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Angle Word Problems Study Guide (page 2)

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Updated on Oct 3, 2011

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.

Angles Formed by Two Parallel Linest by a Transversal

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.

      Corresponding Angles
      Vertical Angles
      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.

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