Properties of the Angles Of A Triangle
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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)
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.
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:
- Evaluate triangle abc, where a = 40° and b = 60°. What is the exterior angle to ∠acb?
- Evaluate triangle abc, where a = 50° and b = 30°. What is the exterior angle to ∠acb?
- Evaluate triangle abc, where a = 90° and b = 40°. What is the exterior angle to ∠acb?
How to use the Exterior Angle Theorem to solve problems involving angles in a triangle?
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?
Parallel Lines and the Triangle Angle-Sum Theorem
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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