Chapter 8.1, Problem 39PPS

To determine

To prove: $\Delta PSQ\sim \Delta PQR,\Delta PQR\sim \Delta QSR\text{and}\Delta PSQ\sim \Delta QSR$

Explanation of Solution

Given:

The given figure is:

  

Calculation:

It is given that triangle PQR is a right triangle.

Definition of altitude shows that $\overline{QS}\perp \overline{RP}$

Therefore, triangles PSQ and QSR are right angles.

As the right angles are congruent, therefore,

  $\angle PQR\cong \angle PSQ\cong \angle QSR$

According to Reflexive property of Congruence,

  $\angle P\cong \angle Pand\angle R\cong \angle R$

AA Similarity postulate shows that,

  $\Delta PSQ\sim \Delta PQR\text{and}\Delta PQR\sim \Delta QSR$

Therefore, According to Transitive property of Congruence,

  $\Delta PSQ\sim \Delta QSR$