Chapter 4.3, Problem 4WE

To determine

Toprove: $\overline{AC}\cong \overline{BD}$.

Explanation of Solution

Given information :

Givenif $M$ is the midpoint of $\overline{AB}$ ; $\angle 1\cong \angle 2;\angle 3\cong \angle 4$.

  

Proof: Since $M$ is the midpoint of $\overline{AB}$ ;

  $\Rightarrow \overline{AM}\cong \overline{BM}$ by the definition of midpoint and it is given by $\angle 1\cong \angle 2;\angle 3\cong \angle 4$

By ASA theorem, the given two triangles are congruent.

Thus, $\Delta AMC\cong \Delta BMD$

  $\Rightarrow$ Pairs of sides and angles of two triangles are equal in measurement.

  $\Rightarrow \angle 1\cong \angle 2$

Hence, $\overline{AC}\cong \overline{BD}$.