Introduction to Angle Word Problems
The composer opens the cage door for arithmetic, the draftsman gives geometry its freedom.
—COCTEAU (1839–1963)
This lesson will review the basic terms of geometry and list the special types of angles and angle pairs found in many geometry word problems.
Basic Figures of Geometry
Knowing the basic terms of geometry can make the study of more complicated shapes much easier. Listed next are a few of these basic terms.
A point is a location in space.
A line is an infinite collection of points extending in opposite directions.
A plane is a neverending flat surface that extends in all directions.
A ray is the set of all points extending in a straight line from one side of an endpoint.
Angles
An angle is made up two rays that meet at a common endpoint. The rays are the sides of the angle, and the common endpoint is the vertex of the angle.
Angles can be named in various ways. They can be named by the letter of the vertex, by three letters with the middle letter being the vertex, or by a number written in the interior of the angle. These three ways to name angles () are shown in the following figure.
Tip:
Here are some important types of angles and their measures:
Acute angles measure less than 90°.
Right angles measure 90°.
Obtuse angles measure greater than 90 but less than 180°.
Straight angles measure 180°.
Reflex angles measure more than 180°.

There are special types of pairs of angles that are common to geometry word problems. These special types are explained next.
Adjacent Angles
Adjacent angles are angles next to each other that share a common ray, or side, and a common vertex. Nonadjacent angles are not next to each other and do not share a common ray, or side. An example of each is shown below.
Vertical Angles
Vertical angles are the nonadjacent angles formed by two intersecting lines. The measures of vertical angles are always equal.
Look at the following example. The measure of angles 1 and 3 are each 60°; they are vertical angles. The measure of angles 2 and 4 are each 120°; they are also a vertical pair of angles. Notice that the adjacent angles have a sum of 120 + 60 = 180°.
Complementary and Supplementary Pairs of Angles
Complementary angles are two angles that have a sum of 90°. They can be adjacent, or nonadjacent angles, as shown next.
Supplementary angles are two angles that have a sum of 180°. They can be adjacent, or nonadjacent angles, as shown next. If two supplementary angles are also adjacent, they are known as a linear pair.
Angles Formed by Two Parallel Lines Cut by a Transversal
When two parallel lines are cut by a transversal, there are many patterns in the angles, as shown next.
Vertical Angles: As explained in the previous section, angle pairs 1 and 4, 2 and 3, 5 and 8, and 6 and 7 are vertical angles.
Alternate Interior Angles: These are the angles on the interior of the parallel lines, but on opposite sides of the transversal. These are the angle pairs 4 and 5 and 3 and 6.
Alternate Exterior Angles: These are the angles on the exterior of the parallel lines, but on opposite sides of the transversal. These are the angle pairs 1 and 8 and 2 and 7.
Corresponding Angles: These are the nonadjacent angles on the same side of the transversal, but one is an interior angle and the other is an exterior angle. Corresponding angles are named by the pairs 1 and 5, 2 and 6, 3 and 7, and 4 and 8.
Supplementary Angles: Any two adjacent angles in the diagram are supplementary. Some of these pairs are 1 and 2, 2 and 4, 3 and 4, 1 and 3, 5 and 6, and so on.
Tip:
When dealing with parallel lines cut by a transversal, the following angle pairs have equal measure.
Alternate Interior Angles
Alternate Exterior Angles
Adjacent angles are supplementary; they add to 180°.

Word Problems With Special Angle Pairs
Vertical Angles
Vertical angle problems can be solved by setting the values for each angle equal to each other.
Example
A pair of vertical angles are represented by the expressions x + 30 and 5x – 10. What is the measure in degrees of each angle?
Read and understand the question. This question is looking for the measure of each vertical angle. The vertical angles formed by two intersecting lines are always congruent.
Make a plan. Set the expressions that represent each angle equal to each other, and solve for x.
Carry out the plan. The equation is x + 30 = 5x – 10. Subtract x from each side to get 30 = 4x – 10. Then, add 10 to each side to simplify the equation to 40 = 4x. Divide each side by 4 to get the variable alone: x = 10. Therefore, the angles are 10 + 30 = 40° each.
