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3. Transformations & Congruence
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United States of AmericaUT
Grade 8

3.06 Angles in triangles

Lesson
Practice
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).

Classifying 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^\circ$90°), the triangle is an acute triangle
  • If one of the angles is a right angle (measures $90^\circ$90°), the triangle is a right triangle
  • If one of the angles is obtuse (measures greater than $90^\circ$90°), 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.

Summary

Note: Equilateral triangles are always acute because they always have three $60^\circ$60° 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 has been highlighted.

 

Interior angles in 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, 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:

Summary

The sum of the angles in a triangle is $180^\circ$180°.

 

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.

Worked example

Question 1

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°

Reflect: 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 2

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 3

Consider the triangle below.

A triangle with three vertices, each marked by a solid dot. The vertex at the right measures $41$41 degrees marked with a yellow arc. The vertex at the top-left measures $49$49 degrees marked with a blue arc; the vertex at the bottom-left is unlabeled.

  1. Is it a right triangle?

    Yes

    A

    No

    B
Question 4

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

Enter each line of working as an equation.

Outcomes

8.G.5

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

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