Unit 3. Parallel and Perpendicular Lines. a. n and m. b. p and q. c. n and p. Sep 14 7:54 AM. Sep 14 7:58 AM. Sep 14 8:07 AM.
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1 3.1 Relationships between lines Unit 3 -- Parallel and Perpendicular Lines Parallel lines lie in the same plane do not intersect Perpendicular lines lie in the same plane intersect to form right angles Sep 14 7:54 AM Sep 14 7:58 AM Skew lines do NOT lie in the same plane never intersect a. n and m b. p and q c. n and p Sep 14 8:04 AM Sep 14 8:07 AM Parallel planes do not intersect Illustration: Line perpendicular to a plane: a line that intersects a plane in a point and that is perpendicular to every line in the plane that intersects it Illustration: Sep 14 8:13 AM Sep 14 8:15 AM 1
2 Very important example!!!!!!!!!!!!!!!!!!! Sep 14 8:18 AM Sep 14 8:18 AM Class Activity Assignment: Sep 14 8:21 AM Sep 14 8:23 AM 3.2 Theorems about Perpendicular Lines Theorem 3.1 All right angles are congruent Illustration: In the diagram, r s and r t. Determine whether enough information is given to conclude that the statement is true. Explain your reasoning. 3 <5 Theorem 3.2 If two lines are perpendicular, then they intersect to form four right angles. Illustration: 4 <5 2 <3 Sep 14 8:23 AM Sep 14 8:26 AM 2
3 In the diagram, g e and g f. Determine whether enough information is given to conclude that the statement is true. Explain. Theorem 3.3 If two lines intersect to form adjacent congruent angles, then the lines are perpendicular. Illustration: Theorem 3.4 If two sides of adjacent acute angles are perpendicular, then the angles are complementary. Illustration: 6 8 Sep 14 8:28 AM Sep 14 8:30 AM In the helicopter at the right, are AXB and CXB right angles? Explain. In the diagram at the right, EF EH and m GEH = 30. Find the value of y. Sep 14 8:32 AM Sep 14 8:33 AM Find the value of the variable. Explain your reasoning. Assignment: EFG HFG AB AD KJ KL, JKM MKL Sep 14 8:34 AM Sep 14 8:39 AM 3
4 Angles formed by Transversals transversal: a line that intersects two or more coplanar lines at different points corresponding angles angles that occupy "corresponding" positions t m n Sep 24 7:43 AM Sep 24 8:15 AM alternate interior angles lie between the two lines on opposite sides of the transversal alternate exterior angles lie outside the two lines on the opposite sides of the transversal Sep 24 8:17 AM Sep 24 8:19 AM same side interior angles... consecutive interior angles lie between the two lines on the same side of the transversal Sep 24 8:21 AM Sep 24 8:22 AM 4
5 Sep 24 8:23 AM Sep 24 8:23 AM 3.4 Parallel Lines and Transversals Assignment: Corresponding Angles Postulate If two parallel lines are cut by a transversal, then corresponding angles are congruent. Sep 24 8:24 AM Sep 24 8:24 AM alternate interior angles theorem If two parallel lines are cut by a transversal, then alternate interior angles are congruent. Sep 24 8:40 AM Sep 24 8:41 AM 5
6 alternate exterior angles theorem if two parallel lines are cut by a transversal, then alternate exterior angles are congruent. Sep 24 8:42 AM Sep 24 8:42 AM Sep 24 8:43 AM Sep 24 8:43 AM Same side interior angles theorem (AKA consecutive interior angles theorem) If two parallel lines are cut by a transversal, then same side interior angles are supplementary. Sep 24 2:34 PM Sep 24 2:36 PM 6
7 Sep 24 2:36 PM Sep 24 2:36 PM Bellwork State whether the angles are corresponding angles, alternate interior angles, alternate exterior angles, same side interior angles, or none of these. Also state whether the angles are congruent, supplementary, or neither. Assignment: Sep 24 2:37 PM Sep 24 2:40 PM Showing Lines are Parallel converse -the statement formed by switching the hypothesis and the conclusion of an if-then statement. example: Statement: If today is Monday, then yesterday was Sunday. Converse: If yesterday was Sunday, then today is Monday. Sep 24 2:41 PM Sep 24 3:23 PM 7
8 Corresponding Angles Converse What did the corresponding angles postulate say? If two parallel lines are cut by a transversal, then corresponding angles are congruent. rlls, so m<1=m<5 So what would the Corresponding Angles Converse Postulate say? If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. m<1=m<5 so rlls Sep 24 3:24 PM Sep 24 3:25 PM Sep 24 3:29 PM Sep 24 3:29 PM Alternate Interior Angles Converse Theorem -If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. Alternate Exterior Angles Converse Theorem -If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel. Sep 24 3:30 PM Sep 24 3:31 PM 8
9 Same-side Interior Angles Converse Theorem -If two lines are cut by a transversal so that same-side interior angles are supplementary, then the lines are parallel. Sep 24 3:32 PM Sep 24 3:32 PM Ticket-out-the-door Sep 24 3:33 PM Sep 24 3:35 PM Translations Assignment: pg 140 #6 28 even,37 43 all transformation an operation that maps, or moves, a figure onto an image translation -one type of transformation -sliding a figure pre-image the figure before the translation image the new figure after the translation Sep 24 3:36 PM Sep 24 3:36 PM 9
10 You try: Sep 28 8:24 AM Sep 28 8:28 AM Labeling Translations Describing Translations Sep 28 8:28 AM Sep 28 8:29 AM Coordinate Notation You try: Example: Sep 28 8:30 AM Sep 28 8:31 AM 10
11 Draw the triangle given by points A(1, -1), B(4, -2), and C(1, -4). Then draw the image of the triangle after the translation given by (x, y) --> (x - 4, y + 2). Sep 28 8:32 AM Sep 28 8:32 AM You try: Class work Help Sep 28 8:33 AM Sep 28 8:34 AM Assignment: pages #1-35, Sep 28 8:36 AM Sep 28 8:39 AM 11
12 y x Oct 3 11:05 AM Oct 3 11:07 AM Mar 7 1:42 PM Mar 8 12:40 PM 12
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