PAPER · REASONING · PRACTICE

Parallel lines: conditions and reasoning

Use angle relations in the correct direction and distinguish a hypothesis from a conclusion.

40 minutes · Year 7, lower volume · text pp. 9–24 · textbook lessons 5–12

Textbook reference: Year 7 lower · Chapter 5 Intersecting and parallel lines · text lessons 5–12

Chapter core practice: use after the referenced textbook lessons, or for chapter review. See the lower-volume plan

One worked example, one guided attempt, eight core questions, four follow-ups and two extensions. Write on paper; the teacher controls the reveal.

Teacher reference Arrange in My Classroom

Lesson routePhase

4′Check prerequisites
5′Observe and predict
8′Worked example and guided attempt
15′Independent practice
5′Discuss and correct
3′Exit questions

Observe · explain

A question before practising

18 minutes using the existing parallels lab: predict which relations survive if parallelism is removed (3 min). Set α=60°, locked parallel, and record corresponding, alternate interior, same-side interior and vertical angles (4 min). Unlock and change only β to 75°, then back to 60°; keep α fixed (5 min). Identify which equalities or sums change and which vertical-angle fact does not (4 min). Give one justified converse argument, distinguishing a test for parallelism from a property that assumes it (2 min).

Organise the key idea

Use angle relations in the correct direction and distinguish a hypothesis from a conclusion.

Open the original chapter Open the interactive inquiry

Use the lesson question here for paper discussion; the original activity opens separately.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: follow the full 18-minute investigation; record predictions before checking.

Worked example1

Coplanar lines a and b are cut by a transversal. A pair of corresponding angles are both 64°. Prove that a is parallel to b.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Both marked angles are given as 64°, so they are equal.
Use the corresponding-angle test for parallel lines, not the property of lines already known parallel.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Guided attempt2

Two parallel coplanar lines are cut by a transversal. One angle is $68^{\circ}$ . Find its same-side interior angle; state the reason.

Parallel lines give same-side interior sum $68+x=\square^{\circ}$ .
Complete $x=180-\square$ .
Write $x=\square^{\circ}$ and name the parallel-line property.

The question gives parallel lines and same-side interior angles.
Same-side interior angles are supplementary when the lines are parallel.
Use 68+x=180; do not set the angles equal.

Use the stated parallel condition before applying this angle relation.
The required angle is $180-68=112$ degrees.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite3

In “If two lines are parallel, their corresponding angles are equal”, identify the hypothesis.

The corresponding angles are equal.
The angles are supplementary.
The two lines are parallel.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

The hypothesis is the information after “if”.
Angle equality is the conclusion of this statement.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite4

One angle is $72^{\circ}$ . Find its vertically opposite angle; state the reason.

Locate both angles and identify their shared lines or vertex.
This relation gives equal angles.
Use the defining equality or angle sum and check the unit.

Use the angle relation stated in the question.
The required angle is $72=72$ degrees.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice5

Two parallel coplanar lines are cut by a transversal. One angle is $57^{\circ}$ . Find its corresponding angle; state the reason.

Locate both angles and identify their shared lines or vertex.
This relation gives equal angles.
Check that parallelism is given; then use the equality or supplementary sum.

Use the stated parallel condition before applying this angle relation.
The required angle is $57=57$ degrees.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice6

Two parallel coplanar lines are cut by a transversal. One angle is $113^{\circ}$ . Find its alternate interior angle; state the reason.

Locate both angles and identify their shared lines or vertex.
This relation gives equal angles.
Check that parallelism is given; then use the equality or supplementary sum.

Use the stated parallel condition before applying this angle relation.
The required angle is $113=113$ degrees.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice7

Two parallel coplanar lines are cut by a transversal. One angle is $124^{\circ}$ . Find its same-side interior angle; state the reason.

Locate both angles and identify their shared lines or vertex.
The two relevant parts have a fixed total of 180 degrees.
Check that parallelism is given; then use the equality or supplementary sum.

