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Lesson 3.1 Lines and Angles You will learn to … * identify relationships between lines * identify angles formed by transversals If two lines are coplanar and do not intersect, then they are _______________ parallel lines Parallel Lines B AB || CD A D C r || p r p If two lines are NONcoplanar and do not intersect, then they skew lines are ____________ If two lines intersect to form one right angle, then they are perpendicular lines ___________________. AB | CD r | p Determine whether the lines are parallel, perpendicular, or neither. 1. t and u neither 2. t and s parallel 3. r and u perpendicular s t r u Determine whether the lines are intersecting or skew. 4. t and u skew t u Postulate 13 Parallel Postulate Given a line and a point not on the line, exactly one line then there is _________________ through the point parallel to the line. Postulate 14 Perpendicular Postulate Given one line and one point, exactly one line through there is ________________ the point perpendicular to the line. transversal – a line that intersects two or more coplanar lines at different points no transversal a b c Identify the transversal: A) B) C) D) E) Line a Line b Line c Lines a and b All 3 lines corresponding angles 1 2 4 3 5 6 7 8 alternate interior angles 1 2 4 3 5 6 7 8 alternate exterior angles 1 2 4 3 5 6 7 8 consecutive interior angles 1 2 4 3 5 6 7 8 same-side interior angles same-side exterior angles 1 2 4 3 5 6 7 8 Describe the relationship between the given angles. 5. 1 and 2 6. 3 and 4 7. 5 and 6 corresponding 1 alternate exterior consecutive interior 5 4 3 2 6 Lesson 3.2 Proof & Perpendicular Lines You will learn to … * write different types of proofs * prove results about perpendicular lines 1. 2. Which angles are congruent? adjacent? 1 2 3 4 1 1 4 4 3 3 22 1 and 4 2 and 4 1 and 3 2 and 3 3. How would the diagram change if adjacent angles were congruent? Theorem 3.1 Perpendicular Lines Theorem If 2 lines intersect to form a linear pair of congruent angles, then the lines are perpendicular _______________. 4. Is RP ST ? How do you know? R Yes, RP ST. 2 1 T P m1 = m2 and m1 + m2 = 180 m1 = 90 m2 = 90 Perpendicular Lines Theorem S Draw two adjacent acute angles such that their uncommon sides are perpendicular. What do you know about the two acute angles? Theorem 3.2 Adjacent Complements Theorem If 2 sides of two adjacent acute angles are perpendicular, then the angles are complementary ________________. 5. AC BD Find x. B (6x + 4) + 20 = 90 x = 11 C A E 20 P D Theorem 3.3 4 Right Angles Theorem If 2 intersecting lines are perpendicular then they _____________, form 4 right angles. Determine whether enough information is given to conclude that the statement is true. d 6. 1 2 yes 7. 2 3 8. 3 4 4 yes no 3 b Paragraph Proofs 9. Given: AB = BC Prove: ½ AC = BC C A B AC = AB + BC by the Segment Addition Postulate. Since AB=BC, AC = BC + BC by substitution. By the Distributive Prop, AC = 2BC. ½ AC = BC by the Division Prop. Look at your Ch 2 Celebration. Paragraph Proofs 10. Given: 1 and 3 are a linear pair 2 and 3 are a 3 linear pair 1 2 Prove: m1 = m2 Since 1 & 3 and 2 & 3 are linear pairs, 1 & 3 are supplementary and 2 & 3 are supplementary by the Linear Pair Postulate. So, 1 2 by the Congruent Supplements Theorem. Look at your By definition of ,Chm2Celebration. 