Interior angle sum of a triangle: 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 triangles to tear, loop 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: triangles to tear (Day 1), one per pair, and loop cards (Day 5), one set for the class.
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

Tear the three corners off any triangle and they fit together along a straight line, so the three angles add to 180°. A line through one corner, parallel to the opposite side, shows why: it makes two alternate angles equal to the other two corners, and the three angles at that corner fill a straight line. So any two angles of a triangle give the third. A polygon with n sides splits into n − 2 triangles from one corner, so its angles add to (n − 2) × 180°: 360° for a quadrilateral, 540° for a pentagon.

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

Write the rule, then the working: x = 180° − 83° − 38° = 59° (angle sum of a triangle). Matching marks on two sides mean they are equal, and a small square is a right angle. The shapes on these sheets are drawn true to size, so a protractor can check an answer. Protractors are needed on Day 1.

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 shows the 180° by tearing and measuring; Day 2 proves it with a parallel line; Day 3 uses it in special triangles and for exterior angles; Day 4 extends it to any polygon; and Day 5 brings everything together in problems. It builds on angles on parallel lines (AC9M7M04) and leads into the angle properties of quadrilaterals and polygons in Year 8.

Materials for the week (one trip)

• From the classroom: pencils, rulers, protractors, scissors, glue sticks, this pack printed.
• Cut-out sheet 1, one per pair: six triangles to tear (Day 1).
• Cut-out sheet 2, one set for the class: 10 loop cards (Day 5).
• Scrap paper for students to draw and tear their own triangles (Day 1).

Words for the word wall

interior angle, angle sum, triangle, acute, obtuse, right-angled, isosceles, equilateral, scalene, apex, base angle, exterior angle, polygon, quadrilateral, pentagon, hexagon, regular, diagonal, proof

Dear families

This week in maths, Year 7 is learning that the three angles of every triangle add to 180°, and using it to find unknown angles in triangles and other shapes.

Try this at home

• Cut any triangle from scrap paper, tear off the three corners and fit them together. Do they make a straight line?
• Find triangles at home: a coat hanger, a roof, a ramp, a sandwich cut corner to corner.
• Ask: if two angles of a triangle are 50° and 45°, what is the third? (85°.)
• Count the sides of a stop sign or a 50c coin. How many triangles do you get by drawing lines from one corner?

Angles this week

Fill one row a day with a shape from class, the angles you knew, and the angle you worked out.

Printed from the free seegongsik Interior angle sum of a triangle teaching pack · seegongsik.com/au/y7/measurement/AC9M7M05/pack

Tear the corners

Tear the corners off any triangle and they line up along a straight line: the three angles add to 180°. Students tear six very different triangles, measure and add the angles of three more, and decide which sets of three angles can make a triangle.

We are learning to

• show by tearing that the angles of a triangle make a straight line,
• measure the angles of a triangle and check their sum,
• decide whether three angles can be the angles of a triangle.

Success criteria

• I can show with torn corners that a triangle’s angles add to 180°.
• I can explain why a measured sum might be 179° or 181°.

You need

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

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Mixing the corners of two triangles. Keep each triangle’s three corners together.
• Expecting a measured sum of exactly 180°. A sum of 179° or 181° is careful measuring, not a different rule.
• Thinking a bigger triangle has a bigger angle sum. Size changes the sides, not the angles.

Answers

• Part A: every set of three corners fills the straight line exactly, whatever the triangle: 180°.
• Part B: (a) 49° + 59° + 72° = 180°; (b) 28° + 48° + 104° = 180°; (c) 90° + 33° + 57° = 180°. Accept each angle within 2°, so sums from 177° to 183°.
• Part C: (a) no, 190°; (b) yes, 180°; (c) yes, 180°; (d) no, 179°.
• Part D: no. Two obtuse angles already add to more than 180°, leaving nothing for the third angle.

Tear the corners

A. Line up the corners

Glue each triangle’s three torn corners on its line, with their points on the dot.

What do the six lines show?   ________________________________________

B. Measure and add

Measure each angle with a protractor, then add the three.

C. Could these be the angles of a triangle?

1. 95°, 45° and 50°   yes / no, because ______________________
2. 47°, 62° and 71°   yes / no, because ______________________
3. 120°, 30° and 30°   yes / no, because ______________________
4. 89°, 46° and 44°   yes / no, because ______________________

D. Stretch

Can a triangle have two obtuse angles? Explain.

Why 180°

Tearing shows the rule; a parallel line proves it. Draw a line through the top corner parallel to the base: the two angles it makes with the sides are alternate to the base angles, and the three angles at the top fill a straight line. Students complete the proof, then use it to find third angles.

