1

Name Due

Homework #9: Pairs of Lines and Angles 3.1 / Using Structure In Exercises 15–18, classify the angle pair as corresponding, alternate interior, alternate exterior, or consecutive interior angles. 15. and 16. and 17. and 18. and 3.2 / In Exercises 3–6, find and Tell which theorems you use in each case. (See Example 1.) Write sentences, not theorem names. *4. *6. In Exercises 7–10, find the value of x. Show your steps. (See Examples 2 and 3.) 7. 8. *13. Error Analysis Describe and correct the error in the student’s reasoning. 1.6 / In Exercises 11–14, find the measure of each angle. (See Example 3.) 11.

5Ð 1Ð 11Ð 13Ð 6Ð 13Ð 2Ð 11Ð

m∠1 m∠2.

2

à3.1 / In Exercises 11–14, identify all pairs of angles of the given type. (See Example 3.) 11. corresponding 12. alternate interior 13. alternate exterior 14. consecutive interior à3.2 / 2. Which One Doesn’t Belong? Which pair of angle measures does not belong with the other three? Explain. In Exercises 3–6, find and Tell which theorems you use in each case. (See Example 1.) Write sentences, not theorem names. *3. *5. In Exercises 7–10, find the value of x. Show your steps. (See Examples 2 and 3.) 9. 10. à2.6 / 29. Mathematical Connections Find the measure of each angle in the diagram. àOn your Unit 2 Journal, write out four theorems that begin with If two parallel lines….

m∠1 m∠2.

3

Name Due

Homework #10: Proving Lines Parallel 3.3 / 1. Vocabulary Two lines are cut by a transversal. Which angle pairs must be congruent for the lines to be parallel? In Exercises 3–8, find the value of x that makes m || n. Explain your reasoning. (See Example 1.) 6. 8. Proof In Exercises 33–36, write a proof. *33. Given Prove Statements Reasons *36. Given Prove Statements Reasons

m∠1=115º, m∠2 = 65º m || n

,|| ba 32 Ð@Ð dc ||

4

à3.3 / In Exercises 3–8, find the value of x making m || n. Explain your reasoning. (See Example 1.) 4. 5. In Exercises 13–18, decide whether there is enough information to prove that m || n. If so, write the theorem (the sentence, not the theorem name) you would use. (See Example 3.) 17. *18. Proof In Exercises 33–36, write a proof. *34. Given and are supplementary. Prove Statements Reasons *35. Given Statements Reasons Prove

àOn your Unit 2 Journal, write out four theorems that end with …then the lines are parallel.

1Ð 3Ð nm ||

,21 Ð@Ð 43 Ð@Ð

CDAB ||

5

Name Due

Homework #11: Interior Angles of a Triangle 3.3 / In Exercises 21–24, are and parallel? Explain your reasoning. *21. 23. 26. Analyzing Relationships Each rung of the ladder is parallel to the rung directly above it. Explain why the top rung is parallel to the bottom rung. *28. Reasoning Use the diagram. Which rays are parallel? Which rays are not parallel? Explain your reasoning. 5.1 / *53. Proving a Theorem Use the diagram to write a proof of the theorem: “The sum of the measures of the interior angles of a triangle is 180º.” Statements Reasons 1. ABC is a triangle. 1. Given 2. Through C, we can draw 2. a line parallel to uniquely.

ACs ruu

DFs ruu

BA

6

à3.3 / In Exercises 21–24, are and parallel? Explain your reasoning. *22. *24. *25. Analyzing Relationships The map shows part of Denver, Colorado. Use the markings on the map. Are the numbered streets parallel to one another? Explain your reasoning. (See Example 4.) *37. Making an Argument Your classmate decided that based on the diagram. Is your classmate correct? Explain your reasoning. Use the Distance Formula to find the distance between the two points. (Section 1.3) 43. (5, –4) and (0, 8) à5.1 / Mathematical Connections In Exercises 49–52, find the values of x and y. 52. 49.

ACs ruu

DFs ruu

BCAD ||

7

Name Due

Homework #12: Slopes of Parallel and Perpendicular Lines 3.2 / 19. Critical Thinking Is it possible for consecutive interior angles to be congruent? Explain.

3.5 / 2. Writing How are the slopes of perpendicular lines related? In Exercises 7 and 8, determine which of the lines are parallel and which of the lines are perpendicular. (See Example 2.) 8. In Exercises 9–12, tell whether the lines through the given points are parallel, perpendicular, or neither. Justify your answer. *11. Line 1: (–9, 3), (–5, 7) Line 2: (–11, 6), (–7, 2) 3.4 / Simplify the ratio. (Skills Review Handbook)

34. 35. 36. 37.

Identify the slope and the y-intercept of the line. (Skills Review Handbook) 38. 41.

