Question

1. complete the proof.

given: point ( m ) is the midpoint of ( overline{bd} ; overline{am} cong overline{cm} ).
prove: ( \triangle abm cong \triangle cdm )

a. ( overline{ab} cong overline{cd} )
b. ( overline{bm} cong overline{dm} )
c. definition of congruent angles
d. vertical angles theorem
e. alternate interior angles theorem

Explanation:

Step1: Determine the reason for $\angle AMB\cong\angle CMD$

$\angle AMB$ and $\angle CMD$ are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent. So, the reason for $\angle AMB\cong\angle CMD$ is the Vertical Angles Theorem.

Step2: Determine the missing segment congruence

Since $M$ is the mid - point of $\overline{BD}$, by the definition of a mid - point, a mid - point divides a segment into two congruent segments. So, $\overline{BM}\cong\overline{DM}$.

Answer:

1. d. Vertical Angles Theorem; 3. b. $\overline{BM}\cong\overline{DM}$