PAPER · REASONING · PRACTICE

Intersections, perpendiculars and distance

Distinguish angle position from angle size, and perpendicular distance from an oblique segment.

40 minutes · Year 7, lower volume · text pp. 1–8 · textbook lessons 1–4

Textbook reference: Year 7 lower · Chapter 5 Intersecting and parallel lines · text lessons 1–4

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, paper and ruler: mark P above a horizontal line l and predict the shortest segment from P to l (3 min). Draw the perpendicular foot H, with A and B on l four equal grid squares to the left and right of H. Use the given lengths PH=3 and PA=PB=5 to compare distance with an arbitrary segment (7 min). Change only the visible segment: its left endpoint is one square right of H and its right endpoint four squares right of H. H is outside that segment but remains on line l (5 min). State the distance definition and explain why measuring a few cases is not a general proof (3 min); no coordinate notation is needed.

Organise the key idea

Distinguish angle position from angle size, and perpendicular distance from an oblique segment.

Open the original chapter

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

One angle is $58^{\circ}$ . 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 $58=58$ degrees.

Check your work

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

Guided attempt2

One angle is $34^{\circ}$ . Find its other part of the right angle; state the reason.

Let the missing angle be x: $x+34=90$ .
Subtract the known part: $x=90-\square$ .
Complete $x=\square^{\circ}$ and check the sum.

The two parts fill a right angle, not a straight angle.
Their total is 90°, and one part is 34°.
Subtract 34 from 90, then add the result back to 34 to check.

Use the angle relation stated in the question.
The required angle is $90-34=56$ degrees.

Check your work

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

Prerequisite3

One angle is $120^{\circ}$ . Find its adjacent supplementary 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.
Use the defining equality or angle sum and check the unit.

Use the angle relation stated in the question.
The required angle is $180-120=60$ degrees.

Check your work

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

Prerequisite4

Two straight lines AB and CD meet at O. Which angle is vertically opposite to $\angle AOC$ ?

$\angle BOD$
$\angle AOD$
$\angle COB$

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.

OA and OB are opposite rays; OC and OD are opposite rays.
The opposite pair is AOC and BOD.

Check your work

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

Independent practice5

One angle is $137^{\circ}$ . 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 $137=137$ degrees.

Check your work

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

Independent practice6

One angle is $31^{\circ}$ . Find its other part of the right angle; state the reason.

Locate both angles and identify their shared lines or vertex.
The two relevant parts have a fixed total of 90 degrees.
Use the defining equality or angle sum and check the unit.

Use the angle relation stated in the question.
The required angle is $90-31=59$ degrees.

Check your work

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

Independent practice7

P is 3 cm above line AB. PH is perpendicular to AB at H, PH=3 cm and PA=5 cm. What is the distance from P to AB? Explain your choice.

Geometry diagram; not drawn to scale except for construction tasks.

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.

Distance to a line is the perpendicular segment length, not an oblique segment.
PH is perpendicular, so the distance is 3 cm; PA is irrelevant to this distance.

Check your work

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

Independent practice8

A transversal cuts two lines. Two marked angles lie between the lines and on opposite sides of the transversal. Name the pair; no parallelism is given.

Same-side interior angles; always supplementary.
Alternate interior angles; equality is not yet justified.
Corresponding angles; always 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.

Position identifies alternate interior angles.
Equal size requires an additional condition such as parallelism.

Check your work

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

Exit question9

Which segment length defines the distance from a point to a line?

Any segment ending on the line.
The longest segment visible in the drawing.
The perpendicular segment to the line.

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.

Distance is defined using the perpendicular foot.
An oblique segment is generally longer.

Check your work

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

Exit question10

One angle is $47^{\circ}$ . Find its adjacent supplementary 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.
Use the defining equality or angle sum and check the unit.

Use the angle relation stated in the question.
The required angle is $180-47=133$ degrees.

Check your work

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

Further practice11

One angle is $82^{\circ}$ . 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 $82=82$ degrees.

Check your work

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

Further practice12

One angle is $151^{\circ}$ . Find its adjacent supplementary 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.
Use the defining equality or angle sum and check the unit.

Use the angle relation stated in the question.
The required angle is $180-151=29$ degrees.

Check your work

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

Further practice13

One angle is $55^{\circ}$ . Find its other part of the right angle; state the reason.

Locate both angles and identify their shared lines or vertex.
The two relevant parts have a fixed total of 90 degrees.
Use the defining equality or angle sum and check the unit.

