Warmup pg. 137 #1-8 in the geo book 6 minutes to finish

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1 Chapter Three Test Friday 2/2 Warmup pg. 137 #1-8 in the geo book 6 minutes to finish 1

2 1 and 5, 2 and 5 3 and 4 1 and 2 1 and 5, 2 and 5 division prop of eq Transitive prop of congruency 16 = 4x x = 4 x + 48 = 109 x = 61 2

3 What are we learning today? Midpoint Formula Pythagorean Theorem Distance Formula Constructions Midpoint 3

4 Midpoint Pythagorean Theorem The Rule of Pythagoras (Pythag. Theorem) The square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs of a right triangle. 4

5 Using Pythagoras Theorem You must remember that you can only use the theorem to find a missing side length if you know 2 sides of a right triangle x x x x 13 Using Pythagoras Theorem You must remember that you can only use the theorem to find a missing side length if you know 2 sides of a right triangle 13 8 b b 2 2 b b 105 5

6 Distance Formula Classwork 6

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9 Homework pg. 46 #7,11,13,15 pg. 54 #12,14,18,24,26,39,49 in the geo book Chapter Three Test Friday 2/2 9

10 Warmup pg. 834 #4,5,8,9,10,12,13 Please have your homework out Problems can be done on the board for extra credit 3n n 27 n 9 4 6y 4 2 y or n 9 8n n or w 5 5 2w 5 w 2w 5 w 5 w 5 10

11 5 p p (40) 5 p 56 5k 2k k 21 k x 9 2x 26 x 13 11

12 What are we learning today? Nonintersecting Lines Transversal Angle Pairs with transversals Parallel lines and transversals Using parallel lines 12

Parallel Lines Never intersect and they are coplanar 2.

13 Nonintersecting Lines There are 2 different types of lines that never intersect. Even though they are similar in the fact they never intersect there is one distinct difference between them. 1. Parallel Lines Never intersect and they are coplanar 2. Skew Lines Never intersect and they are noncoplanar examples of skew lines DC and EH AD and BF Parallel planes are two planes which never intersect example of parallel planes ABC and EFG ABF and DCG Nonintersecting Lines HG DC EF FG BC EF AB EFGH 13

Line t is a transversal which intersects line l and line m Angle Pairs with Transversals

14 Transversals A transversal is a line which intersects 2 or more coplanar lines at different points. Line t is a transversal which intersects line l and line m Angle Pairs with Transversals Alternate Interior Angles 4 and 6, 3 and 5 Alternate Exterior Angles 1 and 7, 2 and 8 Corresponding Angles 1 and 5, 2 and 6, 3 and 7, 4 and 8 Same Side Interior Angles 4 and 5, 3 and 6 14

15 Transversals What type of angle pair are angles 2 and 6? corresponding angles What type of angle pair are angles 1 and 10? alternate interior angles What type of angle pair are angles 12 and 10? vertical angles What type of angle pair are angles 4 and 5? same side interior angles Classwork Pg. 144 #21-23 in geo book 15

16 1 and 2 corresponding 3 and 4 alt interior 5 and 6 - corresponding 1 and 2 same side interior 3 and 4 corresponding 5 and 6 - corresponding 1 and 2 corresponding 3 and 4 same side interior 5 and 6 alt interior Parallel lines and transversals Remember, a transversal is a line that intersects 2 or more lines at different points and special angle pairs exist when a transversal is drawn. If a transversal is drawn through 2 or more parallel lines those angles will have specific properties that go with them. 16

17 Parallel lines and transversals 1) If a transversal intersects 2 parallel lines, then the corresponding angles are congruent (Corresponding angles congruency postulate) Parallel lines and transversals 2) If a transversal intersects 2 parallel lines, then the alternate interior angles are congruent (Alternate interior angles congruency theorem)

18 Parallel lines and transversals 3) If a transversal intersects 2 parallel lines, then the alternate exterior angles are congruent (alternate exterior angles congruency theorem) Parallel lines and transversals 4) If a transversal intersects 2 parallel lines, then the consecutive interior angles are supplementary (Consecutive interior angles theorem) = 180 o = 180 o 18

