Properties of the Angles Of A Triangle

In these lessons, we will give a summary of the properties of the angles of a triangle.

• Triangle Sum Theorem - The sum of the 3 angles in a triangle is always 180°
• The sum of an interior angle and its adjacent exterior angle is 180°
• Exterior Angle Theorem - An exterior angle of a triangle is equal to the sum of the two opposite interior angles
• An equilateral triangle has 3 equal angles that are 60° each. An isosceles triangle has 2 equal angles, which are the angles opposite the 2 equal sides

The following figures show some Angles in Triangle Theorems: Triangle Sum Theorem, Exterior Angle Theorem. Scroll down the page for more examples and solutions.

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The angles of a triangle have the following properties:

Property 1: Triangle Sum Theorem

The sum of the 3 angles in a triangle is always 180°.

Example:

**How to prove the Triangle Sum Theorem?**
The following video shows how to prove that the sum of the angles of a triangle is 180 degrees. (Triangle Sum Theorem)

• Show Step-by-step Solutions

How to Find the Missing Angle in a Triangle Using the Triangle Sum Theorem?
Step 1: Write out the equation by adding all the angles and making them equal to 180°.
Step 2: Solve for x.
Step 3: Substitute to find the missing angles.

• Show Step-by-step Solutions

Property 2:

The sum of an interior angle and its adjacent exterior angle is 180°.

Example:

Property 3: Exterior Angle Theorem

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

Example:

Exterior Angles of a Triangle
Finding the Unknown Angle of a Triangle
Example:

1. Evaluate triangle abc, where a = 40° and b = 60°. What is the exterior angle to ∠acb?
   
2. Evaluate triangle abc, where a = 50° and b = 30°. What is the exterior angle to ∠acb?
   
3. Evaluate triangle abc, where a = 90° and b = 40°. What is the exterior angle to ∠acb?
   

• Show Step-by-step Solutions

How to use the Exterior Angle Theorem to solve problems involving angles in a triangle?

• Show Step-by-step Solutions

Property 4:

An equilateral triangle has 3 equal angles that are 60° each.
An isosceles triangle has 2 equal angles, which are the angles opposite the 2 equal sides.

How to Find the Missing Angle in an Isosceles Triangle?

• Show Step-by-step Solutions

Parallel Lines and the Triangle Angle-Sum Theorem

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The above angle properties can help us to find unknown angles in a triangle.

Example:
Find the value of x in the following triangle.

Solution:
x + 24° + 32° = 180° (sum of angles is 180°)
x + 56° = 180°
x = 180° – 56° = 124°

Example:
Find the values of x and y in the following triangle.

Solution:
x + 50° = 92° (sum of opposite interior angles = exterior angle)
x = 92° – 50° = 42°

y + 92° = 180° (interior angle + adjacent exterior angle = 180°.)
y = 180° – 92° = 88°

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