Use the angle relation stated in the question.
The required angle is $90-55=35$ degrees.

Check your work

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

Further practice14

On the coordinate grid, draw the perpendicular from P(1,3) to the horizontal line through A(−3,0), B(4,0). Label the foot H.

Use the labelled axes and unit grid.

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 horizontal line has a vertical perpendicular through P.
Keep x=1 and set y=0 to reach the line.

Check your work

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

Extension15

A learner says PA=5 cm must be the distance to AB because A lies on AB. Use PH=3 cm, PH perpendicular to AB, to correct the claim.

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.

Compare the perpendicular with the oblique segment.
The given counterexample has lengths 3 and 5; only PH meets the definition.

Check your work

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

Extension16

Q lies on line AB. What is its distance to AB? Explain without inventing a positive gap.

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.

Q already lies on the line.
The minimum distance is zero.

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

Recognise points, rays, straight angles, right angles and degree measures.

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. 1–8 · textbook lessons 1–4

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

Intersections, perpendiculars and distance

Year 7, lower volume · text pp. 1–8 · textbook lessons 1–4 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

One angle is $58^{\circ}$ . Find its vertically opposite angle; state the reason.

Guided attempt2

One angle is $34^{\circ}$ . Find its other part of the right angle; state the reason.

Let the missing angle be x: $x+34=90$ .
Subtract the known part: $x=90-\square$ .
Complete $x=\square^{\circ}$ and check the sum.

Prerequisite3

One angle is $120^{\circ}$ . Find its adjacent supplementary angle; state the reason.

Prerequisite4

Two straight lines AB and CD meet at O. Which angle is vertically opposite to $\angle AOC$ ?

$\angle BOD$
$\angle AOD$
$\angle COB$

Intersections, perpendiculars and distance

Year 7, lower volume · text pp. 1–8 · textbook lessons 1–4 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

One angle is $137^{\circ}$ . Find its vertically opposite angle; state the reason.

Independent practice6

One angle is $31^{\circ}$ . Find its other part of the right angle; state the reason.

Independent practice7

P is 3 cm above line AB. PH is perpendicular to AB at H, PH=3 cm and PA=5 cm. What is the distance from P to AB? Explain your choice.

Geometry diagram; not drawn to scale except for construction tasks.

Independent practice8

A transversal cuts two lines. Two marked angles lie between the lines and on opposite sides of the transversal. Name the pair; no parallelism is given.

Same-side interior angles; always supplementary.
Alternate interior angles; equality is not yet justified.
Corresponding angles; always equal.

Exit question9

Which segment length defines the distance from a point to a line?

Any segment ending on the line.
The longest segment visible in the drawing.
The perpendicular segment to the line.

Exit question10

One angle is $47^{\circ}$ . Find its adjacent supplementary angle; state the reason.

Intersections, perpendiculars and distance — Further practice

Year 7, lower volume · text pp. 1–8 · textbook lessons 1–4

Further practice11

One angle is $82^{\circ}$ . Find its vertically opposite angle; state the reason.

Further practice12

One angle is $151^{\circ}$ . Find its adjacent supplementary angle; state the reason.

Further practice13

One angle is $55^{\circ}$ . Find its other part of the right angle; state the reason.

Further practice14

On the coordinate grid, draw the perpendicular from P(1,3) to the horizontal line through A(−3,0), B(4,0). Label the foot H.

Use the labelled axes and unit grid.

Extension15

A learner says PA=5 cm must be the distance to AB because A lies on AB. Use PH=3 cm, PH perpendicular to AB, to correct the claim.

Extension16

Q lies on line AB. What is its distance to AB? Explain without inventing a positive gap.

Intersections, perpendiculars and distance — Teacher answers

Year 7, lower volume · text pp. 1–8 · textbook lessons 1–4

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: Recognise points, rays, straight angles, right angles and degree measures.

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

Inquiry and discussion: 18 minutes, paper and ruler: mark P above a horizontal line l and predict the shortest segment from P to l (3 min). Draw the perpendicular foot H, with A and B on l four equal grid squares to the left and right of H. Use the given lengths PH=3 and PA=PB=5 to compare distance with an arbitrary segment (7 min). Change only the visible segment: its left endpoint is one square right of H and its right endpoint four squares right of H. H is outside that segment but remains on line l (5 min). State the distance definition and explain why measuring a few cases is not a general proof (3 min); no coordinate notation is needed.

Worked example1

One angle is $58^{\circ}$ . Find its vertically opposite angle; state the reason.

