<meta http-equiv="refresh" content="1; url=/nojavascript/"> Triangle Sum Theorem ( Read ) | Geometry | CK-12 Foundation
Dismiss
Skip Navigation

Triangle Sum Theorem

%
Best Score
Practice Triangle Sum Theorem
Practice
Best Score
%
Practice Now
Triangle Sum Theorem
 0  0  0

What if you knew that two of the angles in a triangle measured 55^\circ ? How could you find the measure of the third angle? After completing this Concept, you'll be able to apply the Triangle Sum Theorem to solve problems like this one.

Watch This

CK-12 Triangle Sum Theorem

James Sousa: Animation of the Sum of the Interior Angles of a Triangle

Now watch this video.

James Sousa: Proving the Triangle Sum Theorem

Guidance

The Triangle Sum Theorem says that the three interior angles of any triangle add up to 180^\circ .

m \angle{1} + m \angle{2} + m\angle{3} = 180^\circ .

Here is one proof of the Triangle Sum Theorem.

Given : \triangle{ABC} with \overleftrightarrow{AD} || \overline{BC}

Prove : m\angle 1 + m\angle 2 + m\angle 3=180^\circ

Statement Reason
1. \triangle{ABC} with \overleftrightarrow{AD}||\overline{BC} Given
2. \angle{1} \cong \angle{4}, \ \angle{2} \cong \angle{5} Alternate Interior Angles Theorem
3. m\angle{1} = m\angle{4}, \ m\angle{2} = m\angle{5} \cong angles have = measures
4. m\angle{4} + m\angle{CAD} = 180^\circ Linear Pair Postulate
5. m\angle{3} + m\angle{5} = m\angle{CAD} Angle Addition Postulate
6. m\angle{4} + m\angle{3} + m\angle{5} = 180^\circ Substitution PoE
7. m\angle{1} + m\angle{3} + m\angle{2} = 180^\circ Substitution PoE

You can use the Triangle Sum Theorem to find missing angles in triangles.

Example A

What is m\angle{T} ?

We know that the three angles in the triangle must add up to 180^\circ . To solve this problem, set up an equation and substitute in the information you know.

m\angle{M} + m\angle{A} + m\angle{T} & = 180^\circ\\82^\circ + 27^\circ + m\angle{T} &= 180^\circ\\109^\circ + m\angle{T} & = 180^\circ\\m\angle{T} & = 71^\circ

Example B

What is the measure of each angle in an equiangular triangle?

To solve, remember that \triangle{ABC} is an equiangular triangle, so all three angles are equal. Write an equation.

m\angle{A}+m\angle{B}+m\angle{C} & = 180^\circ\\m\angle{A}+m\angle{A}+m\angle{A}& = 180^\circ \qquad Substitute, \ all \ angles \ are \ equal.\\3m\angle{A} & = 180^\circ \qquad Combine \ like \ terms.\\m\angle{A} & = 60^\circ

If m\angle{A} = 60^\circ , then m\angle{B} = 60^\circ and m\angle{C} = 60^\circ .

Each angle in an equiangular triangle is 60^\circ .

Example C

Find the measure of the missing angle.

We know that m\angle{O} = 41^\circ and m\angle{G} = 90^\circ because it is a right angle. Set up an equation like in Example A.

m\angle{D} + m\angle{O} + m\angle{G} & = 180^\circ\\m\angle{D} + 41^\circ + 90^\circ & = 180^\circ\\m\angle{D} + 41^\circ & = 90^\circ\\m\angle{D} & = 49^\circ

CK-12 Triangle Sum Theorem

Guided Practice

1. Determine m\angle{1} in this triangle:

2. Two interior angles of a triangle measure 50^\circ and 70^\circ . What is the third interior angle of the triangle?

3. Find the value of x and the measure of each angle.

Answers:

1. 72^\circ + 65^\circ +m\angle{1} = 180^\circ .

Solve this equation and you find that m\angle{1}=43^\circ .

2. 50^\circ + 70^\circ + x = 180^\circ .

Solve this equation and you find that the third angle is 60^\circ .

3. All the angles add up to 180^\circ .

(8x-1)^\circ + (3x+9)^\circ+(3x+4)^\circ&=180^\circ\\(14x+12)^\circ&=180^\circ\\14x = 168\\x =12

Substitute in 12 for x to find each angle.

[3(12) + 9]^\circ = 45^\circ && [3(12) + 4]^\circ = 40^\circ && [8(12) - 1]^\circ = 95^\circ

Practice

Determine m\angle{1} in each triangle.

1.

2.

3.

4.

5.

6.

7.

8. Two interior angles of a triangle measure 32^\circ and 64^\circ . What is the third interior angle of the triangle?

9. Two interior angles of a triangle measure 111^\circ and 12^\circ . What is the third interior angle of the triangle?

10. Two interior angles of a triangle measure 2^\circ and 157^\circ . What is the third interior angle of the triangle?

Find the value of x and the measure of each angle.

11.

12.

13.

14.

15.

Image Attributions

Reviews

Email Verified
Well done! You've successfully verified the email address .
OK
Please wait...
Please wait...

Original text