Angles on a transversal: a week of ready-to-teach maths

Five lessons for Year 7 Measurement. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus wedges to slide and turn, find-the-angle cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets: wedges to slide and turn (Day 2), one per pair, and find-the-angle cards (Day 4), one set per pair.
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

Where two lines cross, four angles meet: the angles opposite each other are equal, and two angles side by side make a straight line, 180°. When a transversal crosses two parallel lines, the crossing at one line is an exact copy of the crossing at the other, slid along the transversal. So corresponding angles, in matching positions (an F), are equal; alternate angles, between the lines on opposite sides of the transversal (a Z), are equal; and co-interior angles, between the lines on the same side (a C), add to 180°. One known angle fixes all eight, and the reason for an answer is the relationship that gave it.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write angles

Matching arrows on two lines mean they are parallel. Write a reason with every angle found: x = 64° (corresponding angles, parallel lines). The angles on these sheets are drawn true to size, so a protractor can check an answer, but the reason is the answer. Protractors are needed on Days 2 and 5.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 revises the angles at one crossing and names the eight angles at two; Day 2 slides one crossing onto the other to find corresponding angles; Day 3 adds alternate and co-interior angles; Day 4 chains the relationships and writes the reasons; and Day 5 turns them round to test whether lines are parallel. It builds on angles on a line and at a point (AC9M6M04) and leads into the angle sum of a triangle (AC9M7M05), which is proved with a parallel line.

Materials for the week (one trip)

• From the classroom: pencils, rulers, protractors, scissors, coloured pencils, small envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: two figures and four wedges to slide and turn (Day 2).
• Cut-out sheet 2, one per pair: 8 find-the-angle cards (Day 4).
• Coloured pencils for tracing the F, Z and C shapes (Days 2 to 4).

Words for the word wall

parallel, transversal, intersect, vertically opposite, adjacent, supplementary, angles on a straight line, angles at a point, interior, corresponding, alternate, co-interior (allied), reason

Dear families

This week in maths, Year 7 is learning the angles made when a line crosses two parallel lines, and how one angle tells you all the others, with a reason for each.

Try this at home

• Look for parallel lines crossed by another line: a road across railway tracks, the rungs and rails of a ladder, a gate with a brace.
• Find an F, a Z and a C in a window frame, a fence or a tiled floor.
• Ask: if one angle where a road crosses a railway line is 75°, what are the others? (75° and 105°.)
• Ask for the reason too: which angles are equal, and which add to 180°?

Angles this week

Fill one row a day with a place you saw parallel lines and a transversal, a pair of angles there, and how they are related.

Printed from the free seegongsik Angles on a transversal teaching pack · seegongsik.com/au/y7/measurement/AC9M7M04/pack

Angles at a crossing

Where two lines cross, the opposite angles are equal and the angles side by side add to 180°. Students find angles at one crossing, on a straight line and at a point, then name the eight angles where a transversal crosses two parallel lines.

We are learning to

• find vertically opposite angles, and angles on a straight line and at a point,
• use the words transversal, parallel and between the lines,
• name the eight angles at two crossings with letters.

Success criteria

• I can find the other three angles at a crossing from one.
• I can explain why vertically opposite angles are equal.

You need

The worksheet, rulers and coloured pencils.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the eight angles; start the second with Part A.

Watch for these ideas

• Taking the angle beside a given angle as equal to it. Side by side, they make a straight line: 180°.
• Adding to 360° for angles on a straight line, or to 180° for angles all the way round a point.
• Lettering the angles in any order. Agree on one order so everyone can talk about angle c.

Answers

• Part A: a = 108°, b = 72°, c = 108°; d = 33°, e = 33°, f = 147°.
• Part B: (a) x = 180° − 41° = 139°; (b) y = 180° − 53° − 45° = 82°; (c) z = 360° − 142° − 96° − 74° = 48°.
• Part C: 1. c; 2. c, d, e and f; 3. f and g; 4. a = 118°, c = 62°, d = 118°; 5. any prediction; Day 2 tests it (f and g).
• Part D: a + b = 180° on the line, and b + d = 180° along the transversal, so a and d are both 180° − b: they are equal.

Angles at a crossing

A. One crossing

Find each lettered angle. The angles are drawn true to size.

B. On a line and round a point

C. Eight angles

D. Stretch

Use angles on a straight line to explain why vertically opposite angles, such as a and d, are always equal.

