Properties of Triangles
Questions designed to stretch thinking, reveal misconceptions, and spark mathematical reasoning.
Convince Me That…
Students must construct a mathematical argument for why each statement is true.
Consider a triangle with angles 45°, 45°, and 90°. It has a right angle (90°), so it is right-angled. It also has two equal angles (both 45°), which means the two sides opposite those angles are equal in length — so it is isosceles. A specific example: a triangle with two short sides each of length 1 and a hypotenuse of \( \sqrt{2} \).
Students often fall for the “mutually exclusive classification” misconception — thinking that triangle categories (right-angled, isosceles, scalene, equilateral, etc.) are completely separate, like putting shapes into different boxes. In fact, some categories describe angles (right-angled, acute, obtuse) and others describe sides (scalene, isosceles, equilateral), so a triangle can belong to one from each group simultaneously.
The angles in any triangle sum to exactly 180°. So if you know two angles, the third must be 180° minus the sum of the other two. For example, if two angles are 65° and 80°, the third is 180° − 65° − 80° = 35°. There is no choice or variation — only one value works.
This challenges the “you must measure every angle” misconception. The third angle is completely determined by the other two. It is never necessary to measure it. This is a consequence of the angle-sum property of triangles, which is not just a pattern observed from a few examples but a theorem that can be proved using parallel lines and alternate angles. By framing it this way, we are essentially solving the algebraic equation \( x + y + z = 180^\circ \), demonstrating that geometry and algebra are intrinsically linked.
For three lengths to form a triangle, every pair of sides must sum to more than the third side (the triangle inequality). Here, 2 + 3 = 5, but 5 < 10. The two shorter sides are too short to “reach” each other around the longest side — if you tried to build this triangle, the two short sides would not meet.
This addresses the “any three lengths form a triangle” misconception. Any time the two shorter sides don’t sum to more than the longest side, a triangle is impossible. For instance, 4, 5, and 9 also fail (4 + 5 = 9, not strictly greater). Students often assume any three positive numbers can make a triangle.
The three angles sum to 180° (1 + 1 + 178 = 180), which is the fundamental requirement for any triangle. This is an extremely flat, elongated triangle — almost like a thin sliver — but it is mathematically valid. The triangle inequality is also satisfied: the two very short sides (opposite the 1° angles) will be tiny, while the side opposite the 178° angle will be very long, but the three sides will still close to form a shape.
Students often have a “prototype” image of what a triangle should look like — roughly equilateral, with a base at the bottom and a point at the top. Any triangle that looks very different from this prototype gets rejected as “not a real triangle.” But a triangle is any closed shape with three straight sides, regardless of how flat, thin, or lopsided it appears.
Give an Example Of…
Think carefully — the fourth box is a trap! Give a non-example that looks right but isn’t.
Example: An isosceles triangle with angles 80°, 50°, 50°.
Another: An isosceles triangle with sides 5 cm, 5 cm, 3 cm.
Creative: An isosceles triangle with angles 20°, 80°, 80° — this is a very tall, narrow triangle. Students might not initially recognise it as having a line of symmetry because it looks so different from the “typical” isosceles shape, but the line runs from the 20° apex to the midpoint of the base.
Trap: An equilateral triangle. A student might think “it’s symmetric, so it has a line of symmetry,” and that is true — but it has three lines of symmetry, not exactly one. The question specifies “exactly one.”
Example: A triangle with angles 90°, 60°, 30°.
Another: A triangle with angles 90°, 50°, 40°.
Creative: A triangle with sides 5 cm, 12 cm, 13 cm — this is a right-angled triangle (by Pythagoras: \( 5^2 + 12^2 = 25 + 144 = 169 = 13^2 \)) and all three sides are different, so it is scalene.
Trap: A triangle with angles 45°, 45°, 90°. A student sees the right angle and thinks the job is done, but the two base angles are both 45° — making this isosceles, not scalene. Students must check that the non-right angles are different.
Example: A triangle with angles 100°, 50°, 30°.
Another: A triangle with angles 95°, 55°, 30°.
Creative: A triangle with angles 91°, 88°, 1° — this is barely obtuse and extremely flat, but it still qualifies. All three angles are different, so it is scalene.
Trap: A triangle with angles 120°, 30°, 30°. A student focuses on the obtuse angle (120°) and forgets to check whether the triangle is scalene. The two 30° angles mean two sides are equal — this is isosceles, not scalene.
Example: 30°, 60°, 90° — here 60° is double 30°. Check: 30 + 60 + 90 = 180°. ✓
Another: 40°, 80°, 60° — here 80° is double 40°. Check: 40 + 80 + 60 = 180°. ✓
Creative: 36°, 72°, 72° — here 72° is double 36°. Check: 36 + 72 + 72 = 180°. ✓ This is also isosceles, showing that the “double angle” relationship can occur in an isosceles triangle.
Trap: A triangle with angles 50°, 100°, 40°. A student correctly spots that 100° is double 50°, but doesn’t verify the angle sum: 50 + 100 + 40 = 190° ≠ 180°. This is NOT a valid triangle. The correct third angle would be 30° (giving 50°, 100°, 30°).
Trick Question! Every attempt is a sneaky non-example! It is mathematically impossible to give a valid example for this.
