Chapter 5, Problem 14CT

To determine

To prove: that $ABCD$ is a parallelogram.

Explanation of Solution

Given information:

Given figure,

  

Given:

  $AT=CT$ and $DT=\frac{1}{2}DB$

Calculation:

Consider the given information,

  $AT=CT$ and $DT=\frac{1}{2}DB$

Consider the following in triangles $ATB$ and $CTD$ .

  $\begin{array}{l}AT=CT\\ DT=BT\end{array}$

Since, $\angle ATB$ and $\angle CTD$ are vertically opposite angles.

  $\angle ATB=\angle CTD$

Use the SAS postulate as,

  $\Delta ATB\cong \Delta CTD$

Since corresponding parts of congruent triangle are congruent.

  $\angle ABT\cong \angle CDT$

Therefore it is proved that $AS=BQ$

Use the converse of alternate interior angle as “if two lines cut by a transversal line and pair of alternate interior on the opposite side of transversal line are congruent then that two lines are parallel. “That is,

  $\overline{AB}\parallel \overline{DC}$

As corresponding parts of congruent triangle are congruent,

  $\angle BAT\cong \angle DCT$

Therefore again by converse of alternate interior angle $ABCD$ is a parallelogram.