proofs of parallel and perpendicular lines · proofs of parallel and perpendicular lines ... and...

Name: ________________________ Class: ___________________ Date: __________ ID: A

1

Proofs of Parallel and Perpendicular Lines

Short Answer

1. Give the missing reasons in this proof of the Alternate Interior Angles Theorem.

Given: l Ä n

Prove: ∠4 ≅ ∠6

Statments Reasons

1. l Ä n

2. ∠2 ≅ ∠6

3. ∠4 ≅ ∠2

4. ∠6 ≅ ∠4

1. Given

a. ?

b. ?

c. ?

Name: ________________________ ID: A

2

2. State the missing reasons in this proof.

Given: ∠1 ≅ ∠5

Prove: p Ä r

Statements Reasons

1. ∠1 ≅ ∠5

2. ∠4 ≅ ∠1

3. ∠4 ≅ ∠5

4. p Ä r

Given

a.____

b.____

c.____

3. The 8 rowers in the racing boat stroke so that the angles formed by their oars with the side of the boat all stay

equal. Explain why their oars on either side of the boat remain parallel.

4. Suppose you have four identical pieces of wood like those shown below. If m∠b = 40° can you construct a

frame with opposite sides parallel? Explain.

5. Find the measure of each interior and exterior angle. The diagram is not to scale.

Name: ________________________ ID: A

3

6. The fireworks technician has two rocket launchers, each with a base and stand in the form of an L. A

diagonal trough on which the technician places a rocket joins the ends of each L. One launcher has a 4-inch

base and 10-inch stand. The other has a 6-inch base and a 15-inch stand. Explain why two rockets launched

from the two devices could follow parallel paths.

Essay

7. Write a paragraph proof of this theorem: In a plane, if two lines are perpendicular to the same line, then they

are parallel to each other.

Given: r ⊥ s, t⊥ s

Prove: r Ä t

8. Write a two-column proof.

Given: ∠2 and ∠5 are supplementary.

Prove: l Ä m

Name: ________________________ ID: A

4

9. Find the values of the variables. Show your work and explain your steps. The diagram is not to scale.

Other

10. Given m∠1 = m∠2, what can you conclude about the lines l, m, and n? Explain.

11. Justify the statement algebraically.

In a triangle, if the sum of the measures of two angles is equal to the measure of the third angle, then the

triangle is a right triangle.

12. Line p contains points A(–1, 4) and B(3, –5). Line q is parallel to line p. Line r is perpendicular to line q.

What is the slope of line r? Explain.

ID: A

1

Proofs of Parallel and Perpendicular Lines

Answer Section

SHORT ANSWER

1. ANS:

a. Corresponding angles.

b. Vertical angles.

c. Transitive Property.

PTS: 1 DIF: L2 REF: 3-1 Properties of Parallel Lines

OBJ: 3-1.2 Properties of Parallel Lines NAT: NAEP 2005 M1f | ADP K.2.1

STA: NJ 4.1.12 B.1 | NJ 4.2.12 A.3a | NJ 4.2.12 A.4a TOP: 3-1 Example 3

KEY: alternate interior angles | Alternate Interior Angles Theorem | proof | reasoning | two-column proof |

multi-part question

2. ANS:

a. Vertical angles.

b. Transitive Property.

c. Alternate Interior Angles Converse.

PTS: 1 DIF: L2 REF: 3-2 Proving Lines Parallel

OBJ: 3-2.1 Using a Transversal NAT: NAEP 2005 M1e | NAEP 2005 M1f | ADP K.2.3

STA: NJ 4.1.12 B.1 | NJ 4.2.12 A.1 | NJ 4.2.12 A.3a | NJ 4.2.12 A.4a

TOP: 3-2 Example 1

KEY: two-column proof | proof | reasoning | corresponding angles | multi-part question

3. ANS:

The rowers keep corresponding angles congruent.




TOP: 3-2 Example 1 KEY: transversal | word problem | reasoning | parallel lines

4. ANS:

No. Explanations may vary. Sample:

Placing three pieces together forms same-side interior angles with measures of 80°. Since 80+ 80 ≠ 180, they

are not supplementary, so the sides are not parallel.

PTS: 1 DIF: L3 REF: 3-3 Parallel and Perpendicular Lines

OBJ: 3-3.1 Relating Parallel and Perpendicular Lines

NAT: NAEP 2005 M1e | NAEP 2005 M1f | ADP K.2.1

STA: NJ 4.2.12 A.1 | NJ 4.2.12 A.3a | NJ 4.2.12 A.4a TOP: 3-3 Example 1

KEY: word problem | problem solving | parallel lines

ID: A

2

5. ANS: m∠1 = m∠2 = m∠3 = 90,m∠4 = 122,m∠5 = m∠6 = 58,m∠8 = 32,m∠7 = m∠9 = 148

PTS: 1 DIF: L3 REF: 3-4 Parallel Lines and the Triangle Angle-Sum Theorem

OBJ: 3-4.2 Using Exterior Angles of Triangles

NAT: NAEP 2005 G3b | NAEP 2005 G3f | ADP J.5.1 | ADP K.1.2

STA: NJ 4.1.12 B.1 | NJ 4.2.12 A.1 | NJ 4.2.12 A.3a | NJ 4.2.12 A.4b

KEY: Triangle Angle-Sum Theorem | exterior angle

6. ANS:

Pointed in the same direction, the two launchers have equal slopes, so the rockets would be set up to follow

parallel paths.