Corresponding angles

On parallel lines, the bottom crossing is a copy of the top one, slid along the transversal. Students slide cut-out wedges, list the corresponding pairs, and measure to compare parallel lines with lines that are not.

We are learning to

• find corresponding angles, in matching positions at the two crossings,
• know that corresponding angles are equal only when the lines are parallel,
• find an unknown angle from its corresponding angle.

Success criteria

• I can trace an F to find a pair of corresponding angles.
• I can explain why corresponding angles on parallel lines are equal.

You need

Cut-out sheet 1 (one per pair), scissors, protractors, coloured pencils and the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after slide and turn; start the second with Part B.

Watch for these ideas

• Calling any two equal-looking angles corresponding. They must sit in the same position at each crossing.
• Using the rule when the lines are not parallel. Without parallel lines the angles differ, as on figure Q.
• Reading the wrong scale on the protractor, which gives 180° minus the angle.

Answers

• Part A: slid, a fits e, b fits f, c fits g, d fits h; turned, a fits d and h, b fits c and g, c fits b and f, d fits a and e. On figure Q nothing fits: the bottom angles are 10° off.
• Part B: (a) p = 57°, q = 57°; (b) p = 57°, q = 49°; accept answers within 2°. The lines in (a) are parallel.
• Part C: (a) x = 46°; (b) x = 113°; (c) x = 78°, each equal to its corresponding angle.
• Part D: draw a line through the point that crosses the given line and measure the angle there; at the point, draw the same angle in the corresponding position. The new line is parallel.

Corresponding angles

A. Slide and turn

Use the wedges and figure P from cut-out sheet 1.

The pairs in the middle column are corresponding angles. What shape do you trace from one to the other?   ______

B. Measure and decide

Measure p and q with a protractor.

Which pair of lines is parallel? How do the angles tell you?   ______________________________

C. Find x

D. Stretch

Explain how you could draw a line through a point, parallel to a line you are given, using only a ruler and a protractor.

Alternate and co-interior

Between the parallel lines, angles on opposite sides of the transversal are alternate and equal (a Z), and angles on the same side are co-interior and add to 180° (a C). Students list the pairs, name pairs, and find unknown angles.

We are learning to

• find alternate angles by a Z and co-interior angles by a C,
• know that alternate angles are equal and co-interior angles add to 180°,
• name any marked pair on parallel lines.

Success criteria

• I can say whether a pair is corresponding, alternate, co-interior or vertically opposite.
• I can find an unknown angle from its alternate or co-interior partner.

You need

The worksheet, coloured pencils and the figures from yesterday.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Copying a co-interior angle as if it were equal. Co-interior angles add to 180°.
• Calling a pair alternate when one angle is outside the lines. Both alternate angles sit between the lines.
• Naming by how big the angles look, not by where they sit.

Answers

• Part A: alternate c and f, d and e; co-interior c and e, d and f.
• Part B: 1 alternate, 2 vertically opposite, 3 corresponding, 4 co-interior, 5 corresponding, 6 alternate.
• Part C: (a) x = 127°, alternate; (b) x = 141°, co-interior: 180° − 39°; (c) x = 84°, alternate; (d) x = 77°, co-interior: 180° − 103°.
• Part D: take d and f. f equals b, its corresponding angle, and b + d = 180° because they make a straight line along the transversal. So d + f = 180°.

Alternate and co-interior

A. The pairs

B. Name the pair

Is each pair p and q corresponding, alternate, co-interior or vertically opposite?

C. Find x

Write x and the name of the pair you used.

D. Stretch

On the figure in Part A, explain why the co-interior angles d and f add to 180°. Use a corresponding pair and a straight line.

Reasons

A found angle is only as good as its reason. Students find angles in one and two steps, write the relationship and the parallel lines behind each step, and work through the find-the-angle cards.

We are learning to

• give a reason for every angle found,
• find an angle in two steps, one reason per step,
• check that the lines are parallel before using the rules.

Success criteria

• I can write x = 72° with a reason such as “alternate angles, parallel lines”.
• I can fill in all eight angles from one.

You need

Find-the-angle cards from cut-out sheet 2 (one set per pair, cut out before the lesson) and the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the card carousel; start the second with Part A.

Watch for these ideas

• A number with no reason, or a reason with no relationship named, such as “they look the same”.
• Jumping two steps at once and naming the wrong pair. Find the angle in between first.
• Using the rules on lines that are not marked parallel, as on card 6.

