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Understanding Parallel Lines, Transversals, and Angle Relationships in Geometry

This section explores the fundamentals of parallel lines, transversals, and their associated angles. It begins with a warm-up that includes questions about collinearity, finding segment measures using the segment addition postulate, and identifying acute, obtuse, and right angles. Complementary angles are also discussed along with the definitions of key concepts such as exterior angles, interior angles, and types of angles related to transversals. Examples illustrate the identification of angles and relationships among parallel lines and transversals effectively.

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Understanding Parallel Lines, Transversals, and Angle Relationships in Geometry

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  1. Chapter 3Section 1 Parallel Lines and Transversals

  2. Warm-Up L M N Q P 1) Are points L, M, and Q collinear? 2) Find the measure of MN if LM = 5x – 4, MN = 6x + 1, and LN = 30? 3) Name an acute angle, an obtuse angle, and a right angle in the figure. 4) Are <LMP and <NMQ complementary?

  3. Warm-Up L M N Q P 1) Are points L, M, and Q collinear? No because they are not on the same line.

  4. Warm-Up L M N Q P 2) Find the measure of MN if LM = 5x – 4, MN = 6x + 1, and LN = 30? Use the segment addition postulate. LN = LM + MN 30 = 5x – 4 + 6x + 1 30 = 11x – 3 33 = 11x 3 = x Plug 3 in for x in the equation for MN. MN = 6x + 1 MN = 6(3) + 1 MN = 18 + 1 MN = 19

  5. Warm-Up L M N Q P 3) Name an acute angle, an obtuse angle, and a right angle in the figure. Acute angle - <LPM or <QMN Obtuse angle- <PMN or <LMQ Right angle- <PMQ 4) Are <LMP and <NMQ complementary? Yes because they form a straight line with <PMQ. So <LMP + <NMQ + <PMQ = 180 <LMP + <NMQ + 90= 180 <LMP + <NMQ = 90 (Definition of complementary)

  6. Vocabulary Parallel lines: Two lines that never meet. (Lines l and m are parallel) Skew lines: Two lines are skew if they do not intersect and are not in the same plane. Transversal: A line that intersects two or more lines in a plane at different points. (Line t is the transversal) Exterior Angles- In the figure, transversal t intersects lines l and m. The exterior angles are <3, <4, <5, and <6. Interior Angles- In the figure, transversal t intersects lines l and m. The interior angles are <1, <2, <7, and <8. t 5 6 l 7 8 1 2 m 3 4

  7. Vocabulary Cont. Consecutive Interior Angles- In the figure, transversal t intersects lines l and m. <7 and <1, and <8 and <2 are consecutive interior angles. Alternate Interior Angles- In the figure, transversal t intersects lines l and m. <7 and <2, and <8 and <1 are alternate interior angles. Alternate Exterior Angles- In the figure, transversal t intersects lines l and m. <5 and <4, and <6 and <3 are alternate exterior angles. Corresponding Angles- In the figure, transversal t intersects lines l and m. <5 and <1, <7 and <3, <6 and <2, and <8 and <4 are corresponding angles. t 5 6 l 7 8 1 2 m 3 4

  8. Example 1: Refer to the figure at the right. • a) Name all planes parallel to the plane ADH. • Plane BCG • b) Name all the segments that intersect AT. • Line segments AB, AC, and AD • c) Name all the segments that are parallel to AT. • Line segments DH, BK, and CG • d) Name all segments that are skew to CG. • Line segments TK, TH, AD, and AB B C A D G K T H

  9. Example 2: Refer to the figure at the right. • a) Name all planes that are parallel to plane ABC. • Plane TKG • b) Name all segments that intersect AB. • Line segments BC, AC, AD, AT, and BK • c) Name all the segments that are parallel to KG. • Line segments BC, AD, and TH • d) Name all segments that are skew to TK. • Line segments CG, DH, AD, AC, and BC B C A D G K T H

  10. Example 3: Identify each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles. • a) <1 and <8 • Alternate Exterior Angles • b) <7 and <10 • Alternate Interior Angles • c) <8 and <12 • Corresponding Angles • d) <1 and <5 • Corresponding Angels • e) <4 and <6 • Consecutive Interior Angles • f) <8 and <9 • Alternate Interior Angles l 1 2 3 4 6 5 8 7 10 9 m 11 12 t

  11. Example 4: Identify each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles. • a) <6 and <10 • Consecutive Interior Angles • b) <9 and <11 • Alternate Interior Angles • c) <1 and <5 • Corresponding Angles • d) <3 and <8 • Alternate Exterior Angels • e) <7 and <12 • Alternate Interior Angles • f) <4 and <8 • Corresponding Angles m l 1 8 10 2 5 t 11 4 12 6 7 3 9

  12. Example 5: Refer to the figure showing three lines and the angles formed by these lines. • A) Identify the transversal to lines l and m. • Line t • B)Identify the special name given to each pair of angles • <7 and <12 • Corresponding Angles • <8 and <10 • Alternate Interior Angles • <2 and <12 • Alternate Exterior Angles l m 1 9 6 11 2 8 4 5 t 3 7 10 12

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