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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 $55^\circ$ ? 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.

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### 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.

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

### 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.

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.

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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.

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### 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.