USING PROPERTIES OF PARALLEL LINES WORKSHEET

Problem 1 :

In the diagram given below, lines m, n and k represent three of the oars. If m||n and n||k, then prove m||k.

Problem 2 :

In the diagram given below, if ∠1 ≅ ∠2, then prove m||n.

Problem 3 :

In the diagram given below, each line is parallel to the next immediate line. Explain why the line K1 is parallel to the line K2.

Problem 4 :

In the diagram given below, lines a and b are perpendicular to the line c. Prove that the lines a and b are parallel.

Problem 5 :

In the diagram given below, if ∠4 and ∠5 are supplementary, then prove g||h.

Problem 6 :

In the diagram given below, find the value of x that makes j||k.

Problem 7 :

If two boats sail at a 45° angle to the wind as shown, and the wind is constant, will their paths ever cross ? Explain.

Problem 8 :

In the diagram given below, decide which rays are parallel.

(i) Is EB parallel to HD?

(ii) Is EA parallel to HC?

1. Answer :

Statements

m||n

∠1 ≅ ∠2

n||k

∠2 ≅ ∠3

∠1  ∠3

m||k

Reasons

Given

Corresponding angles postulate

Given

Corresponding angles postulate

Transitive property of congruence

Corresponding angle converse

2. Answer :

Statements

∠1 ≅ ∠2

∠2 ≅ ∠3

∠1  ∠3

m||n

Reasons

Given

Vertical angles theorem

Transitive property of congruence

Corresponding angles converse

3. Answer :

We are given that K1||K2 and K2||K3.

By transitive property of parallel lines, K1||K3.

Since K1||K3 and K3||K4, again by transitive property, it follows that K1||K4.

4. Answer :

Since the line c cuts both the lines a and b, the line c is transversal. Both the lines a and b are perpendicular to the line c. So,  the measure of both ∠1 and ∠2 in the above diagram is 90° and c is transversal to the lines a and b.

That is,

m∠1 = 9

m∠2 = 9

By corresponding angles converse, the lines a and b are parallel.

5. Answer :

We are given that ∠4 and ∠5 are supplementary. By the linear pair postulate, ∠5 and ∠6 are also supplementary, because they form a linear pair. By the congruence supplements theorem, it follows that ∠4  ∠6. Therefore, by the alternate interior angles converse, g and h are parallel.

6. Answer :

Lines j and k will be parallel if the marked angles are supplementary.

x° + 4x° = 180°

5x = 180

x = 36

So, x = 36 makes j||k.

7. Answer :

Because corresponding angles are congruent, the paths of the boats are parallel. Parallel lines do not intersect. So the paths of the boats will never cross.

8. Answer (i) :

m∠BEH = 58°

m∠DHG = 61° 

∠BEH and ∠DHG are corresponding angles, but they are not congruent. So EB and HD are not parallel. 

8. Answer (ii) :

m∠AEH = 62° + 58° = 120°

m∠CHG = 59° + 61° = 120°

∠AEH and ∠CHG are congruent corresponding angles. So AE and CH are parallel.

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