5-2 Parallel and Perpendicular Lines Warm Up Problem of the Day Lesson Presentation Pre-Algebra

5-2 Parallel and Perpendicular Lines Warm Up Complete each sentence. Pre-Algebra 5-2 Parallel and Perpendicular Lines Warm Up Complete each sentence. 1. Angles whose measures have a sum of 90° are _______________ . 2. Vertical angles have equal measures, so they are ______________. 3. Angles whose measures have a sum of 180° are ______________. 4. A part of a line between two points is called a ____________. complementary congruent supplementary segment

Problem of the Day The square root of 1,813,141,561 is a whole number. Is it odd or even? How do you know? An odd number can only be the product of two odd numbers. Since 1,813,141,561 is an odd number, its square root is also an odd number.

Learn to identify parallel and perpendicular lines and the angles formed by a transversal.

Vocabulary parallel lines perpendicular lines transversal

Parallel lines are two lines in a plane that never meet, like a set of perfectly straight, infinite train tracks. Perpendicular lines are lines that intersect to form 90° angles.

The railroad ties are transversals to the tracks. The tracks are parallel. A transversal is a line that intersects any two or more other lines. Transversals to parallel lines have interesting properties.

Additional Example 1: Identifying Congruent Angles Formed by a Transversal Measure the angles formed by the transversal and parallel lines. Which angles seem to be congruent? 1,  3,  5, and  7 all measure 150°.  2,  4,  6, and  8 all measure 30°.

Additional Example 1 Continued Angles marked in blue appear to be congruent to each other, and angles marked in red appear to be congruent to each other. 1 1 @  3 @  5 @  7 2 3 4 2 @  4 @  6 @  8 6 5 7 8

Try This 1: Example 1 Measure the angles formed by the transversal and parallel lines. Which angles seem to be congruent? 1 2 3 4 5 6 7 8 1,  4,  5, and  8 all measure 36°.  2,  3,  6, and  7 all measure 144°.

Try This 1: Example 1 Continued Angles marked in blue appear to be congruent to each other, and angles marked in red appear to be congruent to each other. 1 @  4 @  5 @  8 2 @  3 @  6 @  7 1 2 3 4 5 6 7 8

PROPERTIES OF TRANSVERSALS TO PARALLEL LINES If two parallel lines are intersected by a transversal, the acute angles that are formed are all congruent, the obtuse angles are all congruent, and any acute angle is supplementary to any obtuse angle. If the transversal is perpendicular to the parallel lines, all of the angles formed are congruent 90° angles.

The symbol for parallel is ||. The symbol for perpendicular is . Writing Math

Additional Example 2A: Finding Angle Measures of Parallel Lines Cut by Transversals In the figure, line l || line m. Find the measure of the angle. A. 4 All obtuse angles in the figure are congruent. m 4 = 124°

Additional Example 2B: Finding Angle Measures of Parallel Lines Cut by Transversals Continued In the figure, line l || line m. Find the measure of the angle. B.  2 2 is supplementary to the angle 124°. m2 + 124° = 180° –124° –124° m 2 = 56°

Additional Example 2C: Finding Angle Measures of Parallel Lines Cut by Transversals Continued In the figure, line l || line m. Find the measure of the angle. C.  6 All acute angles in the figure are congruent. m 6 = 56°

m 7 = 144° Try This: Example 2A In the figure, line n || line m. Find the measure of the angle. A.  7 All obtuse angles in the figure are congruent m 7 = 144° 1 144° 3 4 5 6 7 8 m n

Try This: Example 2B In the figure, line n || line m. Find the measure of the angle. B.  5 5 is supplementary to the angle 144°. 1 144° 3 4 5 6 7 8 m n m 2 + 144° = 180° –144° –144° m 2 = 36°

m 6 = 36° Try This: Example 2C In the figure, line n || line m. Find the measure of the angle. C.  6 All acute angles in the figure are congruent m 6 = 36° 1 144° 3 4 5 6 7 8 m n

If two lines are intersected by a transversal and any of the angle pairs shown below are congruent, then the lines are parallel. This fact is used in the construction of parallel lines.

Lesson Quiz In the figure a || b. 1. Name the angles congruent to 3. 1, 5, 7 2. Name all the angles supplementary to 6. 1, 3, 5, 7 3. If m1 = 105° what is m3? 105° 4. If m5 = 120° what is m2? 60°