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# Triangle Sum Theorem

## Interior angles add to 180 degrees

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Triangle Sum Theorem

What if you wanted to classify the Bermuda Triangle by its sides and angles? You are probably familiar with the myth of this triangle; how several ships and planes passed through and mysteriously disappeared.

The measurements of the sides of the triangle from a map are in the image. What type of triangle is this? Using a protractor, find the measure of each angle in the Bermuda Triangle. What do they add up to? Do you think the three angles in this image are the same as the three angles in the actual Bermuda triangle?

### Triangle Sum Theorem

In polygons, interior angles are the angles inside of a closed figure with straight sides. The vertex is the point where the sides of a polygon meet.

Triangles have three interior angles, three vertices and three sides. A triangle is labeled by its vertices with a \begin{align*}\triangle\end{align*}. This triangle can be labeled ABC,ACB,BCA,BAC,CBA\begin{align*}\triangle ABC, \triangle ACB, \triangle BCA, \triangle BAC, \triangle CBA\end{align*} or CAB\begin{align*}\triangle CAB\end{align*}. Order does not matter. The angles in any polygon are measured in degrees. Each polygon has a different sum of degrees, depending on the number of angles in the polygon. How many degrees are in a triangle?

#### Investigation: Triangle Tear-Up

Tools Needed: paper, ruler, pencil, colored pencils

1. Draw a triangle on a piece of paper. Try to make all three angles different sizes. Color the three interior angles three different colors and label each one, 1,2,\begin{align*}\angle 1, \angle 2,\end{align*} and 3\begin{align*}\angle 3\end{align*}.
2. Tear off the three colored angles, so you have three separate angles.
3. Attempt to line up the angles so their points all match up. What happens? What measure do the three angles add up to?

This investigation shows us that the sum of the angles in a triangle is 180\begin{align*}180^\circ\end{align*} because the three angles fit together to form a straight line. Recall that a line is also a straight angle and all straight angles are 180\begin{align*}180^\circ\end{align*}.

The Triangle Sum Theorem states that the interior angles of a triangle add up to 180\begin{align*}180^\circ\end{align*}. The above investigation is one way to show that the angles in a triangle add up to 180\begin{align*}180^\circ\end{align*}. However, it is not a two-column proof. Here we will prove 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*} above 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

There are two theorems that we can prove as a result of the Triangle Sum Theorem and our knowledge of triangles.

Theorem #1: Each angle in an equiangular triangle measures 60\begin{align*}60^\circ\end{align*}.

Theorem #2: The acute angles in a right triangle are always complementary.

#### Measuring Angles

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

From the Triangle Sum Theorem, we know that the three angles add up to 180\begin{align*}180^\circ\end{align*}. Set up an equation to solve for T\begin{align*}\angle T\end{align*}.

mM+mA+mT82+27+mT109+mTmT=180=180=180=71\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*}

#### Proving Theorem #1

Show why Theorem #1 is true.

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

mA+mB+mCmA+mA+mA3mAmA=180=180=180=60\begin{align*}m \angle A+m \angle B+m \angle C &= 180^\circ\\ m \angle A+m \angle A+m \angle A &= 180^\circ\\ 3m \angle A &= 180^\circ\\ m \angle A &= 60^\circ\end{align*}

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*}.

#### Proving Theorem #2

Use the picture below to show why Theorem #2 is true.

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.

mD+mO+mGmD+41+90mD+41mD=180=180=90=49\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*}

Notice that mD+mO=90\begin{align*}m \angle D + m \angle O = 90^\circ\end{align*} because G\begin{align*}\angle G\end{align*} is a right angle.

#### Bermuda Triangle Problem Revisited

The Bermuda Triangle is an acute scalene triangle. The angle measures are in the picture below. Your measured angles should be within a degree or two of these measures. The angles should add up to 180o\begin{align*}180^{\text{o}}\end{align*}. However, because your measures are estimates using a protractor, they might not exactly add up.

The angle measures in the picture are the measures from a map (which is flat). Because the earth is curved, in real life the measures will be slightly different.

### Examples

#### Example 1

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

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*}.

#### Example 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?

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*}.

#### Example 3

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

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\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 x\begin{align*}x\end{align*} to find each angle.

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

### Review

Determine m1\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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