parallel and perpendicular lines -...

Parallel Lines Chapter Problems

Lines: Intersecting, parallel & skew Class Work – Use image 1, which consists of a rectangular prism and a triangular prism 1. Name all segments parallel toGH :2. Name all segments skew toGH : 3. Name all segments intersecting withGH : 4. Are segmentsGH and BA coplanar? Explain your answer. 5. Are segments GH and BF coplanar? Explain your answer.

Is each statement true always, sometimes, or never?6. Two intersecting lines are skew.7. Two parallel lines are coplanar.8. Two lines in the same plane are parallel.9. Two lines that do not intersect are parallel.10. Two skew lines are coplanar

Lines: Intersecting, parallel & skew Homework – Use Image 1, which consists of a rectangular prism and a triangular prism 11. Name all segments parallel to FE:12. Name all segments skew to FE: 13. Name all segments intersecting with FE:14. Are segments FE and CDcoplanar? Explain your answer.15. Are segments FE and HD coplanar? Explain your answer.

State whether the following statements are always, sometimes, or never true:16. Two coplanar lines are skew.17. Two intersecting lines are in the same plane.18. Two lines in the same plane are parallel.

Lines & Transversals Classify each pair of angles as alternate interior, alternate exterior, same-side interior, same-side exterior, corresponding angles, or none of these.19. ∠11 and ∠16 are20. ∠12 and ∠2 are21. ∠14 and ∠8 are22. ∠6 and ∠16 are23. ∠7 and ∠14 are24. ∠3 and ∠16 are

Geometry – Parallel Lines ~1~ NJCTL.org

Image 1

Classify each pair of angles as alternate interior, alternate exterior, same-side interior, same-side exterior, corresponding angles, or none of these.25. ∠7 and ∠1226. ∠3 and ∠627. ∠6 and ∠1128. ∠7 and ∠1129. ∠4 and ∠1030. ∠14 and ∠1631. ∠2 and ∠332. ∠2 and ∠10

Parallel Lines & ProofsClassworkMatch each expression/equation with the property used to make the conclusion.33. AB = AB34. If m∠A = m∠B and m∠B = m∠C, then

m∠A = m∠C.35. If x + y = 9 and y = 5, then x + 5 = 9.

36. If DE = FG, then FG = DE.a) Substitution Property of Equalityb) Transitive Property of Equalityc) Reflexive Property of Equalityd) Symmetric Property of Equality

PARCC type question:37. Alternate Exterior Angles Proof: Complete the proof by filling in the missing reasons

with the “reasons bank” below.Given: line m || line kProve: ∠2 ≅ ∠8


PARCC type question:38.Same-Side Interior Angles Proof: Complete the proof by filling in the missing reasons

with the “reasons bank” below. Some reasons may be used more than once.Given: line m || line kProve: ∠5 & ∠4 are supplementary


Transitive Property of Congruence

transversal, then the corresponding

Vertical Angles are congruent.

Reasons Banka) Angles that form a linear pair are

supplementary.b) Substitution Property of Equalityc) Definition of supplementary anglesd) If 2 parallel lines are cut by a

transversal, then the corresponding angles are congruent.

e) Definition of congruent anglesf) Given

Statements Reasons1. line m || line k 1.2. ∠2 ≅ ∠6 2.3. ∠6 ≅ ∠8 3.4. ∠2 ≅ ∠8 4.

Statements Reasons1. line m || line k 1.2. ∠1 ≅ ∠5 2.3. m∠1 = m∠5 3.4. ∠1 & ∠4 are supplementary 4.5. m∠1 + m∠4 = 180 5.6. m∠5 + m∠4 = 180 6.7. ∠5 & ∠4 are supplementary 7.

