Supplementary Angles
If the measures of two angles sum up to [latex]180^\circ[/latex], they are called supplementary angles. You’ll notice that when this pair of angles are adjacent, they form a straight angle. Each angle is called a supplement of the other.
Take for instance the diagram above, [latex]\angle AXN[/latex] and [latex]\angle NXF[/latex] are supplementary. If we add their angle measures ([latex]120^\circ + 60^\circ [/latex]), we get [latex]180^\circ[/latex].
But the angles don’t have to be adjacent nor share a common side and vertex to be considered as supplementary angles.
[latex]\angle H[/latex] and [latex]\angle S[/latex] are supplementary. Why? Because even though they are non-adjacent angles, the sum of their measures is [latex]180^\circ[/latex].
[latex]35^\circ + 145^\circ = 180^\circ [/latex]
Example Problems Involving Supplementary Angles
Let’s delve more into the relationship of this angle pair by going through some examples.
Example 1: Are [latex]\angle ERW[/latex] and [latex]\angle WRQ[/latex] supplementary?
We have to add the angle measures of both angles in order to find out if they sum up to [latex]180^\circ[/latex].
And they do! Therefore, [latex]\angle ERW[/latex] and [latex]\angle WRQ[/latex] are supplementary angles.
Example 2: If [latex]\angle CYM[/latex] and [latex]\angle LKG[/latex] are supplementary, what is the measure of [latex]\angle CYM[/latex]?
We are only given the measure of [latex]\angle LKG[/latex]. However, since we know that supplementary angles add up to [latex]180^\circ[/latex], we can simply use subtraction in order to find the measure of [latex]\angle CYM[/latex].
Thus, the measure of [latex]\angle CYM[/latex] is [latex]\textbf{127}^\circ[/latex].
Example 3: [latex]\angle JBU[/latex] and [latex]\angle UBT[/latex] are supplementary. Find the missing angle measure.
This problem is similar to our previous example. The only difference is that the two angles are adjacent to each other. However, the concept stays the same. We can find the missing measure by subtracting the given measure of [latex]\angle UBT[/latex] from [latex]180^\circ[/latex].
The missing angle measure or the measure of [latex]\angle JBU[/latex] is [latex]\textbf{42}^\circ[/latex].
This makes sense because if we add both angle measures, we get [latex]180^\circ[/latex].
[latex]138^\circ + 42^\circ = 180^\circ [/latex]
This proves that both angles are indeed supplementary.
Example 4: What is the value of [latex]x[/latex]?
Just by looking at the diagram, we can tell that [latex]\angle PVH[/latex] and [latex]\angle HVA[/latex] are supplementary. Together, the angle pair form a straight angle while adjacent to each other. A straight angle measures [latex]180^\circ[/latex] and so are supplementary angles.
Both of the angle measures are given but one is expressed in the form of an algebraic expression. It may look challenging but it’s really not. Since we know that they are supplementary, we will set up our equation such that the sum of the angle measures is [latex]180^\circ[/latex]. Then we solve for [latex]x[/latex].
So, the value of [latex]x[/latex] is [latex]\textbf{9}[/latex].
To check if we got the correct answer, let’s plug in the value of [latex]x[/latex] into our original equation. If both sides of the equation equal to [latex]180[/latex], then we got the correct value for [latex]x[/latex].
Perfect! [latex]9[/latex] indeed is the correct value for [latex]x[/latex]. While checking, we also found out that the measure of [latex]\angle HVA[/latex] is [latex]61^\circ[/latex].
Example 5: Suppose [latex]\angle Q[/latex] and [latex]\angle F[/latex] are supplementary. Find the measures of the two angles.
Here we are given two supplementary angles whose measures are expressed in algebraic expressions. Let’s go ahead and set up our equation then solve for the variable [latex]x[/latex].
Now that we know the value of [latex]x[/latex], we can use this to find the measure of each angle. We’ll simply replace [latex]x[/latex] with [latex]7[/latex] on each of the algebraic expressions then simplify.
So the measure of [latex]\angle Q[/latex] is [latex]\textbf{156}^\circ[/latex] and the measure of [latex]\angle F[/latex] is [latex]\textbf{24}^\circ[/latex].
If we add both angle measures, we get [latex]180^\circ[/latex] which means our answers are correct.
[latex]156^\circ + 24^\circ = 180^\circ [/latex]
Example 6: Two supplementary angles are such that the measure of one angle is 3 times the measure of the other. Determine the measure of each angle.
Let [latex]x^\circ[/latex] be the measure of the first angle. Since the second angle measures 3 times than the first, then it will be [latex]3x ^\circ[/latex]. Keep in mind that the angles are supplementary so the right side of the equation must be [latex]180 ^\circ[/latex].
Using the value of [latex]x[/latex], the measure of the second angle will be [latex]3x = 3\left( {45} \right) = 135[/latex].
Therefore, the measures of the angles are [latex]\textbf{45} ^\circ[/latex] and [latex]\textbf{135} ^\circ[/latex] which when added sum up to [latex]180 ^\circ[/latex] .
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