Reference answer

$58$ °.

58

Use the angle relation stated in the question.
The required angle is $58=58$ degrees.

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

Guided attempt2

One angle is $34^{\circ}$ . Find its other part of the right angle; state the reason.

Let the missing angle be x: $x+34=90$ .
Subtract the known part: $x=90-\square$ .
Complete $x=\square^{\circ}$ and check the sum.

Reference answer

$56$ °.

90-34

Use the angle relation stated in the question.
The required angle is $90-34=56$ degrees.

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

Prerequisite3

One angle is $120^{\circ}$ . Find its adjacent supplementary angle; state the reason.

Reference answer

$60$ °.

180-120

Use the angle relation stated in the question.
The required angle is $180-120=60$ degrees.

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

Prerequisite4

Two straight lines AB and CD meet at O. Which angle is vertically opposite to $\angle AOC$ ?

$\angle BOD$
$\angle AOD$
$\angle COB$

Reference answer

$\angle BOD$

OA and OB are opposite rays; OC and OD are opposite rays.
The opposite pair is AOC and BOD.

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

Independent practice5

One angle is $137^{\circ}$ . Find its vertically opposite angle; state the reason.

Reference answer

$137$ °.

137

Use the angle relation stated in the question.
The required angle is $137=137$ degrees.

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

Independent practice6

One angle is $31^{\circ}$ . Find its other part of the right angle; state the reason.

Reference answer

$59$ °.

90-31

Use the angle relation stated in the question.
The required angle is $90-31=59$ degrees.

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

Independent practice7

P is 3 cm above line AB. PH is perpendicular to AB at H, PH=3 cm and PA=5 cm. What is the distance from P to AB? Explain your choice.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

$3$ cm.

3

Distance to a line is the perpendicular segment length, not an oblique segment.
PH is perpendicular, so the distance is 3 cm; PA is irrelevant to this distance.

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

Independent practice8

A transversal cuts two lines. Two marked angles lie between the lines and on opposite sides of the transversal. Name the pair; no parallelism is given.

Same-side interior angles; always supplementary.
Alternate interior angles; equality is not yet justified.
Corresponding angles; always equal.

Reference answer

Alternate interior angles; equality is not yet justified.

Position identifies alternate interior angles.
Equal size requires an additional condition such as parallelism.

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

Exit question9

Which segment length defines the distance from a point to a line?

Any segment ending on the line.
The longest segment visible in the drawing.
The perpendicular segment to the line.

Reference answer

The perpendicular segment to the line.

Distance is defined using the perpendicular foot.
An oblique segment is generally longer.

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

Exit question10

One angle is $47^{\circ}$ . Find its adjacent supplementary angle; state the reason.

Reference answer

$133$ °.

180-47

Use the angle relation stated in the question.
The required angle is $180-47=133$ degrees.

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

Further practice11

One angle is $82^{\circ}$ . Find its vertically opposite angle; state the reason.

Reference answer

$82$ °.

82

Use the angle relation stated in the question.
The required angle is $82=82$ degrees.

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

Further practice12

One angle is $151^{\circ}$ . Find its adjacent supplementary angle; state the reason.

Reference answer

$29$ °.

180-151

Use the angle relation stated in the question.
The required angle is $180-151=29$ degrees.

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

Further practice13

One angle is $55^{\circ}$ . Find its other part of the right angle; state the reason.

Reference answer

$35$ °.

90-55

Use the angle relation stated in the question.
The required angle is $90-55=35$ degrees.

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

Further practice14

On the coordinate grid, draw the perpendicular from P(1,3) to the horizontal line through A(−3,0), B(4,0). Label the foot H.

Use the labelled axes and unit grid.

Reference answer

$H(1,0)$ ; PH is vertical.

A horizontal line has a vertical perpendicular through P.
Keep x=1 and set y=0 to reach the line.

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

Use the labelled axes and unit grid.

Extension15

A learner says PA=5 cm must be the distance to AB because A lies on AB. Use PH=3 cm, PH perpendicular to AB, to correct the claim.

Reference answer

The distance is 3 cm. An endpoint on the line does not make the segment perpendicular.

Compare the perpendicular with the oblique segment.
The given counterexample has lengths 3 and 5; only PH meets the definition.

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

Extension16

Q lies on line AB. What is its distance to AB? Explain without inventing a positive gap.

Reference answer

$0$ cm.

0

Q already lies on the line.
The minimum distance is zero.

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