7.02 Angles in triangles

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

• If all the angles are acute (measure less than 90\degree), the triangle is an acute triangle

• If one of the angles is a right angle (measures 90\degree), the triangle is a right triangle

• If one of the angles is obtuse (measures greater than 90\degree), 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 can have many different combinations of classifications based on their sides and angles. These combinations are outlined in the table below.

Note: Equilateral triangles are always acute because they always have three 60\degree angles.

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, like this:

Using what we learned about  parallel lines and transversals  , let's look at each of these transversals in turn.

Using the first transversal, we can mark two congruent alternate interior angles:

And using the second transversal, we find two congruent corresponding angles:

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 triangle?

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 triangle.

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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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