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Triangles by sides and by angles

Geometry · level 3, lesson 4 of 6

A question to start

Can you draw a triangle with two right angles? Try it now with a ruler: draw a line, put a square corner at each end, and try to close the triangle.

Can you draw a triangle that has all three sides the same length, and also a right angle?

Both sound possible. Both are impossible. In this lesson you will see why, with nothing more than the ideas from the last three lessons.

The idea

The parts of a triangle

A triangle is a flat shape with three straight sides and three corners. ("Tri" means three, as in tricycle; "angle" because it has three angles.) Each corner is called a vertex; more than one vertex are vertices (say "VER-tis-eez"). We often name a triangle by the letters at its vertices, like triangle \(ABC\).

Every triangle has three sides and three angles, one angle at each vertex. We can sort triangles in two different ways: by looking at the sides, or by looking at the angles. Every triangle gets one name from each way, like a first name and a family name.

Sorting by sides

Measure the three sides with a ruler and compare them.

  • Equilateral: all three sides are the same length. ("Equi" means equal and "lateral" means side.)
  • Isosceles (say "eye-SOSS-uh-leez"): two sides are the same length. The word is Greek for "equal legs".
  • Scalene (say "SKAY-leen"): no two sides are the same length. All three are different.

In drawings, equal sides are shown with the same number of small tick marks. Sides with one tick are equal to each other; sides with two ticks are equal to each other.

equilateral isosceles scalene

Is an equilateral triangle also isosceles? It has two equal sides (in fact it has three), so many mathematicians say yes: every equilateral triangle is a special isosceles triangle. It is like saying every square is a special rectangle, which you will meet in the next lesson.

Sorting by angles

Look at the largest angle of the triangle.

  • Acute triangle: all three angles are acute (less than 90°).
  • Right triangle (also called right-angled): one angle is exactly 90°.
  • Obtuse triangle: one angle is obtuse (more than 90°).
acute: all angles acute right: one right angle obtuse: one obtuse angle

Notice the words "one angle" for right and obtuse triangles. Why not two? Here is the reason.

Why a triangle cannot have two right angles

Draw one side of a triangle flat along the bottom. Now suppose the angles at both ends of it are right angles. Then the other two sides both go straight up from the bottom side, at 90°. Two lines that are both perpendicular to the same line are parallel (you met this with the rungs and rails of a ladder in the last lesson). Parallel lines never meet. So the two sides can never come together to make the top corner, and there is no triangle.

never meet two right angles on one side

What if the angles at both ends were bigger than 90°? Then the two sides lean outwards, away from each other, and they are even further from meeting. And if one end is 90° and the other is obtuse, one side goes straight up and the other leans away from it: still no meeting.

So the two sides can only meet if they lean towards each other, and that needs both angles at the ends to be acute. Whichever side you put at the bottom, the same is true. This proves:

  • A triangle has at most one angle that is right or obtuse.
  • So every triangle has at least two acute angles.

That is why we can name a triangle by its biggest angle alone.

A sneak preview: the 180° fact

There is a famous fact you will prove in level 4: the three angles of any triangle add up to exactly 180°, a straight angle. Try it now: draw a triangle, cut it out, tear off the three corners and put their points together. They make a straight line. (If you want the reason, it is in level 4, lesson 3.)

You do not need this fact to sort triangles, but it agrees with everything above: two right angles would already use up \(90 + 90 = 180\) degrees, leaving nothing for the third angle.

What the sides tell you about the angles

Sides and angles are connected. Here are two connections you can check with a ruler and protractor.

In an isosceles triangle, the two angles at the ends of the third side are equal. That third side is called the base (whichever way the triangle is turned), and these two angles are the base angles. Why are they equal? Fold the triangle along a line from the top vertex to the middle of the base. The two equal sides land exactly on top of each other, so the two base angles land on top of each other too. Angles that fit exactly on top of each other are equal.

fold on the dashed line

In an equilateral triangle, all three angles are equal. Every pair of its sides is an equal pair, so folding works from every corner, and every pair of angles is equal. Using the 180° fact, each angle is \(180 \div 3 = 60\) degrees, every time, whatever the size of the triangle.

The biggest angle is always opposite the longest side. "Opposite" means across the triangle from it: the angle at the vertex that is not on that side. The wider a corner opens, the further apart it pushes the ends of the side across from it. In a right triangle, the longest side is the one across from the right angle. It has its own name, the hypotenuse (say "hy-POT-en-yooz"), and in level 6 it gets a famous rule of its own.

