What is a triangle? Parts and notation
A triangle is a closed shape made by joining 3 points with 3 straight lines. The points are called vertices (corners). The lines are the sides. At each corner two sides meet and make an angle.
We name a triangle by its corners, for example triangle ABC, written △ABC. The side opposite corner A is called a (it is BC). The side opposite B is b and the side opposite C is c.
Angle sum: ∠A + ∠B + ∠C = 180°. This is true for every triangle, big or small. If you know two angles you can find the third.
Types of triangles: by sides and by angles
By sides
- Equilateral: all 3 sides equal. All 3 angles are 60°.
- Isosceles: 2 sides equal. The two angles opposite the equal sides are equal too.
- Scalene: all 3 sides different, so all 3 angles are different.
By angles
- Acute-angled: every angle is less than 90°.
- Right-angled: one angle is exactly 90°. The side opposite it is the longest and is called the hypotenuse.
- Obtuse-angled: one angle is more than 90°.
A triangle can have only one right angle or one obtuse angle. Two such angles would already add to 180° or more, and there would be no room for the third.
Sides of a triangle: the triangle inequality
The sum of any two sides of a triangle is greater than the third side. This is the triangle inequality.
Why? The straight path from one corner to another is the shortest. Going through the third corner is a longer way round. If two sticks are too short, they cannot meet, and a gap is left.
Quick check: add the two smaller sides and compare with the biggest side. Sticks 3, 4, 6: 3 + 4 = 7 > 6, so a triangle is possible. Sticks 3, 4, 8: 3 + 4 = 7 < 8, so it is not possible.
Range of the third side: if two sides are 4 and 9, the third side must be bigger than 9 − 4 = 5 and smaller than 9 + 4 = 13.
Medians, angle bisectors and altitudes
Three special lines can be drawn from a corner of a triangle. Every triangle has 3 of each kind.
- Median: joins a corner to the middle point of the opposite side. It cuts that side into two equal parts.
- Angle bisector: cuts the angle at the corner into two equal angles. It meets the opposite side, but not always at its middle.
- Altitude (height): the line from a corner that meets the opposite side at 90°. In an obtuse triangle two altitudes fall outside the triangle.
The altitude is used in the area: Area = ½ × base × altitude. In an isosceles triangle, the median, bisector and altitude from the top corner are the same line.
Why triangles are strong: stability
Make a frame from four sticks joined at the corners. Push it and the corners change their angles, so it leans over. Now make a frame from three sticks. The three sides are fixed, so the angles are fixed. It cannot lean. We say the triangle is rigid (or stable).
A four-sided frame becomes rigid when you add a diagonal brace, because the brace cuts it into two triangles. This is why you see triangles in bridges, cranes, roof frames, gates and cycle frames.
Try it: sticks and straws
Cut straws or sticks of length 3 cm, 4 cm, 5 cm and 9 cm. Try to make triangles with (3, 4, 5), (3, 4, 9) and (4, 5, 9). Which work? Check each with the triangle inequality first, then test it. Then make a square frame from 4 straws and push it, and add a diagonal. Use the free-play step of the 3D to repeat this with sliders.
Key formulas and definitions
- Angle sum: ∠A + ∠B + ∠C = 180°.
- Triangle inequality: a + b > c, b + c > a, a + c > b.
- Third side x of sides p and q: |p − q| < x < p + q.
- Equilateral: 3 equal sides, each angle 60°.
- Isosceles: 2 equal sides, 2 equal angles opposite them.
- Right triangle: one angle 90°; the other two add up to 90°.
- Area of a triangle = ½ × base × altitude.
- Median: corner to middle of opposite side.
Worked examples
1. Two angles of a triangle are 40° and 75°. Find the third angle and say what kind of triangle it is.
Third angle = 180° − 40° − 75° = 65°. All three angles (40°, 75°, 65°) are less than 90°, and all are different. So it is an acute-angled scalene triangle.
2. Can a triangle have sides 5 cm, 7 cm and 13 cm?
Add the two smaller sides: 5 + 7 = 12. Compare with the longest side: 12 is not greater than 13. So the sticks cannot meet and no triangle can be made.
3. Two sides of a triangle are 4 cm and 9 cm. Between which two values must the third side lie?
The third side must be bigger than the difference: 9 − 4 = 5 cm. It must be smaller than the sum: 9 + 4 = 13 cm. So it lies between 5 cm and 13 cm.
4. In an isosceles triangle the angle between the two equal sides is 40°. Find the other two angles.
The other two angles share 180° − 40° = 140°. They are equal (opposite the equal sides), so each is 140° ÷ 2 = 70°.
Common mistakes
- Thinking the median and the altitude are the same. A median goes to the middle; an altitude goes down at 90°. They are the same line only in special cases like an isosceles triangle from its top corner.
- Checking only one pair of sides in the triangle inequality. It is enough to check that the two smaller sides together beat the largest side.
- Thinking the angle bisector always hits the middle of the opposite side. It does so only when the two sides at that corner are equal.
- Forgetting that the angle sum is 180° and using 360° (that is for a four-sided shape).