The measure of the larger acute angle in a right triangle is 10 degrees less than three times the measure of the smaller acute angle. Find the measure of each angle

Answer 1

Bryce,

As you know, a triangle has 180 degrees total. You also know that a right angle is 90 degrees.

Clearly, the two acute angles have to add up to 90 degrees.

180 (the total degrees in triangle) - 90 (the right angle) = 90 (the sum of the two acute angles).

Let's say that the smaller acute angle is x. Knowing that the larger acute angle is 10 degrees less than 3 times the measure of the smaller acute angle, we can say that the larger acute angle is equal to 3x - 10. Going back to where we said that the two acute angles have to add up to 90 degrees, we can say that

x + 3x - 10 = 90. Combining Like Terms simplifies this expression to 4x - 10 = 90. Through algebra, we find that x = 25. So we know the smaller angle is 25 degrees, and the larger angle is "10 less than 3 times" this, which is 65 degrees.

We can check our answer by adding 25 + 65 + 90 to make sure our three angles add up to 180 degrees, which it does! :)

Hope this helps.

Answer 2

This is a system of simultaneous equations in two unknowns, in this case the angles A and B.

in a right triangle

A + B + 90 = 180

=> (1) A + B = 90

The measure of the larger acute angle is 10 degrees less than three times the measure of the smaller acute angle

(2) 3A - 10 = B

Substitute value of B from (2) into (1) to get A + (3A - 10) = 90

=> 4A - 10 = 90 => 4A = 100 => A = 25

Substitute found A in (1) to get 25 + B = 90 => B = 65

Hence our two required angles are 25° and 65°.

Double-check: 25 + 65 + 90 = 180 as required for right angle triangle, and 3x25 = 75 and 65 is 10 smaller as required.