Parallel Lines & ProofsHomeworkFor #39-42 match the description on the left to the name of the property on the right.39. ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C. a) Substitution Property of Equality40. If bc = 77 and b = 11, then 11c = 77. b) Transitive Property of Congruence 41. If ∠P ≅ ∠M, then ∠M ≅ ∠P. c) Reflexive Property of Equality 42. QR = QR d) Symmetric Property of Congruence

PARCC type question:43. Alternate Interior Angles Proof: Complete the proof by filling in the missing reasons

with the “reasons bank” below.Given: line m || line kProve: ∠3 ≅ ∠5


Statements Reasons1. line m || line k 1.2. ∠3 ≅ ∠7 2.3. ∠7 ≅ ∠5 3.4. ∠3 ≅ ∠5 4.

PARCC type question:44.Same-Side Exterior Angles Proof: Complete the proof by filling in the missing reasons

with the “reasons bank” below. Some reasons may be used more than once.Given: line m || line kProve: ∠1 & ∠8 are supplementary


Reasons Banka) Vertical Angles are congruent.b) Givenc) Transitive Property of Congruenced) If 2 parallel lines are cut by a transversal,

then the corresponding angles are congruent.

Statements Reasons1. line m || line k 1.2. ∠1 ≅ ∠5 2.3. m∠1 = m∠5 3.4. ∠5 & ∠8 are supplementary

4.

5. m∠5 + m∠8 = 180 5.6. m∠1 + m∠8 = 180 6.7. ∠1 & ∠8 are supplementary

7.

Properties of Parallel Lines Classwork Use the given diagram to answer problems #33-41.

If m∠9 = 54°, then find the measure the following angles:45.m∠1=46. m∠2=47.m∠4=48.m∠5=49.m∠15=

If m∠2 = (12x-54)° and m∠10 = (7x+26)°, then find the measure the following angles: 50.m∠6=51. m∠11=52. m∠9= 53. m∠16=

Find the values of the unknown variables in each figure. (# 54-58)54. 55.


Reasons Banka) Definition of supplementary anglesb) If 2 parallel lines are cut by a transversal,

then the corresponding angles are congruent.

c) Givend) Definition of congruent anglese) Angles that form a linear pair are

supplementary.f) Substitution Property of Equality

56.

57. 58.

Find measure of the following angles:59. m∠1= 60. m∠2= 61. m∠3=62. m∠4=63. m∠5=

State which segments (if any) are parallel.


64. 65.

66.

Solve for the unknowns67. 68.

Properties of Parallel Lines Homework If m∠9 = 62°, then find the measure the following angles:69.m∠1=70.m∠2=71.m∠4=72.m∠5=73.m∠15=


If m ∠2 = (14x-24)° and m ∠10 = (6x+72)°, then find the measure the following angles: 74.m∠6=75.m∠11=76.m∠9=77.m∠16=

Find the values of the unknown variables in each figure. (#78-82)

78. 79. 80.

81. 82.

Find measure of the following angles:83. m∠1= 84. m∠2= 85. m∠3=86. m∠4=87. m∠5=

State which segments (if any) are parallel.88.


90.

89.

91. 92.

Constructing Parallel LinesClass Work 93. Construct a line m that is parallel to line l that passes thru point C using the stated method.Corresponding Angles


94. Error Analysis: A person was constructing the line n thru point D such that it was parallel to line l using the alternate interior angles method. Using their markings,state their mistake.

95. Use paper- folding techniques to construct parallel lines.

Constructing Parallel LinesHomework

96. Error Analysis: A person was constructing the line n thru point D such that it was parallel to line l using the alternate exterior angles method. Using their markings,state their mistake.


97. Construct parallel lines using a straightedge and compass using alternate interior angles.

98. Construct parallel lines using a straightedge and compass using alternate exterior angles.

PARCC type question:99. The figure shows line j, points C and B are on line j, and point A is not on line j. Also shown is line AB.

Part A:


Consider the partial construction of a line parallel to j through point A. What would be the final step in the construction?

a) Draw a line through points B and Fb) Draw a line through points C and Fc) Draw a line through points A and Fd) Draw a line through points A and G

Part B:Once the construction is complete, which of the following reasons listed contribute to providing the validity of the construction?

a) If two parallel lines are cut by a transversal, then the corresponding angles are congruent.b) If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.c) If two parallel lines are cut by a transversal, then the same-side interior angles are supplementary.d) If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.


