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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 \begin{align*}55^\circ\end{align*}55? 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 \begin{align*}180^\circ\end{align*}180.

\begin{align*}m \angle{1} + m \angle{2} + m\angle{3} = 180^\circ\end{align*}

m1+m2+m3=180
.

Here is one proof of the Triangle Sum Theorem.

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

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

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

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

Example A

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

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

\begin{align*}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\end{align*}

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

\begin{align*}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\end{align*}

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

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

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

Example C

Find the measure of the missing angle.

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

\begin{align*}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\end{align*}

CK-12 Triangle Sum Theorem

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Guided Practice

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

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

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

Answers:

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

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

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

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

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

\begin{align*}(8x-1)^\circ + (3x+9)^\circ+(3x+4)^\circ&=180^\circ\\ (14x+12)^\circ&=180^\circ\\ 14x = 168\\ x =12\end{align*}

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

\begin{align*}[3(12) + 9]^\circ = 45^\circ && [3(12) + 4]^\circ = 40^\circ && [8(12) - 1]^\circ = 95^\circ\end{align*}

Explore More

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

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

9. Two interior angles of a triangle measure \begin{align*}111^\circ\end{align*} and \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

Triangle Sum Theorem

Triangle Sum Theorem

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

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