19 Find the measure of all of the numbered angles m 1, m 5, m 7 = 55 o m 2, m 4, m 6, m 8 = = 125 o Find the measure of all of the numbered angles m 3, m 5, m 6, m 7, m 8 = 105 o m 1, m 2, m 4 = = 75 o 19

20 2x + x 12 = 180 3x = 192 x = 64 Solve for x and y 3y + y + 20 = 180 4y = 160 y = 40 Homework #9 pg.59-60#1-3,6-11,20-24 pg #5-12 in the ch. 3 packet 20

21 Chapter Three Test Friday 2/2 Warmup 21

22 B F D F C H 22

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25 What are we learning today? Proving lines are parallel using transversals Theorem 3-7 Theorem 3-8 Theorem 3-9 Parallel lines and transversals Remember, if a transversal intersects 2 parallel lines, then: 1) alt. interior angles are congruent 2) alt. exterior angles are congruent 3) corresponding angles are congruent 4) consecutive int. angles are supplementary 25

26 Proving Lines are Parallel 7 ways to prove that 2 lines are parallel 1) Show that a pair of alt. interior angles are congruent (alt. int. converse) 2) Show that a pair of alt. exterior angles are congruent (alt. ext. converse) 3) Show that a pair of corresponding angles are congruent (corresponding converse) 4) Show that a pair of consecutive interior angles are supplementary (cons. int angles converse) Is there enough information to prove that p is parallel to q Yes. The same side interior angles are supplementary. SS interior converse proves the lines are parallel Yes. The alternate exterior angles are congruent. alt ext converse proves the lines are parallel No. Vertical angles being congruent is not enough information to prove that the lines are parallel. 26

27 Classwork Pg. 160 #7-11 in the geo book 27

28 Theorem

29 Theorem 3-7 Proving Lines are Parallel 7 ways to prove that 2 lines are parallel 1) Show that a pair of alt. interior angles are congruent (alt. int. converse) 2) Show that a pair of alt. exterior angles are congruent (alt. ext. converse) 3) Show that a pair of corresponding angles are congruent (corresponding converse) 4) Show that a pair of consecutive interior angles are supplementary (cons. int angles converse) 5) Show that both lines are parallel to a third line (theorem 3-7) 6) Show that both lines are perpendicular to a third line (theorem 3-8) 29

30 Homework #10 Parallel Lines activity on the class website and pg. 68 #9-14,21-22 pg #1,5,9 in the ch. 3 packet Chapter Three Test Friday 2/2 30

31 Warmup pg. 169 #33-39 in the geo book 5 minutes to finish Please have your homework out right obtuse acute x = 40 x = 60 x = 20 x = 58 31

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34 What are we learning today? Proof progressions Working with proofs Proof Progressions 1) Given a linear pair angles are supplementary m 1 + m 2 = 180 o (linear pair postulate) (def. of supp angles) 34

35 Proof Progressions 2) 1 3 m 1 = m 3 (def. congruency) Proof Progressions 3) perpendicular lines angle is a right angle m 1 = 90 o (def. of perp. lines) 3b) m 1 = 90 o angle is a right angle perpendicular lines (def. of right angles) (def. of right angles) (def. of perp. lines) 35

36 Given def of perp lines def of perp lines all rt. angles are cong corr. angles converse Given corr ang. cong Given Transitive prop alt ext converse 36

37 Given linear pair postulate def. of supp. angles Given def. of congruent angles substitution property distributive property def. of perpendicular lines def. of a right angle div. prop of equality 37

38 Homework pg. 67 #1-7 in the ch. 3 packet and Complete the last two pages of the packet Chapter Three Test Friday 2/2 38

39 Warmup pg. 69 #1-6 in the ch 3 packet 6 minutes to finish A G C 39

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42 Given def of perp lines def of perp lines all rt. angles are cong corr. angles converse Given corr ang. cong Given Transitive prop alt ext converse 42

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44 What are we learning today? Parallel Postulate Interior Angles of a Triangle Triangle Exterior Angle Theorem Parallel Postulate 1 44

45 Triangle Angle-Sum Theorem The sum of the interior angles of a triangle is 180 o 180 (100+30) = 50 o 180 (90+45) = 45 o Triangle Angle-Sum Theorem find the value of x, y, and z x = 180 (43+59) x = 78 o y = y = 102 o z = 180 (102+49) y =29 o 45