Answers

• Cards: 1 x = 58° (corresponding); 2 x = 132° (alternate); 3 x = 104° (co-interior); 4 x = 144° (vertically opposite, then corresponding); 5 x = 68° (co-interior with the angle above x, then vertically opposite); 6 x cannot be found (the lines are not parallel); 7 x = 73° (two alternate angles, by a parallel line through the bend); 8 x = 54°, y = 126° (x on a straight line, y corresponding); card 6 measures 66°, but no rule gives it.
• Part A: b = 47°, c = 47°, d = 133°, e = 133°, f = 47°, g = 47°, h = 133°.
• Part B: (a) x = 114° (alternate); (b) x = 119° (co-interior); (c) x = 86° (corresponding), each on parallel lines.
• Part C: x = 117°, vertically opposite to the given angle; y = 180° − 117° = 63°, co-interior with x.
• Part D: a parallel line through the bend splits x into 29° and 46° (two Zs), so x = 75°.

Reasons

A. One angle fixes all eight

B. Say why

C. Two steps

D. Stretch

Parallel or not?

Equal corresponding or alternate angles, or co-interior angles that add to 180°, mean the lines are parallel. Students use this test, solve problems with reasons, then play the quiz.

We are learning to

• test whether two lines are parallel from a pair of angles,
• solve problems on parallel lines in steps, with a reason for each,
• use an extra parallel line to find the angle at a bend.

Success criteria

• I can decide whether two lines are parallel and say why.
• I can solve a problem in two steps with a reason for each.

You need

The worksheet, protractors for checking, and the unit’s self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after problem 3; start the second with problem 4.

Watch for these ideas

• Trusting the picture. Lines that look parallel may not be: test them with the angles, as in problem 2.
• Testing corresponding angles by adding to 180°. Only co-interior angles add to 180°.
• Losing track in a two-step problem. Mark the angle in between, then write one reason per step.

Answers with working

1. x = 71°: the angle above x on the right is 71° (corresponding), and x is vertically opposite to it.
2. No: the marked angles are corresponding, and 83° is not 87°.
3. Yes: the marked angles are co-interior, and 107° + 73° = 180°.
4. x = 180° − 37° = 143° (co-interior).
5. x = 62°: the angle above x is 118° (corresponding), and x makes a straight line with it.
6. x = 36° + 52° = 88°.
7. 3x = x + 48 (alternate angles), so x = 24 and each angle is 72°.
8. A line through the bend, parallel to the fences, splits x into two angles, each alternate to a given angle.

Parallel or not?

Arrows mean parallel. Give a reason with every answer.

Problems (7 and 8 are the stretch)

7. Stretch. Two alternate angles on parallel lines are 3x° and (x + 48)°. Find x and the size of each angle.
8. Stretch. Explain why the angle at the bend in problem 6 is the sum of the other two angles.

Slide and turn

Day 2. Cut out the four wedges. Each is a copy of an angle at the top crossing of figure P. Slide each wedge down the transversal to the bottom crossing and find the angle it fits. Then turn it half a turn and find every angle it fits now. Last, try the wedges on figure Q.

Cut along the edges of each wedge. The dot is the corner of the angle.

Find-the-angle cards

Cut out the 8 cards. Day 4: find each unknown angle and write the reason. Arrows on two lines mean the lines are parallel.

What we know: Angles on a transversal

Work on your own. Arrows mean parallel. Give a reason with every angle you find.

Answers and marking guide

Answers with working

1. Co-interior: both between the lines, on the same side of the transversal (a C).
2. Alternate: both between the lines, on opposite sides of the transversal (a Z).
3. x = 69°: corresponding angles on parallel lines are equal.
4. x = 180° − 128° = 52°.
5. x = 180° − 151° = 29° (angles on a straight line); y = 29° (alternate to x, parallel lines).
6. Co-interior angles on parallel lines add to 180°, so x = 180° − 128°.
7. No: the marked angles are co-interior, and 101° + 81° = 182°, not 180°.
8. x = 34° + 45° = 79°: a line through the bend, parallel to the fences, makes two Zs.

A quick three-level guide

Eight questions, four ideas. A student at standard names the three pairs, copies equal angles and subtracts co-interior ones from 180°, gives the relationship and the parallel lines as the reason, and tests whether lines are parallel.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: angles at a crossing, corresponding angles, alternate and co-interior angles, reasons, and problems.