38)4(6

---

1453--

)2(7)3(8

----

)1(2413---

y = 3x + 9 y = −8x − 6

8

à3.5 / In Exercises 9–12, tell whether the lines through the given points are parallel, perpendicular, or neither. Justify your answer. *9. Line 1: (1, 0), (7, 4) *12. Line 1: (10, 5), (–8, 9) Line 2: (7, 0), (3, 6) Line 2: (2, –4), (11, –6) à6.4 / In Exercises 3–6, use the graph of with and (See Example 1.) 3. State the coordinates of points D, E, and F. 4. Show that is parallel to and that à3.4 / Identify the slope and the y-intercept of the line. (Skills Review Handbook)

39. 40.

à5.1 / Mathematical Connections In Exercises 49–52, find the values of x and y. 50. 51. àOn your Unit 2 Journal, explain how the slope formula can be used to determine whether two lines are parallel or perpendicular.

ABCD ,DE ,EF .DF

DECB .2

1CBDE =

y = − 12x + 7 y = 1

6x −8

9

Name Due

Homework #13: Equations of Parallel and Perpendicular Lines 6.4 / In Exercises 3–6, use the graph of with and (See Example 1.) 6. Show that is parallel to and that 3.5 / In Exercises 13–16, write an equation of the line passing through point P that is parallel to the given line. Graph the equations of the lines to check that they are parallel. (See Example 3.) 13. P(0, –1), In Exercises 17–20, write an equation of the line passing through point P that is perpendicular to the given line. Graph the equations of the lines to check that they are perpendicular. (See Example 4.) 17. P(0, 0), 18. P(4, –6), Mathematical Connections In Exercises 43 and 44, find a value for k based on the given description. 43. The line through (–1, k) and (–7, –2) is parallel to the line

ABCD ,DE ,EF .DF

DFAB DF = 1

2 AB.

32 +-= xy

19 --= xy

3-=y

.1+= xy

10

à6.4 / In Exercises 3–6, use the graph of with and (See Example 1.) 5. Show that is parallel to and that à3.5 / In Exercises 13–16, write an equation of the 16. P(4, 0), line passing through point P that is parallel to the given line. Graph the equations of the lines to check that they are parallel. (See Example 3.) 15. P(–2, 6), In Exercises 17–20, write an equation of the line passing through point P that is perpendicular to the given line. Graph the equations of the lines to check that they are perpendicular. (See Example 4.) 19. P(2, 3), Mathematical Connections In Exercises 43 and 44, find a value for k based on the given description. 44. The line through (k, 2) and (7, 0) is perpendicular to the line àOn your Unit 2 Journal, write an equation of a line parallel to a given line through a given point.

ABCD ,DE ,EF .DF

EFAC EF = 1

2 AC.

−x + 2y =12

5-=x

y− 4 = −2 x +3( )

y = x − 285 .

11

Name Due

Homework #14: Writing Equations of Perpendicular Bisectors 3.4 / 1. Complete the Sentence The perpendicular bisector of a segment is the line that passes through the of the segment at a angle. 3.5 / 25. Error Analysis Describe and correct the error in determining whether the lines are parallel, perpendicular, or neither. 26. Error Analysis Describe and correct the error in writing an equation of the line that passes through the point (3, 4) and is parallel to the line In Exercises 27–30, find the midpoint of Then write an equation of the line that passes through the midpoint and is perpendicular to This line is called the perpendicular bisector. 27. P(–4, 3), Q(4, –1) 29. P(0, 2), Q(6, –2)

.12 += xy

.PQ.PQ

12

à3.5 / 32. Reasoning Is quadrilateral QRST a parallelogram? Explain your reasoning. 33. Reasoning A triangle has vertices L(0, 6), M(5, 8), and N(4, –1). Is the triangle a right triangle? Explain your reasoning. 34. Modeling with Mathematics A new road is being constructed parallel to the train tracks through point V. An equation of the line representing the train tracks is Find an equation of the line representing the new road. à6.1 / In Exercises 19–22, write an equation of the perpendicular bisector of the segment with the given endpoints. (See Example 5.) 19. M(1, 5), N(7, –1) 21. U(–3, 4), V(9, 8) àOn your Unit 2 Journal, show an example of how to write an equation of the perpendicular bisector of a segment whose endpoints are given. Be sure to practice using both point-slope and slope-intercept forms.

.2xy =

13

Name Due

Homework #15: Partitioning a Directed Line Segment 3.5 / 1. Complete the Sentence A line segment AB is a segment that represents moving from point A to point B. In Exercises 3–6, find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. (See Example 1.) 3. A(8, 0), B(3, –2); 1 to 4 In Exercises 27–30, find the midpoint of Then write an equation of the line that passes through the midpoint and is perpendicular to This line is called the perpendicular bisector. 28. P(–5, –5), Q(3, 3) 35. Modeling with Mathematics A bike path is being constructed perpendicular to Washington Boulevard through point P(2, 2). An equation of the line representing

Washington Boulevard is Find an equation

of the line representing the bike path.

