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 are ____________ skew lines

If two lines intersect to form one right angle, then they are ___________________. perpendicular lines AB | CD r | p

1. t and u neither 2. t and s parallel 3. r and u perpendicular Determine whether the lines are parallel, perpendicular, or neither. s 1. t and u t neither 2. t and s parallel r 3. r and u perpendicular u

4. t and u skew Determine whether the lines are intersecting or skew.

Postulate 13 Parallel Postulate Given a line and a point not on the line, then there is _________________ through the point parallel to the line. exactly one line

Postulate 14 Perpendicular Postulate Given one line and one point, there is ________________ through the point perpendicular to the line. exactly one line

a line that intersects two or more coplanar lines at different points transversal – a line that intersects two or more coplanar lines at different points no transversal

b a c Identify transversal(s): Line a Line b Line c Lines a and b E) All 3 lines c

Special angles are formed when a transversal intersects 2 lines. corresponding angles Special angles are formed when a transversal intersects 2 lines. 1 1 2 2 4 4 3 3 5 5 6 6 7 7 8 8

alternate interior angles 1 2 3 4 5 6 4 3 5 6 7 8

alternate exterior angles 2 7 2 1 1 8 4 3 5 6 7 8

consecutive interior angles 1 2 3 4 5 6 4 3 5 6 7 8 same-side interior angles

same-side exterior angles 2 2 1 1 3 4 5 6 7 7 8 8

Describe the relationship between the given angles. 1 2 5. 1 and 2 4 3 6. 3 and 4 5 6 7. 5 and 6 corresponding alternate exterior consecutive interior

Lesson 3.2 Proof & Perpendicular Lines You will learn to … * write different types of proofs * prove results about perpendicular lines

2. Which angles are adjacent? 1. Which angles are congruent? 1 and 4 2 and 4 1 and 3 2 and 3 1  2 3  4 2 1 3 4 2 1 3 4 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

Perpendicular Lines Theorem 4. Is RP  ST ? How do you know? R Yes, RP  ST. 1 2 S P T m1 = m2 and m1 + m2 = 180 m1 = 90  m2 = 90  Perpendicular Lines Theorem

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 20 P E (6x + 4) D

Theorem 3.3 4 Right Angles Theorem If 2 intersecting lines are _____________, then they form 4 right angles. perpendicular

Determine whether enough information is given to conclude that the statement is true. 6. 1  2 7. 2  3 8. 3  4 1 c a 2 d yes 4 3 yes b no

Look at your Ch 2 Celebration. Paragraph Proofs 9. Given: AB = BC Prove: ½ AC = BC A B C 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.

Look at your Ch 2 Celebration. Paragraph Proofs 10. Given: 1 and 3 are a linear pair 2 and 3 are a linear pair Prove: m1 = m2 1 2 3 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. By definition of , m 1 = m 2 Look at your Ch 2 Celebration.

Flow Proofs 11. Given: 1 and 2 are a linear pair Prove: m1 = m3 1 3 2

Flow Proofs 12. Given: 5  6 5 and 6 are a linear pair Prove: j  k j 5 6 k

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 Prove: g  h g h 1 2

2. Write a two-column proof of Theorem 3 2. Write a two-column proof of Theorem 3.2 Adjacent Complements Theorem Given: BA  BC Prove: 1 and 2 are complementary A B 1 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

j 4 5 3 6 k 4. Given: j  k , 3 and 4 are complementary Prove:  5   6 j 3 6 k 5 4

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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

If 2 parallel lines are cut by a transversal, then… 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 3 4 5 6 7 8 1 5 2 6 4 8 3 7 1  5 3  7 2  6 4  8

…alternate interior angles are ______________. congruent Alternate Interior Angles Theorem

alternate interior angles 3 4 5 6 4 5 3 6 3  6 4  5

… alternate exterior angles are _______________. congruent Alternate Exterior Angles Theorem

alternate exterior angles 2 7 2 7 1 8 1 8 1  8 2  7

…consecutive interior angles are _______________. supplementary Consecutive Interior Angles Theorem

consecutive interior angles 3 4 5 6 m3 + m5 = 180 m4 + m6 = 180

…same-side exterior angles are _______________. supplementary Same-Side Exterior Angles Theorem

same-side exterior angles 1 2 7 8 m1 + m7 = 180 m2 + m8 = 180

Find the measure of the numbered angle. 110 1. m1 = 110 1 Corresponding s Postulate