Both names at once

Every triangle has one name from each list. Which pairs can happen?

acute right obtuse
equilateral yes: 60°, 60°, 60° no no
isosceles yes: 70°, 70°, 40° yes: 45°, 45°, 90° yes: 30°, 30°, 120°
scalene yes: 50°, 60°, 70° yes: 30°, 60°, 90° yes: 20°, 40°, 120°

Why can an equilateral triangle not be right or obtuse? Because all its angles are equal, and a triangle can only have one right or obtuse angle. If one angle were 90°, all three would be, and we just saw that even two is impossible. So every equilateral triangle is acute.

A right isosceles triangle is half of a square, cut along the diagonal: it has angles 45°, 45° and 90°. The right angle must be the angle between the two equal sides, because the two base angles are equal and they cannot both be 90°.

Worked examples

1. Naming from the sides. A triangle has sides 5 cm, 5 cm and 8 cm. Two sides are equal, so it is isosceles. The longest side is 8 cm, so the biggest angle is the one opposite it, between the two 5 cm sides. (If you draw it carefully and measure, that angle is a little more than 106°, so the triangle is also obtuse: an obtuse isosceles triangle.)

2. Naming from the angles. A triangle has angles 35°, 55° and 90°. One angle is 90°: a right triangle. All three angles are different, so the sides opposite them must all be different: it is scalene. Full name: a right scalene triangle. Its longest side, the hypotenuse, is opposite the 90° angle.

3. Finding a missing base angle. An isosceles triangle has base angles that are equal. One base angle is 65°. What is the other base angle? The base angles are equal, so it is also 65°. (With the 180° fact, the top angle would be \(180 - 65 - 65 = 50\) degrees.)

4. Is it possible? "An obtuse triangle with two obtuse angles." Not possible: a triangle can only have one angle that is right or obtuse. "A right triangle that is isosceles." Possible: half a square, 45°, 45°, 90°.

Try it

  1. Name each triangle by its sides: (a) 4 cm, 4 cm, 4 cm; (b) 3 cm, 5 cm, 6 cm; (c) 7 cm, 2 cm, 7 cm.
  2. Name each triangle by its angles: (a) 60°, 60°, 60°; (b) 25°, 45°, 110°; (c) 90°, 50°, 40°.
  3. An isosceles triangle has one base angle of 72°. What is the other base angle?
  4. Which side of a right triangle is the longest? What is it called?
  5. True or false? (a) Every equilateral triangle is acute. (b) A scalene triangle can be a right triangle. (c) A triangle can have two obtuse angles.
  6. A triangle has angles 50°, 50° and 80°. What can you say about its sides?
  7. Fold a square of paper along its diagonal. What kind of triangle do you get, by sides and by angles?
  8. Explain in your own words why a triangle cannot have two right angles.
Answers
  1. (a) Equilateral (all three equal). (b) Scalene (all different). (c) Isosceles (two sides of 7 cm).
  2. (a) Acute (all less than 90°). It is equilateral too. (b) Obtuse, because of the 110° angle. (c) Right.
  3. 72°. The two base angles of an isosceles triangle are equal.
  4. The side opposite the right angle. It is called the hypotenuse.
  5. (a) True: all its angles are 60°. (b) True: for example 30°, 60°, 90°. (c) False: at most one angle can be right or obtuse.
  6. Two angles are equal (50° and 50°), so the sides opposite them are equal: the triangle is isosceles. The third side, opposite the 80° angle, is the longest of the three, because 80° is the biggest angle. (The fold reason works backwards too: equal base angles mean equal sides.)
  7. Two sides of the triangle were sides of the square, so they are equal: isosceles. The corner of the square is still there: right. A right isosceles triangle, with angles 90°, 45° and 45°.
  8. If both angles at the ends of one side were 90°, the other two sides would both go off at right angles to that side. Lines perpendicular to the same line are parallel, and parallel lines never meet, so the triangle could never close.

Common mistake

Thinking the "base" must be at the bottom. A triangle can be turned any way up. In an isosceles triangle the base is the side that is not one of the equal pair, wherever it happens to be on the page, and the base angles are the angles at its two ends.

Naming a triangle by how it looks. A triangle that looks nearly right-angled might be 88° or 92°. Use the marks on the drawing (tick marks and square corners) or measure. Only a right-angle square or a given 90° makes it a right triangle.

Where it leads