1 = m 2 Flow Proofs 11. Given: 1 and 2 are a linear pair 2 and 3 are a linear pair Prove: m1 = m3 1 2 3 Flow Proofs 12. Given: 5 6 5 and 6 are a linear pair Prove: j k j 5 6 k Practice! The best way for you to get better at writing proofs is to practice. Don’t give up! 1. Write a two-column proof of Theorem 3.1 Perpendicular Lines Theorem Given: 1 2, 1 and 2 are a linear pair g Prove: g h 12 h 2. Write a two-column proof of Theorem 3.2 Adjacent Complements Theorem Given: BA BC Prove: 1 and 2 are complementary A 1 B 2 C 3. Write a two-column proof of Theorem 3.3 4 Right Angles Theorem Given: j k , Prove: 2 is a right angle j 1 2 k 4. Given: j k , 3 and 4 are complementary Prove: 5 6 j 4 5 3 6 k Lesson 3.3 Parallel Lines and Transversals Students need a protractor and straight edge You will learn to … * prove and use results about parallel lines and transversals * use properties of parallel lines Postulate 15 & Theorems 3.4 – 3.6 If 2 parallel lines are cut by a transversal, then… Use a straight edge to create 2 parallel lines cut by a transversal. …corresponding angles are congruent ______________. Corresponding Angles Postulate corresponding angles 1 2 4 3 1 5 2 6 5 6 7 8 3 7 4 8 …alternate interior angles congruent are ______________. Alternate Interior Angles Theorem alternate interior angles 4 3 5 6 3 6 4 5 … alternate exterior angles are congruent _______________. Alternate Exterior Angles Theorem alternate exterior angles 1 2 1 8 2 7 7 8 …consecutive interior angles supplementary are _______________. Consecutive Interior Angles Theorem consecutive interior angles 3 4 5 6 m3 + m5 = 180 m4 + m6 = 180 …same-side exterior angles supplementary are _______________. Same-Side Exterior Angles Theorem same-side exterior angles 1 2 7 8 m1 + m7 = 180 m2 + m8 = 180 Find the measure of the numbered angle. 1. m1 = 110 110 1 Corresponding s Postulate Find the measure of the numbered angle. 2. m2 = 100 100 2 Alt. Ext. s Theorem Find the measure of the numbered angle. 3. m3 = 112 112 3 Alt. Int. s Theorem Find the measure of the numbered angle. 4. m4 = 60 120 4 Cons. Int. s Theorem Find the measure of the numbered angle. 5. m5 = 70 110 5 Same-side Ext. s Theorem 6. Find x. 125 (12x – 5) 12x – 5 + 125 = 180 12x + 120 = 180 12x = 60 x=5 7. Find x. 100 (5x + 40) 5x + 40 = 100 5x = 60 x = 12 Theorem 3.7 Transversal Theorem If a transversal is perpendicular to one of two parallel lines, then … ________________________. it is perpendicular to the other m t n If t n and n || m, then t m A# 3.3 #28 Statements 1) j || k 2) 1 3 3) 2 3 4) 1 2 #29 Statements 1) p q ; q || r 2) 1 is a right angle 3) 1 2 4) 2 is a right angle 5) p r Lesson 3.4 & 3.5 Parallel Lines You will learn to … * prove that two lines are parallel * use properties of parallel lines What is the converse of a conditional if-then statement? Write the converse of the statement. 1. If two parallel lines are cut by a transversal, then corresponding angles are congruent. If corresponding angles are congruent, then the two lines cut by the transversal are parallel. Postulate 16 Corresponding Angles Converse If corresponding angles are___, then… 1 2 3 4 5 6 7 8 Theorem 3.8 – Alternate Interior s Converse If alternate interior angles then… are___, 3 4 5 6 Theorem 3.9 Consecutive Interior s Converse If same-side interior angles are supplementary , then… 3 4 5 6 Theorem 3.10 – Alternate Exterior Angles Converse If alternate exterior angles then… are___, 1 2 7 8 Same-side Exterior s Converse If same-side exterior angles are supplementary , then… 1 2 7 8 …the 2 lines cut by the transversal are parallel Postulate and Theorems IF || lines THEN angles Converse IF angles THEN || lines 2. Can you prove that n || m ? Explain. 