PARCC type question:100. The figure shows line p; points H, K, and M are on line p, and point J is not on line p. Also shown is line JK.

Part A:

Consider the partial construction of a line parallel to p through point J. What would be the final step in the construction?

a) Draw a line through points K and Nb) Draw a line through points J and Nc) Draw a line through points H and Nd) Draw a line through points M and M

Part B:Once the construction is complete, which of the following reasons listed contribute to providing the validity of the construction?

a) If two parallel lines are cut by a transversal, then the corresponding angles are congruent.b) If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.c) If two parallel lines are cut by a transversal, then the same-side interior angles are supplementary.d) If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.


Parallel Lines ReviewMultiple Choice

1. Name the segment parallel to GH and skew to EA.a. FBb.DA c.JId.HD

2. Name the segment parallel to BCand skew toEI .a.FBb.DA c.JId.HD

3. Determine if the statement is always, sometimes, or never true: Two skew lines are coplanar.a. Alwaysb. Sometimes c. Never

4. Determine if the statement is always, sometimes, or never true: Two intersecting lines are coplanara. Alwaysb. Sometimes c. Never

5. Determine if the statement is always, sometimes, or never true: Two lines that do not intersect are skew.a. Alwaysb. Sometimes c. Never

6. Determine the relationship between ∠1 & ∠10.a. Alternate Interiorb. Same-side Interiorc. Corresponding Anglesd. None of these

7. Determine the relationship between ∠5 & ∠15.a. Alternate Exteriorb. Alternate Interior c. Same-side Interiord. None of these


8. Given in the diagram to the right, m∠2=3x-10 and m∠15=2x+30 , what is m∠12?a. 32o

b. 40o

c. 86o

d. 110o

9. Given in the diagram to the right, m∠5= (7x+2)°and m∠11=(5x+14)°, what is m∠14?a. 6°

b. 44°

c. 46°

d. 136°

In 10-11, use the diagram at the right. 10.Given ∠2 ≅ ∠6, what justifies k || m.

a. Converse Alternate Interior Angles Theoremb. Converse Alternate Exterior Angles Theoremc. Converse Corresponding Angles Theoremd. there is not enough info to state parallel

11.Given n || p , what justifies ∠1 ≅ ∠12a. Alternate Interior Angles Theoremb. Alternate Exterior Angles Theoremc. Corresponding Angles Theoremd. there is not enough info to make this statement

Extended Constructed Response

1. Complete the proof by filling in the missing reasons with the “reasons bank” to the right. Some reasons may be used more that once.Given: ∠1 ≅ ∠3; MN || PQProve: ∠2≅∠3

Statements Reasons1. ∠1 ≅ ∠3 1.2. MN || PQ 2.3. ∠1 ≅ ∠2 3.4. ∠2≅∠3 4.


Reasons Banka) Transitive Property of Congruenceb) If 2 parallel lines are cut by a transversal, then the alternate interior angles are congruent.c) Given

2. Complete the proof by filling in the missing reasons with the “reasons bank” to the right. Some reasons may be used more that once.Given: n || p, k || mProve: ∠2 & ∠13 are supplementary

Statements Reasons1. n || p, k || m 1.2. ∠2 ≅ ∠12 2.3. ∠12 ≅ ∠14 3.4. ∠2 ≅ ∠14 4.5. m∠2 = m∠14 5.6. m∠13 & m∠14 are supplementary

6.

7. m∠13 + m∠14 = 180° 7.8. m∠13 + m∠2 = 180° 8.9. ∠2 &∠13 are supplementary 9.