Check your answer. To check this solution, substitute x = 10 into the other expression to be sure it also is equal to 40°: 5(10) – 10 = 50 – 10 = 40°. Each vertical angle is 40°, so this answer is checking.
Complementary Angles
Complementary angle problems can be solved by adding the values of the angles and setting the sum equal to 90°.
Example
One angle of a complementary pair is equal to twice the measure of the other angle. What is the measure in degrees of each angle?
Read and understand the question. This question is looking for the measure of each complementary angle. The measure of two complementary angles is always 90°.
Make a plan. Write the expression for each angle and set the sum equal to 90.
Carry out the plan. Let x = the smaller angle and let 2x = the larger angle. Therefore, the equation is x + 2x = 90. Combine like terms to get 3x = 90. Divide each side of the equation by 3 to get the variable alone: x = 30. Thus, 2x = 60. The two angles measure 30° and 60°, respectively.
Check your answer. To check this problem, make sure that the sum of the measures of the angles is equal to 90° and that one angle is twice the other. The angles are 30 + 60 = 90° and (30)(2) = 60, so this answer is checking.
Supplementary Angles
Supplementary angle problems can be solved by adding the values of the angles and setting the sum equal to 180°.
Example
One of two supplementary angles measures 50° more than the other measures. What is the measure in degrees of each angle?
Read and understand the question. This question is looking for the measure of each supplementary angle. The measure of two supplementary angles is always 180°.
Make a plan.Write the expression for each angle and set the sum equal to 180.
Carry out the plan. Let x = the smaller angle and let x + 50 = the larger angle. Therefore, the equation is x + x + 50 = 180. Combine like terms to get 2x + 50 = 180. Subtract 50 from each side of the equation to simplify it to 2x = 130. Divide each side of the equation by 2 to get the variable alone: x = 65. Thus, x + 50 = 115. The two angles measure 65° and 115°, respectively.
Check your answer. To check this problem, make sure that the sum of the measures of the angles is equal to 180° and that one angle is 50° more than the other. The angles are 65 + 115 = 180°, and 115 – 65 = 50, so this answer is checking.
Alternate Interior and Alternate Exterior Angles
Alternate interior and alternate exterior angle problems can be solved by setting the values for each angle equal to each other.
Example
When parallel lines are cut by a transversal, the measure of one alternate interior angle is equal to 60 less than 3 times the other. Find the measure of both angles.
Read and understand the question. This question is looking for the measure of each alternate interior angle. The alternate interior angles formed by two parallel lines cut by a transversal are congruent.
Make a plan. Write an expression to represent each angle. Then, set the expressions equal to each other and solve for x.
Carry out the plan. Let x = one angle and 3x – 60 = the other angle. The equation is x = 3x – 60. Add 60 to each side of the equation to get x + 60 = 3x. Subtract x from each side to get 60 = 2x. Then, divide each side of the equation by 2 to get the variable alone: x = 30. Therefore, the angles are 30° each.
Check your answer. To check this solution, substitute x = 30 into the other expression to be sure it also is equal to 30°: 3(30) – 60 = 90 – 60 = 30°. Each alternate interior angle is 30°, so this answer is checking.
Corresponding Angles
Corresponding angle problems can be solved by setting the values for each angle equal to each other.
Example
Two parallel lines are cut by a transversal. The sum of two corresponding angles formed is 110°. What is the measure in degrees of each angle?
Read and understand the question. This question is looking for the measure of each corresponding angle. When parallel lines are cut by a transversal, the measures of corresponding angles are equal.
Make a plan. Take the given sum and divide by 2.
Carry out the plan. 110° divided by 2 is 55°. Each corresponding angle is 55°.
Check your answer. To check this solution, make sure that the sum of the two angles is 110°: 55 + 55 = 110, so this answer is checking.
Find practice problems and solutions for these concepts at Angle Word Problems Practice Questions.
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