Chapter 7.3, Problem 11WE

To determine

To prove: Whether the given two polygons are always, sometimes, or never similar.

Explanation of Solution

Given information: A right triangle and an acute triangle.

Definition used:

• A triangle consist of the right angle is called the right triangle.
• A triangle consists of three angles less than the right angle is called an acute triangle.
• The triangles that measure the same angle but not the same size, are called similar triangles.

Proof: In a right-angled triangle, one angle is $90°$ and the other two angles are less than $90°$ .

Here, $<CAB=90°$ . And $<ABC,<BCA$ are less than $90°$

  

And, in an acute triangle, all three angles are less than $90°$

  

Here, $<A\text{'}B\text{'}C\text{'},<B\text{'}C\text{'}A\text{'}$ and $<C\text{'}A\text{'}B\text{'}$ all three angles are less than $90°$ .

We can see the two angles of the right triangle and an acute triangle can be the same. But, it is not possible to have all three angles of the same measurement.

And, for similar triangles, all three angles must be of the same measurement. So, a right triangle and an acute triangle can never be a similar triangle.