Question

Given: $△LMN\cong △RST$_;$\overline{LX}$_{and $\overline{RY}$are altitudes.}

Prove:$\overline{\mathit{L}\mathit{X}}\cong \overline{\mathit{R}\mathit{Y}}$

Short answer

By corresponding parts of congruent triangles, it can be said that$\overline{\mathit{L}\mathit{X}}\cong \overline{\mathit{R}\mathit{Y}}$_.

Step-by-step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Description of step.

It is being given that $△LMN\cong △RST$_.

Therefore, $LM\cong RS$, $\angle LMN\cong \angle RST$.

From the diagram, it can be seen that:$\angle LMN+\angle LMX=180°$

and $\angle RST+\angle RSY=180°$.

Therefore, it can be obtained that:

$\begin{array}{c}\angle LMN+\angle LMX=\angle RST+\angle RSY\\ \angle LMN+\angle LMX=\angle LMN+\angle RSY\\ \angle LMX=\angle RSY\end{array}$

Therefore, $\angle LMX\cong \angle RSY$.

Step 3. Description of step.

From the given diagram it can be noticed that $\angle LXM=\angle RYS=90°$.

In the triangles $△LMX$and $△RSY$, it can be noticed that $\angle LMX\cong \angle RSY$, $\angle LXM\cong \angle RYS$and $△RSY$.

Therefore, the triangles $△LMX$and$△RSY$ are congruent triangles by using the AAS postulate.

Therefore, by corresponding parts of congruent triangles it can be said that $\overline{LX}\cong \overline{RY}$_.

Hence proved.