To find the measure of two supplementary angles, use the defining fact that supplementary angles have a total measure of 180°. Still, if one angle is known, subtract it from 180° to find the other. If the angles are represented by variables or algebraic expressions, set up an equation whose sum equals 180°, solve it, and then check that the two results add up correctly.
What Are Supplementary Angles?
Two angles are supplementary when their measures add up to 180 degrees. This relationship is often connected with a straight angle, which measures 180°. To give you an idea, if one angle measures 70°, the other angle must measure 110° because:
[ 70^\circ + 110^\circ = 180^\circ ]
Supplementary angles do not have to be identical, and they do not need to be the same size. In fact, they are only equal to each other when both measure 90°.
A simple representation of a straight angle divided into two supplementary angles is:
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The two angles on either side of the vertical line form a straight angle. Their combined measure is 180°.
The Basic Formula
When finding the measure of two supplementary angles, use the following formula:
[ \text{Angle 1} + \text{Angle 2} = 180^\circ ]
If the measure of one angle is known, rearrange the formula:
[ \text{Unknown angle} = 180^\circ - \text{Known angle} ]
This formula works whenever the two angles are supplementary. It is useful in geometry, trigonometry, physics, construction, and many real-world situations involving lines and angles.
That said, it actually matters more than it seems. On the flip side, for instance, the pair 40° and 140° is supplementary, but so are 65° and 115° and 89° and 91°. You need an additional piece of information, such as one angle’s measure, an equation, or another relationship between the angles Simple as that..
How to Find the Measure of Two Supplementary Angles
Step 1: Identify the known information
Begin by writing down what you know. Determine whether you have:
- The measure of one angle
- Variables representing the angles
- Algebraic expressions for the angles
- A diagram showing relationships
- A relationship between the two angles, such as one being twice the other
Step 2: Write the supplementary-angle equation
Represent the two angles as (A) and (B). Since they are supplementary:
[ A + B = 180^\circ ]
Substitute the known angle or expressions for (A) and (B).
Step 3: Solve the equation
Use the same algebraic steps you would use to solve any equation. Isolate one variable, perform the required operations, and simplify Most people skip this — try not to..
Step 4: Find the second angle
Once you know the first angle, subtract it from 180°:
[ \text{Second angle} = 180^\circ - \text{First angle} ]
Step 5: Check the answer
Add the two final angle measures. The total must equal 180°. This check helps catch calculation errors and confirms that the angles are supplementary.
Example 1: One Angle Is Known
One of two supplementary angles measures 35°. Find the measure of the other angle.
Use the formula:
[ \text{Unknown angle} = 180^\circ - 35^\circ ]
[ \text{Unknown angle} = 145^\circ ]
Which means, the two angles measure:
[ \boxed{35^\circ \text{ and } 145^\circ} ]
Check:
[ 35^\circ + 145^\circ = 180^\circ ]
Because the sum is
