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Analyzing Sequences, Angles, and Relationships in Geometry

This comprehensive lesson covers several mathematical concepts including identifying patterns in sequences, conjecturing future elements, and understanding angle relationships created by parallel lines and transversals. Students will explore supplementary and complementary angles, conjectures in sequences, and verification through contrapositive statements. Additionally, the lesson will guide students in naming segments that are parallel or skew and classifying relationships between angles within transversal setups, providing a practical application of geometric principles.

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Analyzing Sequences, Angles, and Relationships in Geometry

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  1. Warm ups Make a conjecture about the next number in the sequence, 5, 20, 80, 320. Write the contrapositive of this statement. If you live in Boston, then you live in Massachusetts. 3. Find m<1 and m<2 if m<1 = 8x + 18 and m<2 = 16x – 6 and m<1 and m<2 are supplementary. 4. The measures of two complementary angles are x + 54 and 2x. What is the measure of the smaller angle?

  2. 3-1 Parallel Lines & Transversals Objective: Identify relationships between two lines or two planes. Name angle pairs formed by parallel lines and transversals.

  3. Vocabulary • parallel lines • skew lines • parallel planes • transversal • interior angles • exterior angles • consecutive interior angles • alternate interior angles • alternate exterior angles • corresponding angles

  4. Concept

  5. A. Name all segments parallel to BC. Answer:AD, EH, FG Example 1 Identify Parallel and Skew Relationships

  6. B. Name a segment skew to EH. Answer:AB, CD, BG, or CF Example 1 Identify Parallel and Skew Relationships

  7. Example 1 C. Name a plane parallel to plane ABG. Answer: plane CDE Identify Parallel and Skew Relationships

  8. Example 1a A. Name a plane that is parallel to plane RST. A. plane WTZ B. plane SYZ C. plane WXY D. plane QRX

  9. B. Name a segment that intersects YZ. A.XY B.WX C.QW D.RS Example 1b

  10. C. Name a segment that is parallel to RX. A.ZW B.TZ C.QR D.ST Example 1c

  11. Transversal Angle Pair Relationships

  12. Example 2 A. Classify the relationship between <2 and <6 as alternate interior, alternate exterior, corresponding, or consecutive interior angles. Answer: corresponding Classify Angle Pair Relationships

  13. Example 2 B. Classify the relationship between <1 and <7 as alternate interior, alternate exterior, corresponding, or consecutive interior angles. Answer: alternate exterior Classify Angle Pair Relationships

  14. Example 2 C. Classify the relationship between <3 and <8 as alternate interior, alternate exterior, corresponding, or consecutive interior angles. Answer: consecutive interior Classify Angle Pair Relationships

  15. Example 2 D. Classify the relationship between <3 and <5 as alternate interior, alternate exterior, corresponding, or consecutive interior angles. Answer: alternate interior Classify Angle Pair Relationships

  16. Example 2a A. Classify the relationship between <4 and <5. A. alternate interior B. alternate exterior C. corresponding D. consecutive interior

  17. Example 2b B. Classify the relationship between <7 and <9. A. alternate interior B. alternate exterior C. corresponding D. consecutive interior

  18. Example 2c C. Classify the relationship between <4and <7. A. alternate interior B. alternate exterior C. corresponding D. consecutive interior

  19. Example 2d D. Classify the relationship between <2and <11. A. alternate interior B. alternate exterior C. corresponding D. consecutive interior

  20. Example 3 A. BUS STATION The driveways at a bus station are shown. Identify the transversal connecting <1and <2. Then classify the relationship between the pair of angles. Answer: The transversal connecting <1 and <2 is line v. These are corresponding angles. Identify Transversals and Classify Angle Pairs

  21. Example 3 B. BUS STATION The driveways at a bus station are shown. Identify the transversal connecting <2and <3. Then classify the relationship between the pair of angles. Answer: The transversal connecting <2 and <3 is line v. These are alternate interior angles. Identify Transversals and Classify Angle Pairs

  22. Example 3 C. BUS STATION The driveways at a bus station are shown. Identify the transversal connecting <4and <5. Then classify the relationship between the pair of angles. Answer: The transversal connecting <4 and <5 is line y. These are consecutive interior angles. Identify Transversals and Classify Angle Pairs

  23. Example 3a A. HIKING A group of nature trails is shown. Identify the sets of lines to which line a is a transversal. A.linesc, f B.linesc, d, e C.linesc, d, f D.linesc, d, e, f

  24. Example 3b B. HIKING A group of nature trails is shown. Identify the sets of lines to which line b is a transversal. A. no lines B. lines c, f C. lines c, d, e, f D. lines a, c, d, e, f

  25. Example 3c C. HIKING A group of nature trails is shown. Identify the sets of lines to which line c is a transversal. A. no lines B. lines a, b, d, e, f C. lines a, d, f D. lines a, b, e

  26. Homework • Pg. 176 # 13 – 44, 50

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