At a glance. One lesson of about 60 minutes. Students use the 180° angle sum to find missing angles, classify triangles by angles and by sides, and see the parallel-line proof that the sum is 180° for every triangle.
Calculator: no calculator. Nothing here needs one. The arithmetic is subtraction from 180 and halving, and the checking is the point.
What students will learn
Know
- The angles of a triangle add up to 180°.
- Acute, right-angled and obtuse; scalene, isosceles and equilateral.
- An isosceles triangle has two equal angles opposite its equal sides.
Understand
- The angle sum follows from alternate angles and a straight line, so it holds for every triangle, not just the ones measured.
- The name by sides and the name by angles are independent.
- A measured sum of 178° is a measuring error, not a different triangle.
Do
- Find a missing angle of a triangle, with a reason.
- Name a triangle by its angles and by its sides.
- Combine the triangle fact with angles on a line, and with equations.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: angle sum, classification and missing angles (questions 1 to 7).
- Criterion B. Investigating patterns: the corners figure. Students record several triangles in a table, find the sum, and generalize before the proof.
- Criterion C. Communicating: writing the proof in their own words with the words alternate, parallel and straight line, and giving a reason for each missing angle.
- Criterion D. Applying mathematics in real-life contexts: the folded paper of the hook and the roof frame in question 4, where an angle is found rather than measured.
The one thing to do with the figure
Make the sum surprising before it is explained. Set several triangles and ask for the third angle each time before pressing anything. Then ask what the strip at the bottom shows.
In the second figure, pause at each step and ask: why is this angle equal to that one? The word “alternate” from lesson 3 is the whole proof. Run it with a second triangle to show it did not depend on one picture.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook, with students tearing corners from a paper triangle; the first figure. |
| 10 min | Key idea, then the second figure, one step at a time. |
| 10 min | Worked example; students finish the faded one in pairs. |
| 5 min | Question 1, the check. If many choose A or D, go back to the first figure. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 55°; 57°; 70. |
|---|---|
| Worked example | 78°, acute and scalene. |
| Faded example | 65° and 65°; acute and isosceles. |
| 1. Can these be a triangle's angles? | B. Only B adds up to 180. A is 195, C is 190 and D is 240. |
| 2. The third angle | 61. 180 − (52 + 67) = 61. |
| 3. Naming a triangle | A. B has no right angle in the data; C has no obtuse angle; D needs three equal angles. |
| 4. Isosceles base angle | 70. (180 − 40) ÷ 2 = 70. |
| 5. Angles in a ratio of parts | 90. 6x = 180, x = 30, 3x = 90. |
| 6. Right-angled triangle | 55. 180 − 90 − 35 = 55. |
| 7. Exterior angle | 107. PRQ = 73, then 180 − 73 = 107. |
| 8. Equation in an isosceles triangle | 32. 5x + 20 = 180, 5x = 160, x = 32. |
| 9. Two obtuse angles | B. A and D use the wrong total; C overcorrects, since one obtuse angle is allowed. If many choose D, return to the first figure and watch the line. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Does not add the three angles, or uses 360. | Make them write the sum of each set before choosing. |
| 119, 128 or 113 in question 2 | Stops at the sum, or subtracts only one angle. | Ask them to say what 119 is the sum of, then what is left. |
| 140, 20 or 50 in question 4 | Leaves out halving, halves the wrong angle or uses 90. | Draw the triangle and mark equal angles with the same arc. |
| 30 or 60 in question 5 | Gives x, or divides 180 by 3. | Ask them to substitute x back and read each angle. |
| 145 or 35 in question 6 | Forgets the right angle, or assumes the angles are equal. | List all three angles with a gap for the unknown. |
| 73, 100 or 121 in question 7 | Gives the interior angle, reads the picture (which is not to scale), or uses one angle only. | Ask which side of the line the angle is on, and which fact tells them its size. |
| 36 or 40 in question 8 | Drops the constants, or adds them to the wrong side. | Collect like terms in two columns first. |
Questions worth asking
- Why is the proof for every triangle if we only drew one? Nothing in the steps used the particular angles; only parallel lines and a straight line. Listen for “it would work for any A and B”.
- Can a triangle have two right angles? No: they would use 180° and leave 0 for the third angle.
- Can an equilateral triangle be obtuse? No: all three angles are 60°. It is always acute.
- What is the largest angle a triangle can have? Less than 180°; as close as you like, which is why the figure stops at a fixed range.
Differentiation
A way in for anyone stuck
- Give a paper triangle to tear: seeing the corners meet on a line makes the sum concrete.
- Provide a table with columns for the two known angles, their sum and the third angle.
- Start with right-angled triangles and halve the leftover 90° only afterwards.
Stretch
- The whatIf: prove the exterior angle equals the sum of the two opposite interior angles, and use it for question 7.
- Show that the angles of a four-sided shape add up to 360° by cutting it into two triangles. (Then pentagons: three triangles, 540°. Year 9 picks this up in tessellations and polygon angles.)
Students learning in English as an additional language
Three pairs of words sound alike: acute and obtuse, scalene and isosceles, equal and equilateral. Say each with its picture; “equilateral” contains “equal” and “lateral” (sides).
| Word | Plain meaning | Example |
|---|---|---|
| triangle | a shape with three straight sides and three angles | A roof frame |
| scalene | a triangle with no equal sides | Sides 3, 4 and 5 cm |
| isosceles | a triangle with two equal sides and two equal angles | The gable end of a house, with two equal rafters |
| equilateral | a triangle with three equal sides and three 60° angles | A road sign |
| interior angle | an angle inside a shape at a corner | The three angles of a triangle |
| exterior angle | the angle outside a shape between one side and the extension of the next | PRS in question 7 |
Sentence starters for explaining aloud or in writing:
- The third angle is 180° minus …, so it is …
- The triangle is … because all its angles are …, and … because …
- The line through C is parallel to AB, so … is equal to … because …
Two things not to say
“The angles add up to 180 because I measured them.” Measuring shows it for one triangle with an error. The proof shows it for all.
“Isosceles means it has two equal angles on the bottom.” The equal angles are the ones opposite the equal sides, and the triangle can sit any way up.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Missing angleTwo angles of a triangle are 45° and 85°. Find the third and name the triangle by its angles.
50°, acute. - NamingA triangle has sides 6 cm, 6 cm and 6 cm. State its angles.
60°, 60° and 60°; equilateral.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- TileA triangular floor tile is isosceles, with the angle at its top corner 100°. Find each base angle.
40°. - From Unit 1From Unit 1, lesson 1 (place value and decimals): the angles of a triangle are 60.5°, 59.5° and x. Find x.
60°.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningThe angles of a triangle are in the order x, x + 10 and x + 20. Find the three angles.
50°, 60° and 70°. - GeneralizingFind the angle sum of a four-sided shape by cutting it into two triangles.
360°.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Use a paper triangle and tear off the corners.