topic badge

12.05 Angles within triangles

Lesson

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:

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

Let's look at each of these transversals in turn.

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

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

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

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

In other words: The angle sum of a triangle is 180\degree.

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 1

Consider the triangle below.

Triangle with angles of 49 and 41 degrees.

Is it a right triangle?

Worked Solution
Create a strategy

Find the remaining angle using the angle sum of a triangle.

Apply the idea

Let x be the remaining angle.

\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 triangle.

Example 2

Solve for the value of x in the diagram below.

Triangle with interior angles of 52 and x degrees and opposite 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 56\degreeEvaluate
Idea summary

The angle sum of a triangle is 180\degree.

The exterior angle of a triangle is equal to the sum of the opposite interior angles.

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.

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

The base of both isosceles triangles has been highlighted.

Examples

Example 3

What kind of triangle is this?

A triangle with angles of 41 degrees and 98 degrees.
A
isosceles
B
scalene
Worked Solution
Create a strategy

Find the remaining angle to classify the triangle.

Apply the idea

Let x be the remaining angle, and use the angle sum of a triangle.

\displaystyle x+41+98\displaystyle =\displaystyle 180Add the angles and equate to 180
\displaystyle x+139\displaystyle =\displaystyle 180Evaluate the addition
\displaystyle x\displaystyle =\displaystyle 180-139Subtract 139 from both sides
\displaystyle x\displaystyle =\displaystyle 41\degreeEvaluate

This means that two angles are 41\degree ,so the triangle is isosceles. So, the correct answer is Option A.

Idea summary

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.

Outcomes

MA4-17MG

classifies, describes and uses the properties of triangles and quadrilaterals, and determines congruent triangles to find unknown side lengths and angles

MA4-18MG

identifies and uses angle relationships, including those related to transversals on sets of parallel lines

What is Mathspace

About Mathspace