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Exploring Angles and Theorems in Parallel Lines and Transversals

This session covers essential concepts of angles, parallel lines, and transversals in geometry. Students will learn to identify types of angles formed when parallel lines are intersected by a transversal, including alternate interior, alternate exterior, and consecutive interior angles. The session also includes exercises to solve for unknowns in angle measures and reinforces key geometric theorems. By the end, students should be able to prove relationships between angles in various configurations, aligning with the CCGPS standards for Analytic Geometry.

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Exploring Angles and Theorems in Parallel Lines and Transversals

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  1. Session 3 Daily Check • Name the type of angles. (1 point each) • a) b) 1 2 1 2 • Solve for x. (4 points each) • a) • b) and are complementary. 4x+22 10x-10

  2. Homework Review

  3. CCGPS Analytic GeometryDay 3 (8-9-13) UNIT QUESTION: How do I prove geometric theorems involving lines, angles, triangles and parallelograms? Standards: MCC9-12.G.SRT.1-5, MCC9-12.A.CO.6-13 Today’s Question: Which angles are congruent to each other when parallel lines are cut by a transversal? Standard: MCC9-12.A.CO.9

  4. Parallel Lines and Transversals

  5. Parallel Lines – Two lines are parallel if and only if they are in the same plane and do not intersect. B A D C AB CD

  6. Parallel Planes – Planes that do not intersect.

  7. Skew Lines – two lines that are NOT in the same plane and do NOT intersect

  8. AB, FG, DG, BC Ex 1: Name all the parts of the prism shown below. Assume segments that look parallel are parallel. 1. A plane parallel to plane AFE. F E Plane BGD G D 2. All segments that intersect GB. A C B GD, BC 3. All segments parallel to FE. 4. All segments skew to ED. BG, FA, BC

  9. Transversal – A line, line segment, or ray that intersects two or more lines at different points. a b Line t is a transversal. t

  10. Special Angles 2 1 4 Interior Angles – lie between the two lines (3, 4, 5, and 6) 3 6 5 8 7 Alternate Interior Angles– are on opposite sides of the transversal. (3 & 6 AND 4 and 5) Consecutive Interior Angles – are on the same side of the transversal. (3 & 5 AND 4 & 6)

  11. More Special Angles Exterior Angles – lie outside the two lines (1, 2, 7, and 8) 2 1 4 3 6 5 8 7 Alternate Exterior Angles – are on opposite sides of the transversal (1& 8 AND 2 & 7)

  12. Ex. 2: Identify each pair of angles as alternate interior, alternate exterior, consecutive interior, or vertical. a. 1 and 2 1 Alt. Ext. Angles 6 7 3 b. 6 and 7 4 8 5 Vertical Angles 2 c. 3 and 4 Alt. Int. Angles d. 3 and 8 Consec. Int. Angles

  13. Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then each pair of alternate interior angles are congruent. 8 1 7 2  6 2 6 3 5 4 3  7

  14. Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then each pair of alternate exterior angles are congruent. 8 1 7 1  5 2 6 3 5 4 4  8

  15. Consecutive Interior Angles Theorem If two parallel lines are cut by a transversal, then each pair of consecutive interior angles are supplementary. m2 + m3 = 180° 8 1 7 2 6 m6 + m7 = 180° 3 5 4

  16. Ex. 3 In the figure, pq. If m5 = 28°, find the measure of each angle. 28° q a. m8 = 2 1 p b. m1 = 28° 4 3 c. m2 = 152° 6 5 8 7 d. m3 = 152° e. m4 = 28°

  17. Ex. 4 In the figure, st. Find the mCBG. t A Step 1: Solve for x. G S B D 3x – 5 = 4x - 29 3x -5 C -5 = x - 29 4x -29 E 24 = x F Step 2: mCBG = mABE = 3x -5. 3x-5 = 3(24) – 5 = 72-5 = 67°

  18. Ex: 5Identify each pair of angles as: alt. interior, alt. exterior, consecutive interior, or vertical. 10 11 9 12 16 alternate exterior 8 1 15 13 7 2 6 14 3 5 4

  19. Ex: 6Identify each pair of angles as: alt. interior, alt. exterior, consecutive interior, or vertical. 10 11 9 12 16 consecutive interior 8 1 15 13 7 2 6 14 3 5 4

  20. Ex: 7Identify each pair of angles as: alt. interior, alt. exterior, consecutive interior, or vertical. 10 11 9 12 16 alternate interior 8 1 15 13 7 2 6 14 3 5 4

  21. Ex: 8Identify each pair of angles as: alt. interior, alt. exterior, consecutive interior, or vertical. 10 11 9 12 16 alternate exterior 8 1 15 13 7 2 6 14 3 5 4

  22. Ex: 9Identify each pair of angles as: alt. interior, alt. exterior, consecutive interior, or vertical. 10 11 9 12 16 consecutive interior 8 1 15 13 7 2 6 14 3 5 4

  23. Ex: 10Identify each pair of angles as: alt. interior, alt. exterior, consecutive interior, or vertical. 10 11 9 12 16 vertical 8 1 15 13 7 2 6 14 3 5 4

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