Use the stated parallel condition before applying this angle relation.
The required angle is $180-124=56$ degrees.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice8

Two coplanar lines are cut by a transversal. A pair of same-side interior angles are 72° and 108°. Write a short proof that the lines are parallel.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

72°+108°=180°, so this pair is supplementary.
They have the specified same-side interior positions; apply the converse test.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question9

Without a parallel condition, which statement remains always valid for two intersecting straight lines?

All corresponding angles at two intersections are equal.
Every pair of interior angles is supplementary.
Vertically opposite angles are equal.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Vertically opposite angles use one intersection only.
The other claims need additional conditions.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question10

Two parallel coplanar lines are cut by a transversal. One angle is $39^{\circ}$ . Find its same-side interior angle; state the reason.

Locate both angles and identify their shared lines or vertex.
The two relevant parts have a fixed total of 180 degrees.
Check that parallelism is given; then use the equality or supplementary sum.

Use the stated parallel condition before applying this angle relation.
The required angle is $180-39=141$ degrees.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice11

Two parallel coplanar lines are cut by a transversal. One angle is $46^{\circ}$ . Find its alternate interior angle; state the reason.

Locate both angles and identify their shared lines or vertex.
This relation gives equal angles.
Check that parallelism is given; then use the equality or supplementary sum.

Use the stated parallel condition before applying this angle relation.
The required angle is $46=46$ degrees.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice12

Two parallel coplanar lines are cut by a transversal. One angle is $95^{\circ}$ . Find its same-side interior angle; state the reason.

Locate both angles and identify their shared lines or vertex.
The two relevant parts have a fixed total of 180 degrees.
Check that parallelism is given; then use the equality or supplementary sum.

Use the stated parallel condition before applying this angle relation.
The required angle is $180-95=85$ degrees.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice13

A pair of corresponding angles are 55° and 65°. Can these given angles justify parallelism? Explain.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

The test needs equality, not merely both angles being acute.
55° differs from 65°, so parallelism would contradict the corresponding-angle property.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice14

In one plane, a and b are each perpendicular to the same line c. Explain why a and b are parallel.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Both relevant angles are 90°.
Their corresponding positions give the parallel-line test.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension15

A learner writes “a is parallel to b because corresponding angles are equal; these angles are equal because a is parallel to b.” No equality was given. Diagnose and repair the reasoning.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Neither step supplies independent evidence.
For example, two given corresponding angles of 60° establish equality first, then parallelism.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension16

The true statement “vertically opposite angles are equal” does not imply that every pair of equal angles are vertically opposite. Give a counterexample.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

A converse needs its own justification.
Right angles at (0,0) and (3,0) are both 90° but are not vertically opposite.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Preparation and assessment notes

Prerequisites

Know vertical/supplementary angles and identify corresponding, alternate and same-side interior positions.

Look for in students’ work

Use the worked and guided tasks selectively. The eight core paper tasks are prerequisites, independent tasks and exits; further practice and extensions are optional.

Collect the paper exit questions and note completion and common errors. This page does not grade or store paper answers.

Paper for the class

Download all worksheets (ZIP · 318 PDFs · 47.2 MB)

All 53 Year 7–8 upper- and lower-volume packs in Traditional Chinese and English. Includes teacher answers; student core, further practice and teacher papers are in separate folders.

Core student worksheet: two A4 pages. Further practice is separate. Teacher copies include reference reasoning.

Print edition

Textbook reference and authorship

Original 40-minute core consolidation after the mapped textbook lessons; not a replacement for every first-teaching period. Only the supplied text has been mapped.

大湾区初中数学三维课堂（七年级下册）正文.pdf

Year 7, lower volume · text pp. 9–24 · textbook lessons 5–12

Original questions and explanations; the source textbook and answer PDFs are not reproduced here.

Parallel lines: conditions and reasoning

Year 7, lower volume · text pp. 9–24 · textbook lessons 5–12 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Coplanar lines a and b are cut by a transversal. A pair of corresponding angles are both 64°. Prove that a is parallel to b.

Guided attempt2

Two parallel coplanar lines are cut by a transversal. One angle is $68^{\circ}$ . Find its same-side interior angle; state the reason.

Parallel lines give same-side interior sum $68+x=\square^{\circ}$ .
Complete $x=180-\square$ .
Write $x=\square^{\circ}$ and name the parallel-line property.

