21 1 Angles Formed by Parallel Lines Angles

21. 1 Angles Formed by Parallel Lines Angles Formed by Parallel and Transversals Warm Up Lesson Presentation Lesson Quiz Holt Geometry Holt Mc. Dougal Geometry Lines

21. 1 Angles Formed by Parallel Lines and Transversals Warm Up Identify each angle pair. 1. 1 and 3 corr. s 2. 3 and 6 alt. int. s 3. 4 and 5 alt. ext. s 4. 6 and 7 same-side int s Holt Mc. Dougal Geometry

21. 1 Angles Formed by Parallel Lines and Transversals Objective Prove and use theorems about the angles formed by parallel lines and a transversal. Holt Mc. Dougal Geometry

21. 1 Angles Formed by Parallel Lines and Transversals Holt Mc. Dougal Geometry

21. 1 Angles Formed by Parallel Lines and Transversals Example 1: Using the Corresponding Angles Postulate Find each angle measure. A. m ECF x = 70 Corr. s Post. m ECF = 70° B. m DCE 5 x = 4 x + 22 x = 22 m DCE = 5 x = 5(22) = 110° Holt Mc. Dougal Geometry Corr. s Post. Subtract 4 x from both sides. Substitute 22 for x.

21. 1 Angles Formed by Parallel Lines and Transversals Example 2 Find m QRS. x = 118 Corr. s Post. m QRS + x = 180° m QRS = 180° – x *Def. of Linear Pair* Subtract x from both sides. = 180° – 118° Substitute 118° for x. = 62° Holt Mc. Dougal Geometry

21. 1 Angles Formed by Parallel Lines and Transversals Helpful Hint If a transversal is perpendicular to two parallel lines, all eight angles are congruent. Holt Mc. Dougal Geometry

21. 1 Angles Formed by Parallel Lines and Transversals Remember that postulates are statements that are accepted without proof. Since the Corresponding Angles Postulate is given as a postulate, it can be used to prove the next three theorems. Holt Mc. Dougal Geometry

21. 1 Angles Formed by Parallel Lines and Transversals Holt Mc. Dougal Geometry

21. 1 Angles Formed by Parallel Lines and Transversals Example 3 Find each angle measure. A. m EDG = 75° Alt. Ext. s are Congruent. B. m BDG x – 30° = 75° Alt. Ext. s are congruent. x = 105 Add 30 to both sides. m BDG = 105° Holt Mc. Dougal Geometry

21. 1 Angles Formed by Parallel Lines and Transversals Example 4 Find x and y in the diagram. By the Alternate Interior Angles Theorem, (5 x + 4 y)° = 55°. By the Corresponding Angles Postulate, (5 x + 5 y)° = 60°. 5 x + 5 y = 60 –(5 x + 4 y = 55) y=5 Subtract the first equation from the second equation. 5 x + 5(5) = 60 Substitute 5 for y in 5 x + 5 y = 60. Simplify and solve for x. x = 7, y = 5 Holt Mc. Dougal Geometry

21. 1 Angles Formed by Parallel Lines and Transversals Lesson Quiz State theorem or postulate that is related to the measures of the angles in each pair. Then find the unknown angle measures. 1. m 1 = 120°, m 2 = (60 x)° Alt. Ext. s Thm. ; m 2 = 120° 2. m 2 = (75 x – 30)°, m 3 = (30 x + 60)° Corr. s Post. ; m 2 = 120°, m 3 = 120° 3. m 3 = (50 x + 20)°, m 4= (100 x – 80)° Alt. Int. s Thm. ; m 3 = 120°, m 4 =120° 4. m 3 = (45 x + 30)°, m 5 = (25 x + 10)° Same-Side Int. s Thm. ; m 3 = 120°, m 5 =60° Holt Mc. Dougal Geometry

21. 2 Angles Formed by Parallel Lines Proving Lines Parallel and Transversals Warm Up Lesson Presentation Lesson Quiz Holt Geometry Holt Mc. Dougal Geometry

21. 2 Angles Formed by Parallel Lines and Transversals Warm Up State the converse of each statement. 1. If a = b, then a + c = b + c. If a + c = b + c, then a = b. 2. If m A + m B = 90°, then A and B are complementary. If A and B are complementary, then m A + m AB B =90°. 3. If + BC = AC, then A, B, and C are collinear. If A, B, and C are collinear, then AB + BC = AC. Holt Mc. Dougal Geometry

21. 2 Angles Formed by Parallel Lines and Transversals Objective Use the angles formed by a transversal to prove two lines are parallel. Holt Mc. Dougal Geometry

21. 2 Angles Formed by Parallel Lines and Transversals Recall that the converse of a theorem is found by exchanging the hypothesis and conclusion. The converse of a theorem is not automatically true. If it is true, it must be stated as a postulate or proved as a separate theorem. Holt Mc. Dougal Geometry

21. 2 Angles Formed by Parallel Lines and Transversals Holt Mc. Dougal Geometry

21. 2 Angles Formed by Parallel Lines and Transversals The Converse of the Corresponding Angles Postulate is used to construct parallel lines. The Parallel Postulate guarantees that for any line ℓ, you can always construct a parallel line through a point that is not on ℓ. Holt Mc. Dougal Geometry

21. 2 Angles Formed by Parallel Lines and Transversals Holt Mc. Dougal Geometry

21. 2 Angles Formed by Parallel Lines and Transversals Lesson Quiz: Part I Name the postulate or theorem that proves p || r. 1. 4 5 Conv. of Alt. Int. s Thm. 2 7 Conv. of Alt. Ext. s Thm. 3 7 Conv. of Corr. s Post. 4. 3 and 5 are supplementary. Conv. of Same-Side Int. s Thm. Holt Mc. Dougal Geometry

21. 3 Angles Formed by Parallel Lines Perpendicular Lines and Transversals Warm Up Lesson Presentation Lesson Quiz Holt Geometry Holt Mc. Dougal Geometry

21. 3 Angles Formed by Parallel Lines and Transversals Objective Prove and apply theorems about perpendicular lines. Holt Mc. Dougal Geometry

21. 3 Angles Formed by Parallel Lines and Transversals Vocabulary perpendicular bisector distance from a point to a line Holt Mc. Dougal Geometry

21. 3 Angles Formed by Parallel Lines and Transversals The perpendicular bisector of a segment is a line perpendicular to a segment at the segment’s midpoint. The shortest segment from a point to a line is perpendicular to the line. This fact is used to define the distance from a point to a line as the length of the perpendicular segment from the point to the line. Holt Mc. Dougal Geometry

21. 3 Angles Formed by Parallel Lines and Transversals Example 5: Distance From a Point to a Line A. Name the shortest segment from point A to BC. AP B. Write and solve an inequality for x. AC > AP AP is the shortest segment. x – 8 > 12 Substitute x – 8 for AC and 12 for AP. + 8 Add 8 to both sides of the inequality. x > 20 Holt Mc. Dougal Geometry