Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Technical drawing wikipedia , lookup
Multilateration wikipedia , lookup
Riemannian connection on a surface wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Perspective (graphical) wikipedia , lookup
History of geometry wikipedia , lookup
Perceived visual angle wikipedia , lookup
Trigonometric functions wikipedia , lookup
History of trigonometry wikipedia , lookup
Euler angles wikipedia , lookup
Rational trigonometry wikipedia , lookup
Geometry HS Mathematics Unit: 03 Lesson: 01 Investigating Parallel Lines and Angle Pairs KEY Lines m and n are parallel. Eight angles are formed when transversal t intersects lines m and n. The following postulate and theorems represent the relationships between the angles formed when parallel lines are cut by a transversal. t m n 1 2 3 4 5 6 7 8 Corresponding Angle Postulate: If two parallel lines are cut by a transversal, then the corresponding angles are congruent. ∠1 ≅ ∠5 ∠2 ≅ ∠6 ∠3 ≅ ∠7 ∠4 ≅ ∠8 Alternate Interior Angle Theorem: If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. ∠3 ≅ ∠6 ∠4 ≅ ∠5 Same Side Interior Angle Theorem: If two parallel lines are cut by a transversal, then consecutive (same side) interior angles are supplementary. ∠3 is supplementary to ∠5 ∠ 4 is supplementary to ∠6 Same Side Exterior Angle Theorem: If two parallel lines are cut by a transversal, then same side exterior angles are supplementary. ©2012, TESCCC 07/19/12 page 1 of 7 ∠1 is supplementary to ∠7 ∠2 is supplementary to ∠8 Geometry HS Mathematics Unit: 03 Lesson: 01 Alternate Exterior Angle Theorem: If two parallel lines are cut by a transversal, then alternate exterior angles are congruent. ∠1 ≅ ∠8 ∠2 ≅ ∠7 Perpendicular Transversal Theorem: If a line is perpendicular to one of two parallel lines, then it is perpendicular to the second line. ©2012, TESCCC 07/19/12 page 2 of 7 Geometry HS Mathematics Unit: 03 Lesson: 01 Investigating Parallel Lines and Angle Pairs KEY Guided Practice p r s q 1 3 4 5 6 8 7 2 16 15 14 13 12 11 10 9 62o 62o 62o 62o 118o 118o 118o 118o 90o 90o 90o 90o 90o 90o 90o 90o Given r q. ©2012, TESCCC 07/19/12 page 3 of 7 Geometry HS Mathematics Unit: 03 Lesson: 01 1. Name four pair of corresponding angles. Answers will vary. Sample: 1 & 8, 2 & 5, 3 & 6, 12 & 16 2. Name four pair of alternate interior angles. Answers will vary. Sample: 3 & 8, 4 & 5, 11 & 14, 12 & 13 3. Name four pair of same side interior angles. Answers will vary. Sample: 3 & 5, 4 & 8, 11 & 13, 12 & 14 4. Name four pair of alternate exterior angles. Answers will vary. Sample: 2 & 7, 1 & 6, 9 & 16, 10 & 15 ∠1 = 62o 5. Given , find all the measure of all the other angles. See diagram above. 6. What can be said about line p and line q. Justify this with a theorem. Line p is perpendicular to line q. If a line is perpendicular to one of two parallel lines, then it is perpendicular to the second line. ©2012, TESCCC 07/19/12 page 4 of 7 Geometry HS Mathematics Unit: 03 Lesson: 01 Investigating Parallel Lines and Angle Pairs KEY Practice Problems 1. State the postulate or theorem that allows the conclusion that if a b, then ∠1 ≅ ∠2 . 1 2 a b 1 2 a b 1 2 a b a. b. a. Alt. ext. angle thm 2. In the figure below m n, the other angles. c. b. Corr. Angle post. AC BD c. Alt. int. angle thm. , and the measure of ∠1 = 148o . Find the measure of all A B C D 1 7 6 5 4 3 2 8 m n 148o ©2012, TESCCC 07/19/12 page 5 of 7 Geometry HS Mathematics Unit: 03 Lesson: 01 Obj110 148 148o 148o 32o 32o 32o 32o 148o 3. In the figure k l and s t. Find the values of x, y, and z and justify by theorem. k l t s 98o 14zo (2x+5)o (3y+8)o x = 38.5, y = 30, z = 7 x – Same-side int. angle thm. y – Alt. int. angle thm. z – Alt. int. angle thm. ©2012, TESCCC 07/19/12 page 6 of 7 Geometry HS Mathematics Unit: 03 Lesson: 01 Investigating Parallel Lines and Angle Pairs KEY suur AB suur AB suur TP suur AB 4. Draw and Point T not on . Construct parallel to through Point T. Justify that suur suur AB TP and are parallel using the relationships between angle pairs. See the construction. The corresponding angles are constructed to be congruent, so the lines are parallel. 5. Explain how to create two parallel lines with a perpendicular transversal. Construct the situation. Compare and contrast the relationships between the angle pairs created. Answers may vary. Sample: Draw a line with a given point on the line. Construct a perpendicular line through that point by marking equidistant arcs on the line on either side of the point. Using the intersection points of the line and two arcs, mark equidistant, intersecting arcs above the line. Draw a line through the given point on the line and the intersection of the arcs above the line. This should be a line perpendicular to the original line. Place a point on this perpendicular line. Construct congruent angles using this point and the original given point to draw the parallel line to the original line. See the construction. All angles are equal to 90o, therefore angle pairs are either congruent and supplementary. ©2012, TESCCC 07/19/12 page 7 of 7