Chapter 4.3, Problem 5CE

To determine

Toprove: $\overline{CA}\cong \overline{CB}$

Explanation of Solution

Given information: $\overline{CD}\perp \overline{AB}$ and D is the midpoint of $\overline{AB}$

Proof:

  

Consider $\Delta ADC$ and $\Delta BDC$

  $\overline{CD}\cong \overline{CD}$ (common side of $\Delta ADC$ and $\Delta BDC$ )

s $\left(\because \overline{CD}\perp \overline{AB}\right)$

  $\overline{AD}\cong \overline{BD}$ (as D is a midpoint of AB)

So,

  $\Delta ADC\cong \Delta BDC$ (by SAS congruence criteria)

Now if two triangles are congruent then their corresponding angles and sides are also congruent.

As $\Delta ADC\cong \Delta BDC$ ,

  $\overline{CA}\cong \overline{CB}$ ( $\overline{CA},\overline{CB}$ are corresponding sides of $\Delta ADC,\Delta BDC$ )