We are learning to

• prove the angle sum of a triangle with a parallel line and alternate angles,
• find the third angle of a triangle from the other two,
• write the working with its reason.

Success criteria

• I can explain the proof with a parallel line in my own words.
• I can find any third angle, with whole or decimal degrees.

You need

The worksheet, rulers and yesterday’s torn corners.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the proof; start the second with the board.

Watch for these ideas

• Taking the proof for a check. Tearing shows a few triangles; the parallel line works for any triangle.
• Subtracting only one angle from 180°, or adding the two known angles and stopping.
• Writing the right angle’s 90° as missing. A small square is a known 90°.

Answers

• Part A: p = b and q = c (alternate angles, parallel lines); p + a + q = 180° (a straight line); so a + b + c = 180°.
• Part B: (a) x = 67°; (b) x = 35°; (c) x = 63°; (d) x = 37°.
• Part C: (a) 180° − 17° − 68° = 95°; (b) 180° − 90° − 44.5° = 45.5°; (c) 180° − 33.5° − 104° = 42.5°.
• Part D: (a) x + 2x + 3x = 6x = 180°, so x = 30° and the angles are 30°, 60°, 90°; (b) 3x + 33 = 180, so x = 49° and the angles are 49°, 75° and 56°.

Why 180°

A. The proof

B. Find x

C. No picture

1. Two angles of a triangle are 17° and 68°. Find the third.   ____________
2. Two angles of a triangle are 90° and 44.5°. Find the third.   ____________
3. Two angles of a triangle are 33.5° and 104°. Find the third.   ____________

D. Stretch

(a) The angles of a triangle are x, 2x and 3x. Find each angle.
(b) The angles of a triangle are x, x + 26° and x + 7°. Find each angle.

Special triangles

The angle sum does more with a special triangle: equal sides give equal base angles, a right angle leaves 90° for the other two, and an exterior angle equals the two interior angles opposite it. Students use each, then explain the exterior angle rule.

We are learning to

• find the angles of isosceles, right-angled and equilateral triangles,
• find an exterior angle of a triangle,
• explain each answer from the angle sum.

Success criteria

• I can find both base angles of an isosceles triangle from its apex, and the apex from a base angle.
• I can explain why an exterior angle equals the two opposite interior angles.

You need

The worksheet and rulers.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Halving 180° and stopping. Take the apex away first, then halve what is left.
• Doubling a base angle for the apex, instead of taking both base angles from 180°.
• Taking an exterior angle for the inside angle next to it. The two make a straight line.

Answers

• Part A: (a) x = (180° − 54°) ÷ 2 = 63°; (b) x = 180° − 2 × 71° = 38°; (c) x = (180° − 136°) ÷ 2 = 22°.
• Part B: (a) x = 90° − 41° = 49°; (b) x = 90° ÷ 2 = 45°; (c) each half has 30°, 60° and 90°: the cut halves the 60° at the top and meets the base at a right angle.
• Part C: (a) x = 48° + 75° = 123°; (b) x = 139° − 61° = 78°.
• Part D: the exterior angle and the inside angle next to it make 180°, and so do that inside angle and the other two. Both pairs make 180° with the same angle, so the exterior angle equals the other two together.

Special triangles

Matching marks show equal sides. A small square is a right angle.

A. Isosceles triangles

B. Right angles and equal angles

C. Exterior angles

The base of each triangle is carried on past a corner. Find x.

D. Stretch

Explain why an exterior angle of a triangle always equals the two interior angles opposite it added together.

Other shapes

From one corner, diagonals split a polygon with n sides into n − 2 triangles, so its angles add to (n − 2) × 180°. Students split shapes, build the table, find unknown angles, and find the angles of regular polygons.

We are learning to

• split a polygon into triangles from one corner,
• find the angle sum of a polygon with (n − 2) × 180°,
• find an unknown angle, and each angle of a regular polygon.

Success criteria

• I can explain why a quadrilateral’s angles add to 360°.
• I can find the missing angle of a quadrilateral or a pentagon.

You need

The worksheet, rulers and coloured pencils.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Splitting from a point inside the shape, which adds a full turn of 360° at the centre that is not an angle of the shape.
• Using n triangles instead of n − 2.
• Dividing the angle sum by n for a polygon that is not regular.

