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Middle Years

9.04 Triangles

Lesson

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 are different lengths, the triangle is a scalene triangle
  • If two (or more) 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 are the same length.

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

Summary

Equilateral triangles are always acute isosceles triangles.

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

The base of both isosceles triangles has been highlighted.

Angle sum of a triangle

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, like this:

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:

Summary

The angle sum of a triangle is $180^\circ$180°.

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.

Worked example

Is this triangle scalene, or isosceles?

Think: We have no information about the lengths of the edges. So instead we will try to compare the angles in the triangle. Only two angles are written on the diagram, but we can use the angle sum of a triangle to find the missing one. If two angles are equal, the triangle is isosceles.

Do: The missing angle is

$180^\circ-40^\circ-100^\circ=40^\circ$180°40°100°=40°

Since two of the angles are equal we know that two of the sides must be equal as well. This means that the triangle is isosceles.

 

Practice questions

Question 1

Consider the triangle below.

A triangle with two sides marked with single ticks indicating those two sides are of equal length. All corners have angles that appear to be less than 90 degrees.
  1. Which of the following words describes this triangle?

    scalene

    A

    equilateral

    B

    isosceles

    C
  2. Which of the following words also describes this triangle?

    acute

    A

    right-angled

    B

    obtuse

    C
Question 2

Consider the triangle below.

  1. Is it a right-angled triangle?

    Yes

    A

    No

    B
Question 3

Solve for the value of $x$x in the diagram below.

Enter each line of working as an equation.

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