Triangles

A tri­an­gle is the sim­plest shape with straight sides: just three sides and three cor­ners. Yet it is every­where, in roof frames, bridges and set squares, because a tri­an­gle holds its shape firmly. In this les­son you will learn the parts of a tri­an­gle, the kinds of tri­an­gles, and the two most use­ful facts about their angles. You will also learn which three lengths can make a tri­an­gle and which can­not.

The parts of a tri­an­gle#

Take three points $A$, $B$ and $C$ that are not on one line, and join them. The closed fig­ure you get is the tri­an­gle $ABC$, writ­ten $\triangle ABC$.

• Its three ver­tices are $A$, $B$ and $C$.
• Its three sides are $\overline{AB}$, $\overline{BC}$ and $\overline{CA}$.
• Its three angles are $\angle A$, $\angle B$ and $\angle C$.

So a tri­an­gle has six parts: three sides and three angles.

The side $\overline{BC}$ is called the side oppo­site $\angle A$, because it does not touch $A$. In the same way, $\overline{CA}$ is oppo­site $\angle B$ and $\overline{AB}$ is oppo­site $\angle C$.

Kinds of tri­an­gles by their sides#

Kinds of tri­an­gles by their angles#

In a right-angled tri­an­gle, the side oppo­site the right angle is called the hypotenuse. It is the longest side.

The angle sum prop­erty#

The three angles of a tri­an­gle add up to $180^\circ$.

$$\angle A + \angle B + \angle C = 180^\circ$$

You can see this with paper. Cut out any tri­an­gle, tear off its three cor­ners and place them side by side with their points together. They fit exactly along a straight line, which is $180^\circ$.

Why it is true. Through $A$, draw a line par­al­lel to $\overline{BC}$. At $A$ there are now three angles side by side along this line: one on the left, the tri­an­gle's own $\angle A$ in the mid­dle, and one on the right. Together they make a straight angle, $180^\circ$.

The angle on the left and $\angle B$ are alter­nate inte­rior angles, with $\overline{AB}$ as the trans­ver­sal of the two par­al­lel lines. So they are equal. In the same way, the angle on the right equals $\angle C$, with $\overline{AC}$ as the trans­ver­sal. So $\angle B + \angle A + \angle C = 180^\circ$.

This tells us two more things:

• A tri­an­gle can have at most one right angle, and at most one obtuse angle. Two of them would already add up to $180^\circ$ or more.
• In a right-angled tri­an­gle, the two acute angles add up to $90^\circ$.

Exam­ple 1#

Two angles of a tri­an­gle are $50^\circ$ and $65^\circ$. Find the third angle.

$$180^\circ - 50^\circ - 65^\circ = 65^\circ$$

The third angle is $65^\circ$. Two angles are equal, so this is an isosce­les tri­an­gle.

Exam­ple 2#

The angles of a tri­an­gle are in the ratio $2 : 3 : 4$. Find them.

Let the angles be $2k$, $3k$ and $4k$.

$$2k + 3k + 4k = 180 \quad \text{so} \quad 9k = 180 \quad \text{and} \quad k = 20$$

The angles are $40^\circ$, $60^\circ$ and $80^\circ$. All are acute, so the tri­an­gle is acute-angled.

Exam­ple 3#

The angles of a tri­an­gle are $x^\circ$, $(x + 10)^\circ$ and $(x + 20)^\circ$. Find them.

$$x + (x + 10) + (x + 20) = 180 \quad \text{so} \quad 3x + 30 = 180 \quad \text{and} \quad x = 50$$

The angles are $50^\circ$, $60^\circ$ and $70^\circ$.

Isosce­les and equi­lat­eral tri­an­gles#

In an isosce­les tri­an­gle, the angles oppo­site the two equal sides are equal. These two angles are called the base angles. The other way round is true as well: if two angles of a tri­an­gle are equal, the sides oppo­site them are equal.

In an equi­lat­eral tri­an­gle, all three angles are equal. Each is $180^\circ \div 3 = 60^\circ$.

Exam­ple 4#

In an isosce­les tri­an­gle, the angle between the two equal sides is $40^\circ$. Find the base angles.

The two base angles are equal and together they make $180^\circ - 40^\circ = 140^\circ$. So each base angle is $140^\circ \div 2 = 70^\circ$.

The exte­rior angle prop­erty#

Make side $\overline{BC}$ longer, past $C$, to a point $D$. The angle $\angle ACD$, out­side the tri­an­gle, is called an exte­rior angle of the tri­an­gle. The two angles of the tri­an­gle that are not next to it, $\angle A$ and $\angle B$, are called its inte­rior oppo­site angles.

An exte­rior angle of a tri­an­gle is equal to the sum of its two inte­rior oppo­site angles.

$$\angle ACD = \angle A + \angle B$$

Why. $\angle ACD$ and $\angle ACB$ form a lin­ear pair, so $\angle ACD = 180^\circ - \angle ACB$. By the angle sum prop­erty, $\angle A + \angle B = 180^\circ - \angle ACB$ too. So the two are equal.

Exam­ple 5#

An exte­rior angle of a tri­an­gle is $110^\circ$. One of its inte­rior oppo­site angles is $45^\circ$. Find the other inte­rior oppo­site angle, and the inte­rior angle next to the exte­rior angle.

Other inte­rior oppo­site angle $= 110^\circ - 45^\circ = 65^\circ$.

The inte­rior angle next to the exte­rior angle forms a lin­ear pair with it, so it is $180^\circ - 110^\circ = 70^\circ$.

Check: $45^\circ + 65^\circ + 70^\circ = 180^\circ$.

Which lengths make a tri­an­gle?#

Try mak­ing a tri­an­gle from sticks of 2 cm, 3 cm and 6 cm. You can­not: the two short sticks together are too short to reach across the long one.

The sum of any two sides of a tri­an­gle is greater than the third side.

To test three lengths, it is enough to check that the two shorter ones add up to more than the longest.

Exam­ple 6#

Can these be the sides of a tri­an­gle? (a) 3 cm, 4 cm, 5 cm (b) 2 cm, 3 cm, 6 cm (c) 4 cm, 5 cm, 9 cm

(a) $3 + 4 = 7$, which is more than 5. Yes.

(b) $2 + 3 = 5$, which is less than 6. No.

(c) $4 + 5 = 9$, which is equal to 9, not more. No. The sticks would just lie flat along one line.

Prac­tice#

1. Two angles of a tri­an­gle are $72^\circ$ and $48^\circ$. Find the third angle.
2. One acute angle of a right-angled tri­an­gle is $35^\circ$. Find the other acute angle.
3. The angles of a tri­an­gle are in the ratio $1 : 2 : 3$. Find them. What kind of tri­an­gle is it?
4. In an isosce­les tri­an­gle, each base angle is $65^\circ$. Find the third angle.
5. Can a tri­an­gle have two obtuse angles? Give a rea­son.
6. An exte­rior angle of a tri­an­gle is $125^\circ$ and one inte­rior oppo­site angle is $60^\circ$. Find the other inte­rior oppo­site angle.
7. The angles of a tri­an­gle are $(2x)^\circ$, $(3x)^\circ$ and $(x + 30)^\circ$. Find them.
8. Which of these can be the sides of a tri­an­gle: (a) 5 cm, 6 cm, 10 cm (b) 3 cm, 7 cm, 11 cm?
9. Each angle of a tri­an­gle is $60^\circ$. What kind of tri­an­gle is it, by its sides?

Answers#