12.05 Angles within triangles

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:

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 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.

Examples

Example 1

Consider the triangle below.

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 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 triangle.

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Example 2

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\degree	Evaluate

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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.

Examples

Example 3

What kind of triangle is this?

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

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

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