Question

GEOMETRY! Using Postulates and/or Theorems learned in Unit 1, determine whether PRQ ~ MRN. Show all your work and explain why the triangles are similar or why they are not.

Answer 1

Answer:  No, they are not similar.

Explanation:

There are three ways to check whether two triangle are similar or not,

AAA similarity postulate: when three corresponding angles are congruent.

SSS similarity postulate: when all sides are in same proportion

SAS similarity postulate : If an angle of a triangle is congruent to an angle of another triangle and if the sides including these angles are proportional,

Since, Here all three angles or all three sides are not given,

That's why we will only check the SAS postulate to prove these triangles similar,

Here, PRQ and MRN are two triangles Shown in the diagram,

By the given diagram,

$\angle PRQ\cong\angle MRN$     ( Right angles )

$\frac{PR}{MR}=\frac{PM+MR}{MR}=\frac{8+10}{10}=\frac{18}{10}=\frac{9}{5}$

$\frac{RQ}{RN}=\frac{RN+NQ}{RN}=\frac{7+14}{7}=\frac{21}{7}=\frac{3}{1}$

$\implies \frac{PR}{MR}\neq \frac{RQ}{RN}$

Thus, however the angle of triangle PRQ  is congruent to the angle of triangle MRN but the sides including these angles are not proportional,

⇒ Triangle PRQ is not similar to the triangle MRN.