Chapter 7.3, Problem 12WE

To determine

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

Explanation of Solution

Given information: An isosceles triangle and a scalene triangle.

Proof: An isosceles triangle has two equal angle measurements and two equal side lengths.

Here, $<CAB,<ABC$ are equal angles and $<ACB$ would be $180°-2<ABC$ .

  

And, scalene triangles have three different angle measurements and three different side lengths.

  

Here, $<A\text{'}B\text{'}C\text{'},<B\text{'}C\text{'}A\text{'}$ and $<C\text{'}A\text{'}B\text{'}$ all three angles are of different measurements. And when added gives $<A\text{'}B\text{'}C\text{'}+<B\text{'}C\text{'}A\text{'}+<C\text{'}A\text{'}B\text{'}=180°$

We can see the two angles of the isosceles triangle and only one angle of the scalene triangle can be the same. Or, one angle of the isosceles triangle and one angle of the scalene 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, an isosceles triangle and a scalene triangle can never be a similar triangle.