PTS: 1 DIF: L3 REF: 3-7 Slopes of Parallel and Perpendicular Lines

OBJ: 3-7.1 Slope and Parallel Lines

NAT: NAEP 2005 A1h | NAEP 2005 A2a | ADP J.4.1 | ADP J.4.2 | ADP K.10.2

STA: NJ 4.1.12 B.1 | NJ 4.2.12 A.1 | NJ 4.2.12 A.3a | NJ 4.2.12 C.1b | NJ 4.2.12 C.1d | NJ 4.2.12 C.1e | NJ

4.3.12 C.1a | NJ 4.3.12 C.2

KEY: slope | slopes of parallel lines | word problem | problem solving | writing in math

ESSAY

7. ANS:

[4] By the definition of ⊥, r ⊥ s implies m∠2 = 90, and t ⊥ s implies m∠6 = 90. Line s

is a transversal. ∠2 and ∠6 are corresponding angles. By the Converse of the

Corresponding Angles Postulate, r || t.

[3] correct idea, some details inaccurate

[2] correct idea, not well organized

[1] correct idea, one or more significant steps omitted

PTS: 1 DIF: L4 REF: 3-3 Parallel and Perpendicular Lines

OBJ: 3-3.1 Relating Parallel and Perpendicular Lines

NAT: NAEP 2005 M1e | NAEP 2005 M1f | ADP K.2.1

STA: NJ 4.2.12 A.1 | NJ 4.2.12 A.3a | NJ 4.2.12 A.4a TOP: 3-3 Example 2

KEY: paragraph proof | proof | reasoning | extended response | rubric-based question | perpendicular lines

ID: A

3

8. ANS:

[4] Statements Reasons

1. ∠2 and ∠5 are supplementary

2. ∠3 ≅ ∠2

3. ∠3 and ∠5 are supplementary

4. l Ä m

1. Given

2. Vertical angles

3. Substitution

4. Converse of Same-Side

Interior Angles Theorem

[3] correct idea, some details inaccurate

[2] correct idea, some statements missing

[1] correct idea, several steps omitted




KEY: two-column proof | proof | extended response | rubric-based question | parallel lines | supplementary

angles

9. ANS:

[4] w + 31 + 90 = 180, so w = 59º. Since vertical angles are congruent, y = 59º. Since

supplementary angles have measures with sum 180, x = v = 121º. z + 68 + y = z

+ 68 + 59 = 180, so z = 53º.

[3] small error leading to one incorrect answer

[2] three correct answers, work shown

[1] two correct answers, work shown


OBJ: 3-4.2 Using Exterior Angles of Triangles



KEY: Triangle Angle-Sum Theorem | vertical angles | supplementary angles | extended response |

rubric-based question

OTHER

10. ANS:

l and m are both perpendicular to n. Explanation: Since l and m are parallel, ∠1 and ∠2 are supplementary

by the Same-Side Interior Angles Theorem. It is given that m∠1 = m∠2, so

180 = m∠1 + m∠2 = m∠1 + m∠1 = 2m∠1, and m∠1 = 90 = m∠2. Since ∠1 and ∠2 are right angles, l is

perpendicular to n and m is perpendicular to n.

PTS: 1 DIF: L3 REF: 3-1 Properties of Parallel Lines

OBJ: 3-1.2 Properties of Parallel Lines NAT: NAEP 2005 M1f | ADP K.2.1

STA: NJ 4.1.12 B.1 | NJ 4.2.12 A.3a | NJ 4.2.12 A.4a

KEY: perpendicular lines | reasoning | writing in math

ID: A

4

11. ANS:

m∠1 + m∠2 + m∠3 = 180. Given m∠1 + m∠2 = m∠3, by substitution, m∠3 + m∠3 = 180. 2m∠3 = 180, and

m∠3 = 90. Thus, ∠3 is a right angle, and the triangle is a right triangle.


OBJ: 3-4.1 Finding Angle Measures in Triangles



KEY: Triangle Angle-Sum Theorem | reasoning | writing in math

12. ANS:

4

9; Line r is perpendicular to line p because a line perpendicular to one of two parallel lines is also

perpendicular to the other. Thus, the slope of line r is the opposite reciprocal of the slope of line p.

PTS: 1 DIF: L3 REF: 3-7 Slopes of Parallel and Perpendicular Lines

OBJ: 3-7.2 Slope and Perpendicular Lines

NAT: NAEP 2005 A1h | NAEP 2005 A2a | ADP J.4.1 | ADP J.4.2 | ADP K.10.2

STA: NJ 4.1.12 B.1 | NJ 4.2.12 A.1 | NJ 4.2.12 A.3a | NJ 4.2.12 C.1b | NJ 4.2.12 C.1d | NJ 4.2.12 C.1e | NJ

4.3.12 C.1a | NJ 4.3.12 C.2

KEY: perpendicular lines | parallel lines | slopes of parallel lines | slopes of perpendicular lines | reasoning |

writing in math

proofs of parallel and perpendicular lines · proofs of parallel and perpendicular lines ... and...

Documents