112 n 112 m Yes, Corresponding s Converse 3. Can you prove that n || m ? Explain. 78 n 78 m Yes, Alternate Ext. s Converse 4. Can you prove that n || m ? Explain. 72 108 n m Yes, Consecutive Interior s Converse 5. Can you prove that n || m ? Explain. n m 102 102 Yes, Alternate Interior s Converse 6. Can you prove that n || m ? Explain. 123 n 47 m NO 123 + 47 180 7. Can you prove that n || m ? Explain. 100 100 n m NO p || n If p || k and n || k, then…? p k n Theorem 3.11 3 Parallel Lines Theorem If 2 lines are parallel to the same line, then they are parallel to each other. ____________ r || t. If r || s and s || t, then ____ p || n If p k and n k, then…? n Theorem 3.12 || Theorem If 2 lines are perpendicular to the same line, then they are parallel to each other. _________ r || t If r s and t s, then ____ A# 3.4 #30 Statements 1) 4 5 2) 4 & 6 are vertical angles 3) 4 6 4) 5 6 5) g || h A# 3.4 #32 Statements 1) B BEA 2) BEA CED 3) CED C 4) B C 5) AB || CD A# 3.4 #34 Statements 1) m 7 = 125°; m 8 = 55° 2) m 7 + m 8 = 125° + m 8 3) m 7 + m 8 = 125° + 55° 4) m 7 + m 8 = 180° 5) 7 & 8 are supplementary 6) j || k Do Practice Proofs… Students need scissors, glue, and 2 sheets of paper. Lesson 3.6 & 3.7 Parallel and Perpendicular Lines You will learn to … * find slopes of lines and use slope to identify parallel lines and perpendicular lines * write equation of parallel lines * write equations of perpendicular lines The slope of a line is the ratio of the vertical change (rise) to the horizontal change (run). rise y2 – y1 Slope = = x –x run 2 1 Find the slope of the line that passes through the given points. 1. (-5,7) (-2,4) m = -1 2. (3, -2) (-5, -2) m=0 3. (-6, 2) (-6, -2) m = undefined Postulate 17 Slopes of Parallel Lines Lines are parallel if and only if they have the same slope. All vertical lines are parallel. All horizontal lines are parallel. h horizontal lines vertical v lines v x=# y=# Slope is 0 Slope is undefined slope-intercept form y = mx + b slope y–intercept (0,b) Identify the slope of the line. 4. y = -5x + 14 m=-5 5. 2x – 4y = -3 – 4y = – 2x – 3 -4 -4 -4 y=½x+¾ m=½ Write the equation of the line with the given slope and y-intercept. 6. slope = - 2 y-int = 3 y = -2x + 3 7. slope = 0 y-int = -5 y=-5 Write the equation of the line that has a y-intercept of -7 and is parallel to the given line. 8. y = - ½ x + 10 m=- y=-½x–7 ½ point-slope form y – y1 = m (x – x1) 9. Use the point-slope form to write the equation of the line through the point (2, 3) that has a slope of 5. y – y1 = m (x – x1) y - 3 = 5 (x - 2) y - 3 = 5x - 10 y = 5x - 7 10. Write the equation of the line through the point (-2, -4) that is parallel to y = - ½ x + 5. y – y1 = m (x – x1) y - 4 = - ½ (x - 2) y + 4 = - ½ (x + 2) y+4=-½x-1 y=-½x-5 Postulate 18 Slopes of Perpendicular Lines Lines are perpendicular if and only if the product of their slopes is -1. Opposite & Reciprocals Identify the slope of the line that is perpendicular to the given line. 1 11. y = -5x + 14 m = 5 12. y = ½ x - 7 13. y = - 8 m=-2 m = undefined Find the equation of the line that is perpendicular to the given line and passes through the given point. 14. y = 3x – 2 (9,4) m 1 y – 4 = - /3 (x – 9) y–4=- y=- 1/ x 3 1/ +3 x + 7 3 1 = -? /3 Find the equation of the line that is perpendicular to the given line and passes through the given point. 15. y = m = -?7 – 11 (5, -10) y + 10 = - 7 (x – 5) 1/ x 7 y + 10 = - 7x + 35 y = - 7x + 25