Prerequisite3

In “If two lines are parallel, their corresponding angles are equal”, identify the hypothesis.

The corresponding angles are equal.
The angles are supplementary.
The two lines are parallel.

Prerequisite4

One angle is $72^{\circ}$ . Find its vertically opposite angle; state the reason.

Parallel lines: conditions and reasoning

Year 7, lower volume · text pp. 9–24 · textbook lessons 5–12 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Two parallel coplanar lines are cut by a transversal. One angle is $57^{\circ}$ . Find its corresponding angle; state the reason.

Independent practice6

Two parallel coplanar lines are cut by a transversal. One angle is $113^{\circ}$ . Find its alternate interior angle; state the reason.

Independent practice7

Two parallel coplanar lines are cut by a transversal. One angle is $124^{\circ}$ . Find its same-side interior angle; state the reason.

Independent practice8

Two coplanar lines are cut by a transversal. A pair of same-side interior angles are 72° and 108°. Write a short proof that the lines are parallel.

Exit question9

Without a parallel condition, which statement remains always valid for two intersecting straight lines?

All corresponding angles at two intersections are equal.
Every pair of interior angles is supplementary.
Vertically opposite angles are equal.

Exit question10

Two parallel coplanar lines are cut by a transversal. One angle is $39^{\circ}$ . Find its same-side interior angle; state the reason.

Parallel lines: conditions and reasoning — Further practice

Year 7, lower volume · text pp. 9–24 · textbook lessons 5–12

Further practice11

Two parallel coplanar lines are cut by a transversal. One angle is $46^{\circ}$ . Find its alternate interior angle; state the reason.

Further practice12

Two parallel coplanar lines are cut by a transversal. One angle is $95^{\circ}$ . Find its same-side interior angle; state the reason.

Further practice13

A pair of corresponding angles are 55° and 65°. Can these given angles justify parallelism? Explain.

Further practice14

In one plane, a and b are each perpendicular to the same line c. Explain why a and b are parallel.

Extension15

A learner writes “a is parallel to b because corresponding angles are equal; these angles are equal because a is parallel to b.” No equality was given. Diagnose and repair the reasoning.

Extension16

The true statement “vertically opposite angles are equal” does not imply that every pair of equal angles are vertically opposite. Give a counterexample.

Parallel lines: conditions and reasoning — Teacher answers

Year 7, lower volume · text pp. 9–24 · textbook lessons 5–12

Use the worked and guided tasks selectively. The eight core paper tasks are prerequisites, independent tasks and exits; further practice and extensions are optional.

Ordinary route: prerequisites 4, short inquiry 5, worked/guided 8, independent 15, correction 5, exit 3 minutes. Advanced route: inquiry 18, organise 4, selected practice 13, exit 5 minutes.

Prerequisites: Know vertical/supplementary angles and identify corresponding, alternate and same-side interior positions.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: complete the full investigation.

Inquiry and discussion: 18 minutes using the existing parallels lab: predict which relations survive if parallelism is removed (3 min). Set α=60°, locked parallel, and record corresponding, alternate interior, same-side interior and vertical angles (4 min). Unlock and change only β to 75°, then back to 60°; keep α fixed (5 min). Identify which equalities or sums change and which vertical-angle fact does not (4 min). Give one justified converse argument, distinguishing a test for parallelism from a property that assumes it (2 min).

Worked example1

Coplanar lines a and b are cut by a transversal. A pair of corresponding angles are both 64°. Prove that a is parallel to b.

Reference answer

The corresponding angles are equal, so a is parallel to b.

Both marked angles are given as 64°, so they are equal.
Use the corresponding-angle test for parallel lines, not the property of lines already known parallel.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Guided attempt2

Two parallel coplanar lines are cut by a transversal. One angle is $68^{\circ}$ . Find its same-side interior angle; state the reason.

Parallel lines give same-side interior sum $68+x=\square^{\circ}$ .
Complete $x=180-\square$ .
Write $x=\square^{\circ}$ and name the parallel-line property.

Reference answer

$112$ °.