.PQ.PQ

y = − 23x.

14

à3.5 / In Exercises 3–6, find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. (See Example 1.) 5. A(1, 6), B(–2, –3); 5 to 1 In Exercises 27–30, find the midpoint of Then write an equation of the line that passes through the midpoint and is perpendicular to This line is called the perpendicular bisector. 30. P(–7, 0), Q(1, 8) 31. Modeling with Mathematics Your school lies directly between your house and the movie theater. The distance from your house to the school is one- fourth of the distance from the school to the movie theater. What point on the graph represents your school? 39. Critical Thinking Suppose point P divides the directed line segment XY so that the ratio of XP to PY is 3 to 5. Describe the point that divides the directed line segment YX so that the ratio of YP to PX is 5 to 3.

.PQ.PQ

15

àOn your Unit 2 Journal, show an example of how to partition a directed line segment into two pieces whose lengths are in a given ratio. Name Due Monday, 10.15

Homework #16: Test 2 Review

1. Transversal intersects (1) and as shown in the diagram (2) at the right. Which statement could (3) and are supplementary always be used to prove (4) and are supplementary

2. What is an equation of a line which passes through (6, 9) (1)

and is perpendicular to the line whose equation is 4x – 6y = 15? (2)

(3)

(4)

3. Given shown at the right, with M(–6, 1) and N(3, –5), what is an equation of the line that passes through point P(6, 1) and is parallel to

(1)

(2)

(3)

(4)

4. Write an equation of the perpendicular bisector of the segment with endpoints M(1, 5) and N(7, –1). 5. Point P is on the directed line segment from point X(–6, –2) to point Y(6, 7) and divides the segment in the ratio What are the coordinates of point

EF! "##

ABs ruu

∠2 ≅∠4CD! "##, ∠7 ≅∠8

∠3 ∠6AB! "##||CD! "##? ∠1 ∠5

y− 9 = − 32x − 6( )

y− 9 = 23x − 6( )

y+ 9 = − 32x + 6( )

y+ 9 = 23x + 6( )

MN

MN ?

y = − 23x + 5

y = − 23x −3

y = 32x + 7

y = 32x −8

1: 5. P?

16

6. In with vertices A(–6, –2), B(2, 8), and C(6, –2),

has midpoint D, has midpoint E, and has midpoint F. Find the coordinates of E and F.

Show that and that 7. In the diagram below, and bisects The measure of angle DFB is (1) 36º (2) 54º (3) 72º (4) 82º 8. Prove the theorem: “The sum of the measures of the interior angles of a triangle is 180º.” Statements Reasons 1. ABC is a triangle. 1. Given

ΔABCAB BCAC

FE || AB FE = 12 AB.

DE || BC, m∠C = 26º, m∠A = 82º, DF ∠BDE.

17

Name Due Score (out of 20)

Unit 2 Journal ☐ Know and apply four relationships between pairs of angles when parallel lines are cut by a transversal. Write out four theorems that begin with If two parallel lines…. Be sure to use one or more diagrams to explain what each type of angle pair refers to. (4 points) ☐ Know and apply four theorems that prove two lines parallel. Write out four theorems that end with …then the lines are parallel. (2 points) ☐ Find and apply the slope of a line given the coordinates of two points on the line. Explain how the slope formula can be used to determine whether two lines are parallel or perpendicular. Then write out the slope formula enough times that you’re sure you have it memorized. (3 points) ☐ Find and apply the slope of a line given its equation in any form. Give an example of an equation of a line from which the slope is not immediately obvious. Then show how to determine the slope of that line. (2 points)

18

☐ Write an equation of a line passing through a given point, parallel or perpendicular to a given line. Show an example of how to write an equation of a line parallel to a given line passing through a given point. Express your answer either in slope-intercept form or point-slope form. (2 points) ☐ Write an equation of the perpendicular bisector of a segment whose endpoints are given. Show an example of how to write an equation of the perpendicular bisector of a segment whose endpoints are given. Express your answer in whichever form you didn’t use in the previous example (slope-intercept form or point-slope form). (4 points) ☐ Partition a directed line segment into two pieces whose lengths are in a given ratio. Show an example of how to partition a directed line segment into two pieces whose lengths are in a given ratio. (3 points) ☐ Solve multistep problems with the interior angles of a triangle. ☐ Grow through mistakes. What can you learn from your mistakes (if there were any) on Test 1? ☐ Know and apply Unit 2 Definitions, Theorems and Properties: Perpendicular lines form right angles. All right angles are congruent. If two lines intersect to form a linear pair of congruent angles, then they are perpendicular. Through a given point, there exists exactly one line parallel to a given line. Through a given point, there exists exactly one line perpendicular to a given line. The sum of the measures of the interior angles of a triangle is 180º. Transitive Property of Parallel Lines (If two lines are parallel to the same line, then they are parallel to each other.)