9.04 Triangles

Whenever three points do not lie on the same line, we can connect them together with three segments. This three-sided shape is called a triangle. Three angles are formed at the same time (which is how the shape gets its name).

The kinds of angles that are formed lets us classify different types of triangles:

• If all the angles are acute, the triangle is an acute triangle.

• If one of the angles is a right angle, the triangle is a right-angled triangle.

• If one of the angles is obtuse, the triangle is an obtuse triangle.

The lengths of the sides allow us to classify different types of triangles in a completely different way:

• If all the sides have different lengths, the triangle is a scalene triangle

• If at least two sides have the same length, the triangle is an isosceles triangle

• A special kind of isosceles triangle is the equilateral triangle, where all three sides have the same length.

Triangles for each combination of types exist, and this is summarised in the diagram below:

Equilateral triangles are always acute isosceles triangles.

Isosceles triangles have a special property. If two sides have the same length, the angles formed with the third side (called the base) are always equal in measure. The reverse is true as well.

The base of both isosceles triangles in the image has been highlighted.

Examples

Example 1

Consider the triangle below.

a

Which of the following words describes this triangle?

A

Scalene

B

Equilateral

C

Isosceles

Worked Solution

Create a strategy

Take note of the sides marked as equal.

Apply the idea

The answer is option C: isosceles, because two sides of the triangle were marked equal.

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b

Which of the following words also describes this triangle?

A

acute

B

right-angled

C

obtuse

Worked Solution

Create a strategy

We can use the diagram below:

Apply the idea

All the angles of the triangle are acute angles. So the correct answer is A: acute.

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For any triangle, we can draw a line through one point that is parallel to the opposite side.

Extending all the sides then creates a diagram with two parallel lines and two transversals.

Using what we learned in the  last lesson  , let's look at each of these transversals in turn.

Using the first transversal, the angle inside the triangle forms an alternate angle pair.

And using the second transversal, the other angle inside the triangle forms a corresponding angle pair.

This means that the three angles inside the triangle add together to form a straight angle.

In other words: the sum of the angles in a triangle is 180\degree.

Examples

Example 2

Consider the triangle below.

Is it a right-angled triangle?

A

Yes

B

No

Worked Solution

Create a strategy

We need to find the remaining angle. Let x be the remaining angle, add it to the given angles, and equate the sum to 180\degree.

Apply the idea

\displaystyle x+49+41	\displaystyle =	\displaystyle 180	Add the angles and equate to 180
\displaystyle x+90	\displaystyle =	\displaystyle 180	Evaluate the addition
\displaystyle x+90-90	\displaystyle =	\displaystyle 180-90	Subtract 90 from both sides
\displaystyle x	\displaystyle =	\displaystyle 90\degree	Evaluate

Yes, it is a right-angled triangle. So the correct answer is A.

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The angles formed outside the triangle by extending the sides are called exterior angles. The size of an exterior angle is always equal to the sum of the internal angles on the opposite side.

Examples

Example 3

Solve for the value of x in the diagram below.

Worked Solution

Create a strategy

Equate the sum of the interior angles to the opposite exterior angle.

Apply the idea

\displaystyle x+52	\displaystyle =	\displaystyle 108	Add the interior angles and equate to 108
\displaystyle x+52-52	\displaystyle =	\displaystyle 108-52	Subtract 52 from both sides
\displaystyle x	\displaystyle =	\displaystyle 56	Evaluate

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