Learn
Finding Angle Measures
As you can see it can be a little tricky when multiple angles are formed in one diagram.
Using the previous information we can now find missing angle degree measures in a figure. Here we have a diagram of two intersecting lines. We are only given one degree measure but we will be able to find the other three using our knowledge of vertical angles and linear pairs.
Since ∠ a and ∠c are vertical angles we know they are congruent to each other. Hence, m∠c = 120°.
Since ∠a and ∠b are a linear pair we know their sum is 180°.
Hence m∠b = 180° – 120° = 60°.
Since ∠b and ∠d are vertical angles we now know m∠d is also 60°.
Quick Quiz
Now you try.
m∠1 =
1 and 2 form a linear pair, so they sum to 180°.
180° - 30° = 150°
m∠2 =
1 and 3 are vertical angles, so they are congruent.
m∠1 = m∠3 = 30°
m∠3 =
∠1 and ∠4 form a linear pair, so they sum to 180°.
180° - 30° = 150°
m∠4 =
Solution
Now let us incorporate some algebra to our geometry knowledge.
Solve for x and y using the diagram.
Remember, when solving for a variable you must first set up an equation. In this problem we will have two different equations since there are two variables x and y. You must determine the type of angles that have the same variable. In our diagram the angles with the variable x are vertical. We know that vertical angles are equal to each other. Hence our equation will look like this:
3x + 5 = x + 75
2x + 5 = 75
2x = 70
x = 35
Now we must solve for y. Only one angle has the variable y in it but that does not help us set up an equation. We do know that the angle vertical to it is 70 which gives us enough information to set up an equation because we know vertical angles are equal to each other.
y + 20 = 70
y =
Since the problem only ask us to solve for x and y we have completed the problem.
Important note: Many times we will also need to find each angle measure. In that case we would have substituted the value for each variable back into the expression to find the degree measure. See below.
3x + 5
3 ( ) + 5 =
+ 5
y + 20
+ 20
Since vertical angles are equal we would not have to do it again for the other angles.
Important Note!
It is extremely important that you look at the information carefully to determine your equation.
Notice this time that the angles are not vertical angles but instead they are a linear pair. Linear pair angles have a sum of 180°. In this case our equation would look like this:
(2x + 20°) + (x + 40°) = 180°
So do not get caught thinking all problems are worked the same way. The given information is what determines how you will solve the problem.
Quick Quiz
Solve for x and y.
Let's solve for x first. Two angles have x as part of their measures, the angle with
measure 2x + 8 and the angle with measure 3x + 17.
Are these angles vertical or do they form a linear pair?
OK, now that we know the angles form a linear pair, let's set up an equation expressing their relationship. Click on the correct equation.
2x + 8 = 3x + 17
(2x + 8) + (3x + 17) = 180°
(2x + 8) + (3x + 17) = 180° is the correct equation. Let's work through the steps.
x + = 180°
5x =
x =
Now, let's solve for y. Only one angle has y as a part of the measure, 2y. Let's use this and the angle with a measure of 70° to find y.
Are these angles vertical or do they form a linear pair?
OK, now that we know the angles form a linear pair, let's set up an equation expressing their relationship. Click on the correct equation.
2y = 70°
(2y) + 70 = 180°
(2y) + (70) = 180° is the correct equation. Let's work through the steps.
2y =
y =