3. Using a compass and straightedge, construct parallel lines. You can use any method of your choice.


Reasons Banka) Transitive Property of Congruenceb) Definition of supplementary anglesc) If 2 parallel lines are cut by a transversal, then the alternate interior angles are congruent. d) Definition of Congruent Anglese) Givenf) If 2 parallel lines are cut by a transversal, then the alternate exterior angles are congruent.g) Angles that form a linear pair are supplementaryh) Substitution Property of Equality

Answers

Segments JI,FE, BA,CD Segments JF,IE, FB, EAXSegments GJ ,GF, GC, HI ,HE,HDYes, because these segments are parallelNo, these lines are skew, so they are not coplanar.NeverAlwaysSometimesSometimesNeverSegments JI, GH , BA,CD Segments HI , GJ , CG,DH Segments FG, FB, FJ ,EA, FB, EH , EI Yes, because they are parallelNo, these lines are skew, so they are not coplanarNeverAlwaysSometimesSame side interiorNone of theseAlternate interiorCorrespondingSame-side interiorNone of theseCorrespondingSame-sideAlternate interiorCorrespondingCorrespondingSame-side interiorNone of theseNone of thesec. Reflexive Property of Equalityb. Transitive Property of Equalitya. Substitution

Reasons


Property of Equalityd. Symmetric Property of EqualityProof reasons should be:Statements1. line m || line k 1. d.2. ∠2 ≅ ∠6 2. b.3. ∠6 ≅ ∠8 3. c.4. ∠2 ≅ ∠8 4. a.

1. Proof reasons should be:

Statements Reasons1. line m || line k 1. f.2. ∠1 ≅ ∠5 2. d.3. m∠1 = m∠5 3. e.4. ∠1 & ∠4 are supplementary

4. a.

5. m∠1 + m∠4 = 180°

5. c.

6. m∠5 + m∠4 = 180°

6. b.

7. ∠5 & ∠4 are supplementary

7. c.

2. b. Transitive Property of Congruence3. a. Substitution Property of Equality4. d. Symmetric Property of Congruence5. c. Reflexive Property of Equality6. Proof reasons should be:

Statements Reasons1. line m || line k 1. b.2. ∠3 ≅ ∠7 2. d.3. ∠7 ≅ ∠5 3. a.4. ∠3 ≅ ∠5 4. c.

7. Proof reasons should be:

Statements Reasons1. line m || line k 1. c.


2. ∠1 ≅ ∠5 2. b.3. m∠1 = m∠5 3. d.4. ∠5 & ∠8 are supplementary

4. e.

5. m∠5 + m∠8 = 180

5. a.

6. m∠1 + m∠8 = 180

6. f.

7. ∠1 & ∠8 are supplementary

7. a.

8. 54°9. 126°10.126°11.54°12.54°13.138°14.42°15.42°16.138°17.x= 144°18.x= 64° and y= 49/419.x=6; z=220.x=24, y=11; z=22/521.x=33; y=222.44°23.107°24.29°25.29°26.136°27.Segments AD and BCare parallel28.Segments OPand RSare parallel29.None of these30.x=9 and y=8 and z=731.x=8 and y=732.62°33.118°34.118°35.62°36.62°37.144°38.36°


39.36°40.144°41.x=55°42.x=86° and y=743.x=9; y=6; z=744.x=15; y=10; z=845.x=25; y=346.41°47.106°48.33°49.33°50.129°51.cannot be determined52.SegmentsNK and MLare parallel53.Segments QP and TSare parallel54.x=6; y=12; z=755.x=18; y=756.See student work57.made same side interior the same58.See student work59.Made angles congruent that should be supplementary. 60.see student work61.see student work62.Part A: c & Part B: d63.Part A: b & Part B: b

REVIEW1. c2. b3. c4. a5. b6. c7. a8. c9. d10.c11.d

EXTENDED CONSTRUCTED RESPONSE 1.


Statements Reasons∠1 ≅ ∠3 c. GivenMN ¿∨PQ c. Given∠1 ≅ ∠2 b. Alternate

Interior Angles Theorem∠2≅∠3 a. Transitive Property of congruence

Statements Reasons1. n || p, k || m 1. e2. ∠2≅∠12 2. f3. ∠12≅∠14 3. c4. ∠2≅∠14 4. a5. m∠2+m∠14 5. d6. ∠13 & ∠14 are supplementary

6. g

7. m∠13 = m∠14 = 180° 7. b8. m∠13 + m∠2 = 180° 8. h9. ∠2 & ∠13 are supplementary

9. b

3. See student work


parallel and perpendicular lines -...

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