Chapter 7.3, Problem 5CE

To determine

To check: Whether the polygons are similar if the corresponding angles of two polygons are congruent.

Answer to Problem 5CE

No, if the corresponding angles of two polygons are congruent, it is not necessary that the polygons are also similar.

Explanation of Solution

Given information: corresponding angles of two polygons are congruent

Two polygons are said to be similar if the following conditions are satisfied:

1. Corresponding sides of the polygons are proportional.
2. Corresponding angles of the polygons are equal.

It’s given that corresponding angles of two polygons are congruent. But it is not necessary that sides will also be proportional also.

So,

If the corresponding angles of two polygons are congruent, it is not necessary that the polygons are also similar.

For example:

Consider the following figures:

  

Here,

  $\begin{array}{l}\angle A=\angle P=30°\\ \angle C=\angle R=60°\\ \angle B=\angle Q=90°\end{array}$

So, corresponding angles are equal.

Now check whether the sides are proportional.

  $\begin{array}{l}\frac{AB}{PQ}=\frac{4}{12}=\frac{1}{3}\\ \frac{BC}{QR}=\frac{3}{5}\\ \frac{AC}{PR}=\frac{5}{13}\\ \therefore \frac{AB}{PQ}\ne \frac{BC}{QR}\ne \frac{AC}{PR}\end{array}$

In this case although corresponding angles are equal but as the corresponding sides are not proportional, triangles ABC and PQR are not similar.