# Problem Solving With Complementary and Supplementary Angles

Solving problems regarding complementary and supplementary angles could be fun, easy, and challenging if you know how to translate very basic mathematical expression to symbols.

Before getting into the proper problem solving, let us first recall some basic facts.

• Sum of complementary angles is equal to 90 degrees. Meaning, the complement of angle X is 90-X.
• Sum of supplementary angles is equal to 180 degrees. Meaning, the supplement of angle X is 180-X.

Technique:

We will demonstrate our technique used here by giving a specific problem.

Twice the complement of an angle is 24 degrees less than its supplement. What is the measure of the angle?

Solution:

To make it easy for you, we will use color coding to exploit the problem easily.

Twice the complement of an angle is 24 degrees less than its supplement. What is the measure of the angle?

Our goal is to translate the word problems into mathematical equation. Thinking that the angle we are looking for is X, then the phrase in red can be expressed as,

2(90-X)

The word is is the word for equal(=) symbol.

The phrase in brown can be expressed as follows,

(180-x)-24

Relating the three we have,

2(90-X) = (180-x)-24

Solving for X we have,

2(90-X) (180-x)-24

180-2X=180-X-24

X=24°

Worked Problem 1:

Six times the complement of an angle is 40 degree less than its supplement. What is the supplement of an angle?

Solution:

Let x be the angle

Complement of x: 90-x

Supplement of x: 180-x

Translating the word to a mathematical equation we have, $6(90-x)=(180-x)-40$ $540-6x=140-x$ $5x=400$ $x=80^\circ$

The supplement of 80° is 180-80=100°.

Worked Problem 2: 2013 MMC Grade 7 Elimination

Two angles are supplementary. One angle is 24° less than thrice the other. What is the complement of smaller angle?

Solution:

Let x be the angle.

Let 180-x be the other angle

Translating the word problem into mathematical equation we have, $x=3(180-x)-24$ $x=540-3x-24$ $4x=516$ $x=129^\circ$

The other angle can be easily obtained by subtracting the answer from 180°. Thus, the other angle is 180-129=51°  which is smaller than 129°.

Worked Problem 3:

The product of the angle and its complement in degrees is 2016. What is the larger angle?

Solution:

Let x be the angle

Let 90-x be the other angle

Relating the angles we have, $x(90-x)=2016$ $90x-x^2=2016$ $x^2-90x+2016=0$

Gees! This is a quadratic equation with huge coefficients. Quadratic equations can be solved using factoring, completing the square, and using quadratic formula.  For me, it would be more convenient to use completing the square since the leading coefficient is 1. $x^2-90x+2016=0$ $x^2-90x=-2016$ $x^2-90x+45^2=-2016+45^2$ $(x-45)^2=9$ $\sqrt{(x-45)^2}=\sqrt{9}$ $x-45=\pm 3$ $x=45\pm 3$ $x=42,48$

The bigger angle is obviously 48°.