Answers

• Part A: 4 sides, 2 triangles, 360°; 5 sides, 3 triangles, 540°; 6 sides, 4 triangles, 720°; 7 sides, 5 triangles, 900°; 10 sides, 8 triangles, 1440°; n sides, n − 2 triangles, (n − 2) × 180°.
• Part B: (a) x = 360° − 84° − 107° − 93° = 76°; (b) x = 360° − 90° − 118° − 64° = 88°; (c) x = 540° − 98° − 121° − 110° − 87° = 124°.
• Part C: pentagon 540° ÷ 5 = 108°; nonagon 1260° ÷ 9 = 140°; decagon 1440° ÷ 10 = 144°.
• Part D: (a) 1620 ÷ 180 = 9 triangles, so 11 sides; (b) a diagonal from one corner goes to every corner except itself and its two neighbours, n − 3 diagonals, which cut the shape into n − 2 triangles.

Other shapes

A. Split and count

Draw diagonals from one corner of each shape. Then fill the table.

B. Find x

C. Regular polygons

All the sides of a regular polygon are equal, and so are all its angles.

1. Find the size of each angle of a regular pentagon.   ____________
2. Find the size of each angle of a regular nonagon (9 sides).   ____________
3. Find the size of each angle of a regular decagon (10 sides).   ____________

D. Stretch

(a) The angles of a polygon add to 1620°. How many sides does it have?
(b) Explain why diagonals from one corner always make n − 2 triangles.

Unknown angles

The week comes together in problems about roofs, ladders, shelves and coins. Students play the loop cards, solve the problems with reasons, then play the quiz.

We are learning to

• choose the angle fact that fits each problem,
• solve a problem in two steps, with a reason for each,
• check an answer by adding the angles again.

Success criteria

• I can find an unknown angle in a real problem and say which fact I used.
• I can find each angle of a regular polygon.

You need

Loop cards from cut-out sheet 2 (one set, cut out before the lesson), the worksheet, and the unit’s self-check quiz.

Lesson flow (about 60 minutes)

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

Watch for these ideas

• Taking an exterior angle for the interior angle beside it, as in problems 1 and 6.
• Forgetting the right angle that the wall makes with the ground in problem 3.
• Dividing 360° by 12 for the coin, which gives the exterior angle, not the interior one.

Answers with working

Loop cards, in playing order: E, B, H, D, J, A, F, I, C, G.

1. x = 128° − 54° = 74° (an exterior angle equals the two opposite interior angles).
2. x = (180° − 128°) ÷ 2 = 26° (equal rafters, equal base angles).
3. x = 90° − 72° = 18° (the wall meets the ground at a right angle).
4. x = 180° − 47° − 64° = 69° (a straight line along the shelf).
5. (12 − 2) × 180° = 1800°, and 1800° ÷ 12 = 150°.
6. The left angle is 180° − 141° = 39°, so x = 180° − 39° − 58° = 83°.
7. 10x = 360°, so x = 36°: 36°, 72°, 108°, 144°.
8. Each angle is 156°; the exterior angles make one full turn, so 360° ÷ 24° = 15 sides.

Unknown angles

The shapes are drawn true to size. Matching marks show equal sides; a small square is a right angle.

Problems (7 and 8 are the stretch)

7. Stretch. The angles of a quadrilateral are x, 2x, 3x and 4x. Find each angle.
8. Stretch. Each exterior angle of a regular polygon is 24°. How many sides does it have?

Triangles to tear

Day 1. Cut out each triangle along its edges. Tear off its three coloured corners, then lay the corners side by side with their points together on a straight line drawn on your worksheet. Each triangle has a letter in the middle, so keep its corners together.

Loop cards

Day 5. Cut out the 10 cards and deal them out. The student with “I have 76°” starts by reading the question on that card; whoever holds the answer reads “I have …” and the next question, until the loop comes back to the start.

What we know: Interior angle sum of a triangle

Work on your own. Show your working. Matching marks show equal sides; a small square is a right angle.

Answers and marking guide

Answers with working

1. x = 180° − 57° − 81° = 42°.
2. x = (180° − 34°) ÷ 2 = 73°: the equal sides make the base angles equal.
3. x = 90° − 28° = 62°.
4. x = 360° − 72° − 115° − 98° = 75°.
5. (8 − 2) × 180° = 1080°.
6. x = 39° + 66° = 105°: the exterior angle equals the two opposite interior angles.
7. Tear the corners and they make a straight line; or draw a line through one corner parallel to the opposite side, so the other two angles appear there as alternate angles and the three fill a straight line, 180°.
8. Each base angle is 180° − 124° = 56°, so x = 180° − 2 × 56° = 68°.

A quick three-level guide

Eight questions, four ideas. A student at standard shows why a triangle’s angles add to 180°, uses it in special triangles and for exterior angles, and finds the angle sum of any polygon.

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: tearing the corners, the proof, special triangles, other shapes, and unknown angles.