Understanding Relationships Between Lines: Parallel, Perpendicular, and Skew
This chapter explores the various relationships between lines, including parallel, perpendicular, and skew lines. It includes definitions and theorems essential for understanding these concepts. Learn how to identify and classify lines based on their relationships within a plane, and examine the properties of right angles. The chapter features practical examples and exercises, aiding in the application of theorems pertaining to perpendicular lines. Whether you are solving for angles or determining line relationships, this guide enhances your understanding of geometry.
Understanding Relationships Between Lines: Parallel, Perpendicular, and Skew
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Presentation Transcript
CHAPTER 3 CHAPTER REVIEW
Relationships Between Lines: GOAL: Identify relationships between lines Two lines are parallel lines if they lie in the same plane and do not intersect. Two lines are perpendicular lines if they intersect to form a right angle. p m n q
Two lines are skew lines if they do not lie in the same plane. Skew Lines never intersect. c b ● A
All segments in the diagram are part of a line and all corners of the cube form right angles. R V Name a line that is skew to VW. Name a plane that appears parallel to plane VWX. Name a line that is perpendicular to plane VWX. Q U W S T X
THEOREMS ABOUT PERPENDICULAR LINES: GOAL: Use theorems about perpendicular lines Theorem 3.1 Words: All right angles are congruent. A Symbols: If the m A = 90° and m B = 90°, then A B B
Theorem 3.2 Words: If two lines are perpendicular, then they intersect to form four right angles. Symbols: If n m, then m 1 = 90°, m 2 = 90°, m 3 = 90°, and m 4 = 90°. 1 4 2 3
Theorem 3.3 Words: If two lines intersect to form adjacent congruent angles, then the lines are perpendicular. ● B A 1 2 ● D ● C Symbols: If 1 2, then AC BD
Theorem 3.4 Words: If two sides of adjacent acute angles are perpendicular, then the angles are complementary. F ● ● G 3 4 ● H ● E Symbols: If EF EH, then m 3 + m 4 = 90°
In the diagram at the right, EF EH and m GEH = 30 °. Find the value of y. F ● 2y – 12 E ● 30° ● G ● H
Name a pair of alternate interior angles Name a pair of corresponding angles. Name a pair of same-side interior angles. Name a pair of alternate exterior angles. 1 3 2 4 5 7 6 8
If j ǁ k then find: m<1 m<2 m<3 m<4 m<5 m<6 m<7 j 72° 1 2 3 k 4 5 6 7
120° 2x + 32 Solve for x.
45° 3x – 15 Solve for x.
74° 5x + 14 Solve for x.
103° 6y – 23 Solve for y.
Showing Lines are Parallel Goal: Show that two lines are parallel. The converse of an if-then statement is the statement formed by switching the hypothesis and the conclusion. Write the converse for the given if-then statement: 1. If two angles have the same measure, then the two angles are congruent.
Examples: Determine the postulate that proves the lines are parallel. 63° 2. 1. 55° 125° 63°
138° 3. 4. 56° 138° 56° 145° 5. 145°
Theorem 3.11 Words: If two lines are parallel to the same line, then they are parallel to each other. Symbols: If q ǁ r and r ǁ s, then q ǁ s. q r s
Theorem 3.12 Words: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. Symbols: If m p and n p then m ǁ n. n m p
Determine whether if the given picture is a translation. 2. 1. 3. 4.
PRACTICE PAGE 160-163 1-32
