Algebra in Geometry Examples

An angle measures 15° more than its supplement. What is its measure?

Supplementary angles add up to 180°. We can label the measures of the two angles x and x + 15. The supplementary knowledge that the angles are supplementary gives us an equation.

x + (x + 15) = 180
2x + 15 = 180

Solving this equation yields x = 82.5. Since the question asks for the larger of the two angles, we plug 82.5 into x + 15 to get 97.5°.

An angle is one fifth as large as its complement. What is its measure?

This time, we can call the measures of the two angles x and 5x (since x is one fifth as large as 5x). Complementary angles give us the equation.

x + 5x = 90

Solving it gives us x = 15. The question asks for the smaller angle, which is just x, so we have an answer of 15°.

Suppose AB = 2x + 3, BC = 3x – 2, and AC = x + 10. If B is between A and C, what is the length of AC?

Since B is between A and C, we know AB + BC = AC. Substituting our given algebraic values, we have the equation:

(2x + 3) + (3x – 2) = (x + 10)

At this point, algebra should be a piece of cake. Solving this equation for x gives us:

x = 2.25

The question asks for AC, so we plug 2.25 into the equation for AC.

AC = x + 10
AC = 2.25 + 10

That means that AC has a length of 12.25 units.

Suppose AB = 2x + 3, BC = 3x – 2, and AC = x + 10. If B is between A and C, is B the midpoint of AC?

We still know x = 2.25 from the last example, so we can use it to find AB and BC.

AB = 2x + 3
AB = 2(2.25) + 3
AB = 7.5

That's AB. Now for BC.

BC = 3x – 2
BC = 3(2.25) – 2
BC = 4.75

Since they're not equal, we know that B is not the midpoint of AC.