Each and every triangle has a common characteristic: the sum of all its internal angles is always equal to 180°. Based on this principle, if you have the measure of two angles of a given triangle, finding the measure of the third is an easy task. However, in some cases you will have variables in place of measurements or even the measurement of just one of the angles. In this tutorial, learn what to do to determine the angles of a triangle in any of these situations.
Method 1 of 3: Using the Measure of the Two Other Angles
Step 1. Add the measurements of the two other angles together
The sum of the interior angles of a triangle always equals 180°. So, if you have the measure of two of the three angles, a few calculations are enough to determine the measure of the missing angle. Start by adding the two known angles: suppose these two angles are 80° and 65°. By adding them together (80° + 65°) you get the result 145°.
Step 2. Subtract this result from 180°
Since the sum of the three angles must result in 180°, by subtracting from this total the sum of the two known angles we obtain the measure of the third. Thus, 180° - 145° = 35°.
Step 3. Check your answer
You have found the measurement of the third angle, which in this example measures 35°. If you have doubts about your calculations, you can check your answer by adding all the known angles: the result must be 180° in order to comply with the condition of existence of a triangle. In this example, we have the angles 80° + 65° + 35° = 180°. So the answer is correct.
Method 2 of 3: Using Variables
Step 1. Write down the problem
Sometimes you won't have the measure of two angles, but some variables and the measure of just one of the angles (in some cases, just variables). Suppose the problem is as follows: "Find the angle measure x of a triangle whose angles measure x, 2x and 24°". Before starting, make a note of this problem.
Step 2. Add up all these measurements
Here the principle is similar to the previous method: just add all the measures (in this case, add the numerical measures and combine the variables). Thus, x + 2x + 24° = 3x + 24°.
Step 3. Subtract this result from 180°
Then subtract that sum from 180°, equating the equation to zero. Thus, the equation will be expressed as 180° - 3x + 24° = 0. After some operations, the new equation will be 156° - 3x = 0.
Step 4. Isolate variable "x" from the equation
Put the variable on one side of the equality and the independent terms on the other. The equation will be in the format 3x = 156°. Then divide both sides of the equation by the number that multiplies the variable (in this example, three) and you get the result x = 52°. This means that one of the angles of this triangle measures 52°. Thus, the other unknown angle, 2x, will measure 52° twice, that is, 104°.
Step 5. Check your answer
As in the previous method, you can add the three angles you got and then check if this triangle is valid. Adding the angles of this example we will have 52° + 104° + 24° = 180°. So your calculations are correct and your answer is right.
Method 3 of 3: Special Cases
Step 1. Determine the measure of the third angle of an isosceles triangle
The isosceles triangle has two equal sides and two equal angles. This type of triangle usually has a stripe on two of its sides to indicate that these are equal sides. If you have the measurement of one of your two similar angles, you can easily determine the remaining angles. Take a look at the following example to better understand:
- Suppose one of two equal angles measures 40°: because it is isosceles, one of the unknown angles also measures 40°. To find the third angle, add these two angles together and then subtract that sum from 180°. The sum of the two angles is 40° + 40° = 80°. Then, by subtracting this result from 180°, we have 180° - 80° = 100°. This is a measure of the missing angle.
Step 2. Determine the measure of the third angle of an equilateral triangle
The equilateral triangle has all its sides and angles equal. You will usually find two scratches in the middle of each of its sides, indicating that this triangle is equilateral. Since all three angles are the same, each one measures 60°. By adding these three angles, we can prove that this triangle exists: 60° + 60° + 60° = 180°.
Step 3. Determine the measure of the third angle of a right triangle
Suppose you have the measure of one of the angles of a right triangle and that it holds 30°: because it is a rectangle, this triangle has a right angle, that is, the second angle measures 90°. To determine the third angle, just apply the same principle as in the examples above: add up the known measurements and subtract the result from 180°. Adding the two known angles together we get 30° + 90° = 120°. Finally, by subtracting this sum from the total of 180° we get 180° - 120° = 60°. So the third angle measures 60°.