180-68

Use the stated parallel condition before applying this angle relation.
The required angle is $180-68=112$ degrees.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite3

In “If two lines are parallel, their corresponding angles are equal”, identify the hypothesis.

The corresponding angles are equal.
The angles are supplementary.
The two lines are parallel.

Reference answer

The two lines are parallel.

The hypothesis is the information after “if”.
Angle equality is the conclusion of this statement.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite4

One angle is $72^{\circ}$ . Find its vertically opposite angle; state the reason.

Reference answer

$72$ °.

72

Use the angle relation stated in the question.
The required angle is $72=72$ degrees.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice5

Two parallel coplanar lines are cut by a transversal. One angle is $57^{\circ}$ . Find its corresponding angle; state the reason.

Reference answer

$57$ °.

57

Use the stated parallel condition before applying this angle relation.
The required angle is $57=57$ degrees.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice6

Two parallel coplanar lines are cut by a transversal. One angle is $113^{\circ}$ . Find its alternate interior angle; state the reason.

Reference answer

$113$ °.

113

Use the stated parallel condition before applying this angle relation.
The required angle is $113=113$ degrees.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice7

Two parallel coplanar lines are cut by a transversal. One angle is $124^{\circ}$ . Find its same-side interior angle; state the reason.

Reference answer

$56$ °.

180-124

Use the stated parallel condition before applying this angle relation.
The required angle is $180-124=56$ degrees.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice8

Two coplanar lines are cut by a transversal. A pair of same-side interior angles are 72° and 108°. Write a short proof that the lines are parallel.

Reference answer

Their sum is 180°, so the lines are parallel by the same-side-interior-angle test.

72°+108°=180°, so this pair is supplementary.
They have the specified same-side interior positions; apply the converse test.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question9

Without a parallel condition, which statement remains always valid for two intersecting straight lines?

All corresponding angles at two intersections are equal.
Every pair of interior angles is supplementary.
Vertically opposite angles are equal.

Reference answer

Vertically opposite angles are equal.

Vertically opposite angles use one intersection only.
The other claims need additional conditions.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question10

Two parallel coplanar lines are cut by a transversal. One angle is $39^{\circ}$ . Find its same-side interior angle; state the reason.

Reference answer

$141$ °.

180-39

Use the stated parallel condition before applying this angle relation.
The required angle is $180-39=141$ degrees.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice11

Two parallel coplanar lines are cut by a transversal. One angle is $46^{\circ}$ . Find its alternate interior angle; state the reason.

Reference answer

$46$ °.

46

Use the stated parallel condition before applying this angle relation.
The required angle is $46=46$ degrees.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice12

Two parallel coplanar lines are cut by a transversal. One angle is $95^{\circ}$ . Find its same-side interior angle; state the reason.

Reference answer

$85$ °.

180-95

Use the stated parallel condition before applying this angle relation.
The required angle is $180-95=85$ degrees.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice13

A pair of corresponding angles are 55° and 65°. Can these given angles justify parallelism? Explain.

Reference answer

No. They are unequal; in this configuration the lines are not parallel.

The test needs equality, not merely both angles being acute.
55° differs from 65°, so parallelism would contradict the corresponding-angle property.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice14

In one plane, a and b are each perpendicular to the same line c. Explain why a and b are parallel.

Reference answer

Corresponding right angles are equal; therefore a is parallel to b.

Both relevant angles are 90°.
Their corresponding positions give the parallel-line test.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension15

A learner writes “a is parallel to b because corresponding angles are equal; these angles are equal because a is parallel to b.” No equality was given. Diagnose and repair the reasoning.

Reference answer

This is circular. Supply an independent given angle equality or another valid parallel test before claiming parallelism.

Neither step supplies independent evidence.
For example, two given corresponding angles of 60° establish equality first, then parallelism.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension16

The true statement “vertically opposite angles are equal” does not imply that every pair of equal angles are vertically opposite. Give a counterexample.

Reference answer

No. Two separate right angles are equal but need not share a vertex.

A converse needs its own justification.
Right angles at (0,0) and (3,0) are both 90° but are not vertically opposite.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.