Algebra

Find the measures of two complementary angles if the measure of the larger angle is five more than four times the measure of the smaller angle.

Step-by-step solution with explanation

Final Answer

The smaller angle is and the larger angle is .

Step-by-step solution

1

Define variables for both angles

= smaller angle, = larger angle
We name the two unknown angles with variables so we can write equations. The smaller angle is and the larger is .
2

Write the complementary angles equation

Complementary angles add up to 90 degrees. This gives us our first equation.
3

Write the relationship equation

The larger angle is five more than four times the smaller angle. 'Four times the smaller' is , and 'five more' means we add 5.
4

Substitute into the first equation

We replace with in the complementary equation. Now we have one equation with one unknown.
5

Combine like terms and simplify

Add the terms together: . The equation becomes .
6

Solve for the smaller angle

Subtract 5 from both sides to get , then divide both sides by 5 to find degrees.
7

Solve for the larger angle

Plug back into . Multiply first, then add 5 to get degrees.
8

Verify both conditions are satisfied

Check that the angles add to 90 (they do) and that the larger equals four times the smaller plus five (it does). Both conditions are met.

Understanding this problem

Learning Insight

Complementary angles always sum to 90°. When a word problem gives two conditions about the same pair of angles, you can set up a system of two equations. Substitution lets you collapse that system into one equation you can solve directly.

Quick Tip

Spot the words 'complementary' (sum = 90°) vs. 'supplementary' (sum = 180°) right away — that single fact gives you your first equation every time.

Common Mistake

Students often write the relationship equation backwards, saying instead of . Always re-read which angle is described as larger before assigning the expression.