Chapter 8.3, Problem 16PSB

a

To determine

To find : The statement “If two triangles are similar, then they are congruent.” is always, sometimes or never be true.

Answer to Problem 16PSB

The Given statement is sometimes possible.

Explanation of Solution

Given information : If two triangles are similar, then they are congruent.

For similarity of triangles, we just need its corresponding angles to be congruent while for congruency of triangles, we need its sides as well as angles to be congruent.

  $\Rightarrow$ It is sometimes possible, that two similar triangles are congruent.

b

To determine

To find : The statement “If two triangles are congruent, then they are similar.” is always, sometimes or never be true.

Answer to Problem 16PSB

The Given statement is always possible.

Explanation of Solution

Given information : If two triangles are congruent, then they are similar.

For similarity of triangles, we just need its corresponding angles to be congruent, which will be proved automatically when those two triangles are congruent.

  $\Rightarrow$ It is always possible, that two congruent triangles are similar.

c

To determine

To find : The statement “An obtuse triangle is similar to an acute triangle.” is always, sometimes or never be true.

Answer to Problem 16PSB

The Given statement is never possible.

Explanation of Solution

Given information : An obtuse triangle is similar to an acute triangle.

An obtuse triangle is a triangle whose any one angle is greater than $90°$ and an acute triangle is a triangle whose all angles are less than $90°$ .

  $\therefore$ An obtuse angle can have two acute angles but no angle in acute triangle can be greater than $90°$ .

  $\Rightarrow$ It is never possible, that an obtuse triangle is similar to an acute triangle.

d

To determine

To find : The statement “Two right triangles are similar.” is always, sometimes or never be true.

Answer to Problem 16PSB

The Given statement is sometimes possible.

Explanation of Solution

Given information : Two right triangles are similar.

A right triangle is a triangle whose one angle is of $90°$ . For two right triangles to be similar, we need ratio of any two corresponding sides equal, which is not always possible.

  $\Rightarrow$ It is sometimes possible, that two right triangles are similar.

e

To determine

To find : The statement “Two equilateral triangles are similar.” is always, sometimes or never be true.

Answer to Problem 16PSB

The Given statement is sometimes possible.

Explanation of Solution

Given information : Two equilateral polygons are similar.

Two equilateral triangles are similar, if both of them have same number of sides, which is not always possible.

  $\Rightarrow$ It is sometimes possible, that two equilateral polygons are similar.

f

To determine

To find : The statement “Two equilateral triangles are similar.” is always, sometimes or never be true.

Answer to Problem 16PSB

The Given statement is always possible.

Explanation of Solution

Given information : Two equilateral triangles are similar.

Two equilateral triangles are similar, when all its angles are equal.

As, all angles in equilateral triangle are of $60°$ ,

  $\Rightarrow$ It is always possible, that two equilateral triangles are similar.

g

To determine

To find : The statement “Two rectangles are similar if neither is a square” is always, sometimes or never be true.

Answer to Problem 16PSB

The Given statement is always possible.

Explanation of Solution

Given information : Two rectangles are similar, if neither is a square.

Two rectangles are similar, when ratio of all its corresponding sides is equal, which is sometimes possible.

  $\Rightarrow$ It is sometimes possible, that rectangles are similar, if neither is square.