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9.04 Triangles

Lesson

Classifications of 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).

First box has 3 points. Second box has the points connected to form a triangle. Last box shows a triangle with angle arcs.

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:

This image shows a table of classifications of triangles. Ask your teacher for more information.

Equilateral triangles are always acute isosceles triangles.

This image shows two isoscles triangles. Ask your teacher for more information.

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.

Exploration

Use the following applet to explore more on the different types of triangles.

Loading interactive...

A triangle classification can be changed by adjusting the lengths of two sides.

Examples

Example 1

Consider the triangle below.

Triangle with the same markings on two sides.
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.

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:

This image shows a table of classifications of triangles. Ask your teacher for more information.
Apply the idea

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

Idea summary

Triangles can be classified by their angles and their sides as shown in the diagram below:

This image shows a table of classifications of triangles. Ask your teacher for more information.

Equilateral triangles are always acute isosceles triangles.

Angle sum of a triangle

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

This image shows two parallel lines and two transversals forming a triangle. Ask your teacher for more information.

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.

This image shows parallel lines, two transversals, and alternate interior angles. Ask your teacher for more information.

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

This image shows parallel lines, two transversals, and corresponding angles. Ask your teacher for more information.

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

This image shows parallel lines, two transversals, and straight angle. Ask your teacher for more information.

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.

Triangle with angles of 49 and 41 degrees.

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

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

Idea summary

The sum of the angles in a triangle is 180\degree.

Exterior angles in a triangle

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.

This image shows exterior angles of a triangle. Ask your teacher for more information.

Examples

Example 3

Solve for the value of x in the diagram below.

Triangle with interior angles of 52 and x degrees and exterior angle of 108 degrees.
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 108Add the interior angles and equate to 108
\displaystyle x+52-52\displaystyle =\displaystyle 108-52Subtract 52 from both sides
\displaystyle x\displaystyle =\displaystyle 56Evaluate
Idea summary

An exterior angle is equal to the sum of the opposite interior angles.

Outcomes

VCMMG263

Demonstrate that the angle sum of a triangle is 180° and use this to find the angle sum of a quadrilateral

VCMMG262

Demonstrate that the angle sum of a triangle is 180° and use this to find the angle sum of a quadrilateral

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