An isosceles triangle has two equal base angles. If they were both obtuse (e.g., 91°), their sum would be \( 91^\circ + 91^\circ = 182^\circ \). This already exceeds the 180° limit for a triangle before we even add the third angle. Therefore, the equal base angles in an isosceles triangle must always be acute.
Always, Sometimes, Never
Is the statement always true, sometimes true, or never true? Students should justify their decision with examples.
This is a fundamental property of all triangles: the larger the angle, the longer the opposite side, and vice versa. In an equilateral triangle all sides and angles are equal, so the property holds trivially. In a 30°–60°–90° triangle, the longest side (hypotenuse) is opposite the 90° angle.
This challenges the “side length is unrelated to angle size” misconception — students sometimes think which side is longest depends on which side is at the bottom, not which angle is opposite. No counterexample exists: this follows from the sine rule.
TRUE case: A triangle with sides 5 cm, 5 cm, 3 cm is isosceles and has exactly one line of symmetry (from the apex to the midpoint of the unequal base). FALSE case: An equilateral triangle (e.g. sides 5 cm, 5 cm, 5 cm) is isosceles — it has at least two equal sides — but has three lines of symmetry, not one.
This targets the “equilateral is not a type of isosceles” misconception. Students who treat equilateral and isosceles as entirely separate categories will say ALWAYS, forgetting the equilateral special case.
TRUE case: A triangle with angles 90°, 60°, 30° contains a right angle. FALSE case: An equilateral triangle (60°, 60°, 60°) contains no right angle. Nor does a triangle with angles 100°, 50°, 30°.
This exposes the “all triangles have a right angle” misconception. Because textbooks heavily feature right-angled triangles (especially for Pythagoras and trigonometry), students can over-associate triangles with right angles and assume every triangle has one.
An obtuse angle is greater than 90°. If a triangle had two obtuse angles, their sum would already exceed 90° + 90° = 180° — before even adding the third angle. Since the three angles must sum to exactly 180°, there is no room for two angles that are each greater than 90°.
This addresses the “small obtuse angles could fit” misconception — students sometimes think that two angles of, say, 91° could work because they’re “only just” obtuse, but 91° + 91° = 182°, which already exceeds 180°.
It is always larger than the two opposite interior angles (because the exterior angle exactly equals their sum). However, it is not always larger than the adjacent interior angle.
For example, in a triangle with angles 10°, 10°, and 160°, the exterior angle next to the 160° angle is only 20° (since \( 180^\circ – 160^\circ = 20^\circ \)). 20° is much smaller than the 160° interior angle sitting right next to it.
Odd One Out
Which is the odd one out? Can you make a case for each one? There’s no single right answer!
By Angle Type:
By Side Length:
Explain the Mistake
Each example contains a deliberate error targeting a common misconception. Can you find where and why the reasoning goes wrong?
Answer: Equilateral
Reasoning: “It has two angles the same, and equilateral means the angles are equal.”
The student has confused equilateral with isosceles. The word “equilateral” means all three sides (and therefore all three angles) are equal — an equilateral triangle has three 60° angles. A triangle with exactly two equal angles is isosceles.
This triangle has two 70° angles and one 40° angle, so it is isosceles, not equilateral. The confusion likely arises because both words contain “equal,” but equilateral means “equal sides” (all three), while isosceles means “equal legs” (exactly two).
Answer: 50° ✓
Reasoning: “I added up the angles of a triangle on the board and they came to 180, so I just did 180 take away 55 and take away 75 to get 50.”
The answer is correct — the missing angle is indeed 50° (since 55 + 75 + 50 = 180). However, the student’s justification is based on generalising from a single measured example. Measuring one triangle on the board and getting 180° doesn’t prove it works for all triangles — it could be coincidence, or there could be measurement error. Ask the student: “What if you drew this triangle perfectly with a ruler and protractor, but when you measured the third angle, your protractor said 49°? Who is right — the math theorem, or your protractor?” This forces them to realize that geometric laws overrule empirical measurement.
The angle sum property is a geometric theorem that can be proved using properties of parallel lines (e.g. drawing a line through one vertex parallel to the opposite side and using alternate angles). A student who relies on “I measured it once” may not trust the property when confronted with unusual triangles, like a very flat one.
Answer: No
Reasoning: “Triangles can’t have angles bigger than 90°. That’s more like a corner of a square. Proper triangles don’t look like that.”
The student believes triangles cannot contain obtuse angles. This is a common misconception reinforced by textbook diagrams that predominantly show acute or right-angled triangles. In fact, a triangle can have one obtuse angle — these are called obtuse-angled triangles.
The angles here sum to 180° (110 + 40 + 30 = 180), so this is a perfectly valid triangle. The student’s reasoning is driven by a prototype image of what triangles “should” look like, rather than the mathematical definition (a closed shape with three straight sides whose angles sum to 180°).
Answer: 1
Reasoning: “It’s got two sides the same so it’s isosceles, and isosceles triangles have one line of symmetry down the middle.”
The student has correctly noticed that the triangle has equal sides, but has failed to recognise it as equilateral — all three sides are 6 cm, not just two of them. While it is true that an equilateral triangle is a special case of isosceles (it has “at least two” equal sides), the key point is that having all three sides equal gives it three lines of symmetry, not one.
The student spotted the first pair of equal sides and stopped looking, rather than examining all three sides systematically. Each line of symmetry runs from one vertex to the midpoint of the opposite side. The correct answer is 3 lines of symmetry.