46 Exterior angles of a triangle The sum of the exterior angles of a triangle are 360 o Exterior angles of a triangle 46

47 Classwork pg. 209 #25-29 in the geo book 7 minutes to finish x+60 = 120; x = 60 y = ; y = 60 x+60 = 120; x = 60 y = ; y = 60 x+2x+3x = 180; x = 30 x+10 + x-20 + x+25 = 180 x = 55 20x x-2 + 7x+1 = 180 x = 3 47

48 Homework pg #1-10, in the ch. 3 packet Chapter Three Test Friday 2/2 48

49 Warmup pg. 77 #1-4 in the ch 3 packet 6 minutes to finish w+14+30=180 w+44=180 w=136 o x =180 x+126=180 x=54 o y+54=180 y=126 o z =180 z+145=180 z=35 o 49

50 1/7s+3+1/7s-3+s=180 9/7s=180 s=180(7/9)=140 o e=60+69=129 o 50

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the equation of a line Slope (gradient) The slope or gradient of a

53 What are we learning today? Slope (gradient) Slopes of parallel and perpendicular lines Writing the equation of a line Slope (gradient) The slope or gradient of a line is the measure of its steepness. It is also the rate at which the line rises or falls. Slope(gradient) = rise/run 53

54 Slope (gradient) Slopes of parallel and perpendicular lines If 2 lines are parallel them they have equal slopes. In order to verify whether 2 graphed lines are parallel or not you must check the slopes of those lines. If 2 lines are perpendicular then the slopes are negative reciprocals. In order to verify whether 2 graphed lines are perpendicular or not you must check the slopes of those lines. 54

lines are perpendicular Writing the equation of a line There are 3 different forms for

55 Slopes of parallel and perpendicular lines the slope of both lines is 3/1 therefore they are parallel the slope of line 1 is 3/1 the slope of line 2 is -1/3 therefore the lines are perpendicular Writing the equation of a line There are 3 different forms for the equation of a line. Point-Slope Form y-y 1 = m(x x 1 ) Standard Form Ax + By = C 55

56 Writing the equation of a line To write the equation of a line you must: 1) Find the slope of the line you are writing the equation for. 2a) If you are given the y-intercept simply plug in your y- intercept and slope into the slope-intercept form of a line. 2b) If you are given the a point on the line then plug in your slope and the point into the point-slope form for a line. 3) Rewrite the equation into the proper form, usually slopeintercept form. Writing the equation of a line To write the equation of a line you must: 1) Figure out the slope of the line you are writing the equation for. Use the slope formula if given two points and remember parallel lines have equal slopes and perpendicular lines have negative reciprocal slopes. 2a) If you are given the y-intercept simply plug in your y- intercept and slope into the slope-intercept form of a line. 2b) If you are given the a point on the line then plug in your slope and the point into the point-slope form for a line. 3) Rewrite the equation into the proper form, usually slopeintercept form. 56

57 Writing the equation of a line Write the equation of the line with a slope of 2 and a y- intercept of (0, 3), then graph the line y x Writing the equation of a line Write the equation of the line that is parallel to y = 3x 2 and passes through (2, 4) 57

58 Writing the equation of a line Write the equation of the line that is perpendicular to y = 1/4x 8 and passes through (-1, 9) Homework pg. 83 #9-14 pg #2-14 even in the ch. 3 packet 58

59 Chapter Three Test Tomorrow Warmup pg. 210 #36-38, 45, 46 in the geo book 6 minutes to finish Please have your homework out 59

60 m = 2 b = -1 m = -2 (-5, 3) y = ½ x + 12 m = 8 x 1 = -6 y 1 = 2 y 2 = 8(x + 6) y = 8x + 50 m = -6 x 1 = 3 y 1 = -3 y + 3 = -6(x - 3) y = -6x

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2 and 6 4 and 8 1 and 5 3 and 7

2 and 6 4 and 8 1 and 5 3 and 7 Geo Ch 3 Angles formed by Lines Parallel lines are two coplanar lines that do not intersect. Skew lines are that are not coplanar and do not intersect. Transversal is a line that two or more lines at different