The acute angles of a right-angled triangle are 1: 5. In another right-angled triangle

The acute angles of a right-angled triangle are 1: 5. In another right-angled triangle, the difference in acute angles is 60 degrees. Are these triangles similar and why?

Triangles are called rectangular if one of the angles is 90 °.

In this case, similar will be triangles in which the angles are respectively equal.

In order to determine whether the triangles ΔАВС and ΔА1В1С1 are similar, it is necessary to calculate the degree measures of their angles.

Consider the triangle ΔABS.

Since the sum of all the angles of the triangle is 180º, and the acute angles ∠A and ∠B are related as 1: 5, then we express:

x – degree measure ∠A;

5x – degree measure ∠В;

90º is the degree measure of the angle ∠С;

180º is the sum of all the angles of the triangle;

x + 5x + 90 = 180;

x + 5x = 180 – 90;

6x = 90;

x = 90/6 = 15;

∠А = 15º;

∠В = 15º · 5 = 75º.

Consider a triangle ΔА1В1С1.

Since the sum of all the angles of the triangle is 180º, and the difference between the acute angles ∠A and ∠B is 60º, we express:

x – degree measure ∠A1;

x + 60 – degree measure of angle ∠В1;

90º – degree measure ∠С1;

180º is the sum of all the angles of the triangle:

x + x + 60 + 90 = 180;

x + x = 180 – 90 – 60;

2x = 30;

x = 30/2 = 15;

∠A1 = 15º;

∠В1 = 15º + 60º = 75º.

Answer: triangles ΔАВС and ΔА1В1С1 are similar since their angles are respectively equal: ∠А = ∠А1 = 15º; ∠В = ∠В1 = 75º; ∠С = ∠С1 = 90º.