Exercise 53 Page 21

First Congruence

The first congruence can be shown using the Reflexive Property. It states that $a=a.$ Therefore, we can write the following expression.

Second Congruence

We are told that $\overline{AB}\cong \overline{CD}.$ Also by the Reflexive Property, we know that $\overline{CD}\cong \overline{DC},$ since it's the same side written in two different ways. Therefore, by using the Transitive Property we can write the following expressions.

Third Congruence

To prove $\overline{BD}\cong \overline{AC},$ let's use two congruences that we are given: $\overline{AB}\cong \overline{BD}$ and $\overline{AB}\cong \overline{AC}.$ First, using the Reflexive Property we can rewrite the first congruence as $\overline{BD}\cong \overline{AB}.$ Now, from the Transitive Property, we get the last part of what we needed.

Conclusion

Because we were able to show the three required congruences, we know that Pedro is able to use the Reflexive and Transitive Properties to show that the triangles are congruent.