Chapter 6, Problem 3CUR

To determine

To determine which congruence postulate or theorem will be useful to show the triangles as congruent.

Explanation of Solution

Given:

  

We have four options:

  $PC$ and $AL$

1. bisects each other.
2. $\angle P\cong \angle C;U$ is the midpoint of $PC$ .
3. $PA$ is parallel to $LC$ .
4. $PA\perp AL;LC\perp AL;PU\cong UC$ .

Option (a): $PC$ and $AL$ bisects each other would be a better option to choose.

Proof:

  $PC$ and $AL$ bisects each other.

  $\Rightarrow AU=UL$ ….(i) and

  $PU=UC$ ….(ii).

Now in $\Delta PAU$ and $\Delta CLU$ ,

  $AU=UL$ (given)

  $\angle PUA=\angle CUL$ (vertically opposite angles are equal)

  $PU=UC$ (given)

  $\therefore \Delta PAU\cong \Delta CLU$ by Side-Angle-Side (SAS) congruency criteria.

Conclusion:

Therefore, option (a) is a better option to choose.