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Ch. 1 - Trigonometric Functions
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 1 - Trigonometric FunctionsProblem 15
Chapter 2, Problem 15

Find the measure of each marked angle. In Exercises 19–22, m and n are parallel.
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1
Identify the given information: lines m and n are parallel, and there are marked angles formed by a transversal crossing these parallel lines.
Recall the properties of angles formed by a transversal with parallel lines, such as corresponding angles, alternate interior angles, and consecutive interior angles, which are either equal or supplementary.
Use the fact that corresponding angles are equal when two lines are parallel and cut by a transversal. Set the measure of the marked angle equal to its corresponding angle if applicable.
If the marked angle is an alternate interior angle to a known angle, use the property that alternate interior angles are equal to find its measure.
If the marked angle forms a linear pair with a known angle, use the fact that the sum of angles in a linear pair is \(180^\circ\) to set up an equation and solve for the marked angle.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Properties of Parallel Lines and Transversals

When two parallel lines are cut by a transversal, several angle relationships arise, such as corresponding angles being equal, alternate interior angles being equal, and consecutive interior angles being supplementary. Recognizing these relationships helps determine unknown angle measures.
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Angle Relationships (Corresponding, Alternate Interior, and Supplementary Angles)

Corresponding angles lie on the same side of the transversal and in corresponding positions; they are equal when lines are parallel. Alternate interior angles are on opposite sides of the transversal but inside the parallel lines and are equal. Consecutive interior angles add up to 180 degrees.
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Using Algebra to Solve for Unknown Angles

Often, marked angles are expressed in terms of variables. Setting up equations based on angle relationships allows solving for these variables, which then helps find the exact measure of each angle. This combines geometric reasoning with algebraic manipulation.
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Related Practice
Textbook Question

Find the measure of (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3. 54°

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Textbook Question

Find the measure of each marked angle. In Exercises 19–22, m and n are parallel. See Examples 1 and 2 .

Textbook Question

Sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of θ. Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. See Examples 1, 2, and 4. (―8 , 15)

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Textbook Question

Find the measure of each marked angle.

Textbook Question

Find the measure of (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3. 10°

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Textbook Question

Solve each problem. Length of a Road A camera is located on a satellite with its lens positioned at C in the figure. Length PC represents the distance from the lens to the film PQ, and BA represents a straight road on the ground. Use the measurements given in the figure to find the length of the road. (Data from Kastner, B., Space Mathematics, NASA.)