Find the measure of the numbered angle. 100 2. m2 = 100 2 Alt. Ext. s Theorem

Find the measure of the numbered angle. 112 3. m3 = 112 3 Alt. Int. s Theorem

Find the measure of the numbered angle. 60 4. m4 = 120 4 Cons. Int. s Theorem

Find the measure of the numbered angle. 70 5. m5 = 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

it is perpendicular to the other 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 5 2 6 3 7 4 8

Theorem 3.8 – Alternate Interior s Converse If alternate interior angles are___, then…  5 6 3 4

Theorem 3.9 Consecutive Interior s Converse If same-side interior angles are supplementary , then… 5 6 3 4

Theorem 3.10 – Alternate Exterior Angles Converse If alternate exterior angles are___, then…  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

|| lines || lines IF THEN angles IF THEN angles Postulate and Theorems Converse || lines IF THEN angles

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 n 108 m Yes, Consecutive Interior s Converse

5. Can you prove that n || m ? Explain. 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

If p || k and n || k, then…? p || n p k n

If r || s and s || t, then ____ 3 Parallel Lines Theorem Theorem 3.11 If 2 lines are parallel to the same line, then they are ____________ to each other. parallel r || t. If r || s and s || t, then ____

If p  k and n  k, then…? p || n p n k

  || Theorem Theorem 3.12 If 2 lines are perpendicular to the same line, then they are _________ to each other. parallel 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

Students need scissors, glue, and 2 sheets of paper. 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

y2 – y1 x2 – x1 Slope = rise run = The slope of a line is the ratio of the vertical change (rise) to the horizontal change (run). y2 – y1 x2 – x1 Slope = rise run =

m = -1 m = 0 m = undefined 1. (-5,7) (-2,4) 2. (3, -2) (-5, -2) Find the slope of the line that passes through the given points. m = -1 1. (-5,7) (-2,4) m = 0 2. (3, -2) (-5, -2) 3. (-6, 2) (-6, -2) m = undefined

All vertical lines are parallel. All horizontal lines are parallel. 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 Vertical Lines v Horizontal Lines v x = # y = # Slope is 0 Slope is undefined

slope-intercept form y = mx + b y–intercept (0,b) slope

m = - 5 m = ½ 4. y = -5x + 14 5. 2x – 4y = -3 – 4y = – 2x – 3 Identify the slope of the line. m = - 5 4. y = -5x + 14 5. 2x – 4y = -3 – 4y = – 2x – 3 -4 m = ½ y = ½ x + ¾

y = -2x + 3 y = - 5 6. slope = - 2 y-int = 3 7. slope = 0 y-int = -5 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 - ½ m = y = - ½ x – 7

point-slope form y – y1 = m (x – x1)

y – y1 = m (x – x1) y - 3 = 5 (x - 2) y - 3 = 5x - 10 y = 5x - 7 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

y – y1 = m (x – x1) y - -4 = - ½ (x - -2) y + 4 = - ½ (x + 2) 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 = - ½ x - 1 y = - ½ x - 5

Opposite & Reciprocals Postulate 18 Slopes of Perpendicular Lines Lines are perpendicular if and only if the product of their slopes is -1. Opposite & Reciprocals

1 m = 5 m = - 2 m = undefined 11. y = -5x + 14 12. y = ½ x - 7 Identify the slope of the line that is perpendicular to the given line. m = 1 5 11. y = -5x + 14 12. y = ½ x - 7 m = - 2 13. y = - 8 m = undefined

y = - 1/3 x + 7 14. y = 3x – 2 (9,4) m = - 1/3 ? y – 4 = - 1/3 (x – 9) 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 = - 1/3 ? y – 4 = - 1/3 (x – 9) y – 4 = - 1/3 x + 3 y = - 1/3 x + 7

y = - 7x + 25 15. y = 1/7x – 11 (5, -10) m = -7 ? y + 10 = - 7 (x – 5) Find the equation of the line that is perpendicular to the given line and passes through the given point. 15. y = 1/7x – 11 (5, -10) m = -7 ? y + 10 = - 7 (x – 5) y + 10 = - 7x + 35 y = - 7x + 25

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