing a diagram of triangle ACG. We will also mark the angles we are asked to compare.
We can prove the claim in two steps. Let's focus on parallelogram BDHA first.
According to Theorem 6.4, opposite angles of this parallelogram are congruent. In this case, the opposite angles are ∠ BDH and ∠ A.
∠ BDH≅ ∠ A
Let's now shift our focus on triangle △ ACG. We are going to mark the congruent sides, CA and CG.
Now, recall the Isosceles Triangle Theorem. This theorem tells us that in a triangle angles opposite the congruent sides are also congruent.
CG≅CA
⇓
∠ A≅ ∠ G
We can now list what we found. We will form a sequence of angles, where each of the angles is congruent to the next one.
∠ BDH≅ ∠ A≅ ∠ G
Using the Transitive Property of Congruence, we can conclude that angles ∠ BDH and ∠ G are congruent.
Completed Proof
Let's summarize our findings in a two-column proof.
&Given:&& BDHAis a parallelogram.
& && CG≅CA
&Prove:&& ∠ BDH≅ ∠ G
Proof:
