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# Triangle Sum Theorem

## Interior angles add to 180 degrees

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Triangle Sum Theorem

What if you knew that two of the angles in a triangle measured 55\begin{align*}55^\circ\end{align*}? 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\begin{align*}180^\circ\end{align*}.

m1+m2+m3=180
.

Here is one proof of the Triangle Sum Theorem.

Given: ABC\begin{align*}\triangle{ABC}\end{align*} with AD||BC¯¯¯¯¯\begin{align*}\overleftrightarrow{AD} || \overline{BC}\end{align*}

Prove: m1+m2+m3=180\begin{align*}m\angle 1 + m\angle 2 + m\angle 3=180^\circ\end{align*}

Statement Reason
1. ABC\begin{align*}\triangle{ABC}\end{align*} with AD||BC¯¯¯¯¯\begin{align*}\overleftrightarrow{AD}||\overline{BC}\end{align*} Given
2. 14, 25\begin{align*}\angle{1} \cong \angle{4}, \ \angle{2} \cong \angle{5}\end{align*} Alternate Interior Angles Theorem
3. m1=m4, m2=m5\begin{align*}m\angle{1} = m\angle{4}, \ m\angle{2} = m\angle{5}\end{align*} \begin{align*}\cong\end{align*} angles have = measures
4. m4+mCAD=180\begin{align*}m\angle{4} + m\angle{CAD} = 180^\circ\end{align*} Linear Pair Postulate
5. m3+m5=mCAD\begin{align*}m\angle{3} + m\angle{5} = m\angle{CAD}\end{align*} Angle Addition Postulate
6. m4+m3+m5=180\begin{align*}m\angle{4} + m\angle{3} + m\angle{5} = 180^\circ\end{align*} Substitution PoE
7. m1+m3+m2=180\begin{align*}m\angle{1} + m\angle{3} + m\angle{2} = 180^\circ\end{align*} Substitution PoE

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

#### Example A

What is mT\begin{align*}m\angle{T}\end{align*}?

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

mM+mA+mT82+27+mT109+mTmT=180=180=180=71

#### Example B

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

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

mA+mB+mCmA+mA+mA3mAmA=180=180Substitute, all angles are equal.=180Combine like terms.=60

If mA=60\begin{align*}m\angle{A} = 60^\circ\end{align*}, then mB=60\begin{align*}m\angle{B} = 60^\circ\end{align*} and mC=60\begin{align*}m\angle{C} = 60^\circ\end{align*}.

Each angle in an equiangular triangle is 60\begin{align*}60^\circ\end{align*}.

#### Example C

Find the measure of the missing angle.

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

mD+mO+mGmD+41+90mD+41mD=180=180=90=49

CK-12 Triangle Sum Theorem

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### Guided Practice

1. Determine m1\begin{align*}m\angle{1}\end{align*} in this triangle:

2. Two interior angles of a triangle measure 50\begin{align*}50^\circ\end{align*} and 70\begin{align*}70^\circ\end{align*}. What is the third interior angle of the triangle?

3. Find the value of x\begin{align*}x\end{align*} and the measure of each angle.

1. 72+65+m1=180\begin{align*}72^\circ + 65^\circ +m\angle{1} = 180^\circ \end{align*}.

Solve this equation and you find that m1=43\begin{align*}m\angle{1}=43^\circ\end{align*}.

2. 50+70+x=180\begin{align*}50^\circ + 70^\circ + x = 180^\circ \end{align*}.

Solve this equation and you find that the third angle is 60\begin{align*}60^\circ\end{align*}.

3. All the angles add up to 180\begin{align*}180^\circ\end{align*}.

(8x1)+(3x+9)+(3x+4)(14x+12)14x=168x=12=180=180

Substitute in 12 for x\begin{align*}x\end{align*} to find each angle.

[3(12)+9]=45[3(12)+4]=40[8(12)1]=95

### Explore More

Determine m1\begin{align*}m\angle{1}\end{align*} in each triangle.

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8. Two interior angles of a triangle measure 32\begin{align*}32^\circ\end{align*} and 64\begin{align*}64^\circ\end{align*}. What is the third interior angle of the triangle?

9. Two interior angles of a triangle measure 111\begin{align*}111^\circ\end{align*} and 12\begin{align*}12^\circ\end{align*}. What is the third interior angle of the triangle?

10. Two interior angles of a triangle measure \begin{align*}2^\circ\end{align*} and \begin{align*}157^\circ\end{align*}. What is the third interior angle of the triangle?

Find the value of \begin{align*}x\end{align*} and the measure of each angle.

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### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 4.1.

### Vocabulary Language: English

Triangle Sum Theorem

Triangle Sum Theorem

The Triangle Sum Theorem states that the three interior angles of any triangle add up to 180 degrees.