Finding the Measure of the Third Angle in a Triangle

What do you already know about triangles? Triangle have three sides and three angles. Triangles are named by their vertices. The triangle below is called [latex]\Delta ABC[/latex], read ‘triangle [latex]\text{ABC}[/latex] ’. We label each side with a lower case letter to match the upper case letter of the opposite vertex.

The three angles of a triangle are related in a special way. The sum of their measures is [latex]\text{180}^ \circ[/latex].

The measures of two angles of a triangle are [latex]55^\circ[/latex] and [latex]82^\circ[/latex]. Find the measure of the third angle.

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Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	The measure of the third angle in the triangle.
Step 3. Name. Choose a variable to represent it.	Let [latex]x=[/latex]the measure of the angle.
Step 4. Translate. Write the appropriate formula and substitute.	[latex]m\angle A+m\angle B+m\angle C=180[/latex]
Step 5. Solve the equation.	[latex]55+82+x=180[/latex] [latex]137+x=180[/latex] [latex]x=43[/latex]
Step 6. Check.	[latex]55+82+43\stackrel{?}{=}180[/latex] [latex]180=180\checkmark[/latex]
Step 7. Answer the question.	The measure of the third angle is [latex]43°[/latex]

Right Triangles

Sometimes you will be asked to find the unknown angle in a right triangle. As we just learned, a right triangle has one [latex]90^\circ[/latex] angle, which is often marked with the symbol shown in the triangle below.

If we know that a triangle is a right triangle, we know that one angle measures [latex]90^\circ[/latex] so we only need the measure of one of the other angles in order to determine the measure of the third angle.

One angle of a right triangle measures [latex]28^\circ[/latex]. What is the measure of the third angle?

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Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	The measure of an angle.
Step 3. Name. Choose a variable to represent it.	Let [latex]x=[/latex]the measure of the angle.
Step 4. Translate. Write the appropriate formula and substitute.	[latex]m\angle A+m\angle B+m\angle C=180[/latex]
Step 5. Solve the equation.	[latex]x+90+28=180[/latex] [latex]x+118=180[/latex] [latex]x=62[/latex]
Step 6. Check.	[latex]180\stackrel{?}{=}90+28+62[/latex] [latex]180=180\checkmark[/latex]
Step 7. Answer the question.	The measure of the third angle is [latex]62°[/latex]

In the examples so far, we could draw a figure and label it directly after reading the problem. In the next example, we will have to define one angle in terms of another. So we will wait to draw the figure until we write expressions for all the angles we are looking for.

The measure of one angle of a right triangle is [latex]20^\circ[/latex] more than the measure of the smallest angle. Find the measures of all three angles.

Show Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the measures of all three angles
Step 3. Name. Choose a variable to represent it. Now draw the figure and label it with the given information.	Let [latex]a={1}^{st}[/latex] angle. [latex]a+20={2}^{nd}[/latex] angle. [latex]90={3}^{rd}[/latex] angle (the right angle).
Step 4. Translate. Write the appropriate formula and substitute into the formula.	[latex]m\angle A+m\angle B+m\angle C=180[/latex] [latex]a+(a+20)+90=180[/latex]
Step 5. Solve the equation.	[latex]2a+110=180[/latex] [latex]2a=70[/latex] [latex]a=35[/latex] first angle [latex]a+20[/latex] second angle [latex]\color{red}{35}+20[/latex] [latex]55[/latex] second angle [latex]90[/latex] third angle.
Step 6. Check.	[latex]35+55+90\stackrel{?}{=}180[/latex] [latex]180=180\checkmark[/latex]
Step 7. Answer the question.	The three angles measure [latex]35°, 55°, 90°[/latex]