3.06 Angles in 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).
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.
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:
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.
Which of the following words describes this triangle?
scalene
Aequilateral
Bisosceles
CWhich of the following words also describes this triangle?
acute
Aright-angled
Bobtuse
C
Question 3
Consider the triangle below.
Is it a right triangle?
Yes
ANo
B
Question 4
Solve for the value of $x$x in the diagram below.
Enter each line of working as an equation.
