One of the corners of a right-angled triangle is 4 times smaller than the other. In another right-angled triangle

One of the corners of a right-angled triangle is 4 times smaller than the other. In another right-angled triangle, the difference in acute angles is 54 degrees. Are triangles similar? Why?

1. Take as x the value of one of the angles of the first given triangle. Another acute angle is 4x, since, according to the problem statement, one of the corners of the first triangle is 4 times larger than the other.

2. Their sum is equal to:

180 ° – 90 ° = 90 °.

3. Taking this into account, we compose the equation:

x + 4x = 90 °;

x = 18 °.

The second acute angle is 18 x 4 = 72 °.

4. We take as x (degrees) the value of one of the angles of the second given triangle.

Another acute angle (x + 54) degrees.

5. Let’s make the equation:

2x + 54 = 90 °;

x = 18 °

Another acute angle is 18 ° + 54 ° = 72 °.

Based on the above calculations, we come to the conclusion that the two acute angles of both triangles are equal. Hence, triangles are similar.

Answer: The given triangles are similar in two equal angles.