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# 4.1: Triangle Sums

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Understand the Triangle Sum Theorem.
• Identify interior and exterior angles in a triangle.
• Use the Exterior Angle Theorem.

## Review Queue

Classify the triangles below by their angles and sides.

1. Draw and label a straight angle, ABC\begin{align*}\angle{ABC}\end{align*}. Which point is the vertex? How many degrees does a straight angle have?

Know What? Below is a map of the Bermuda Triangle. The myth of this triangle is that ships and planes have passed through and mysteriously disappeared.

The measurements of the sides of the triangle are in the picture. Classify the Bermuda triangle by its sides and angles. Then, using a protractor, find the measure of each angle. What do they add up to?

Recall that a triangle can be classified by its sides...

and its angles...

Interior Angles: The angles inside of a polygon.

Vertex: 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*}.

Triangle Sum Theorem The interior angles in a polygon are measured in degrees. How many degrees are there in a triangle?

Investigation 4-1: Triangle Tear-Up

Tools Needed: paper, ruler, pencil, colored pencils

1. Draw a triangle on a piece of paper. 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. Line up the angles so the vertices 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 angle where all the vertices meet.

Triangle Sum Theorem: The interior angles of a triangle add up to 180\begin{align*}180^\circ\end{align*}.

m1+m2+m3=180

Example 1: What mT\begin{align*}m\angle{T}\end{align*}?

Solution: Set up an equation.

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

Even thought Investigation 4-1 is a way to show that the angles in a triangle add up to 180\begin{align*}180^\circ\end{align*}, it is not a proof. Here is the 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*} 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

Example 2: What is the measure of each angle in an equiangular triangle?

Solution: ABC\begin{align*}\triangle{ABC}\end{align*} is an equiangular triangle, where 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 3: Find the measure of the missing angle.

Solution: 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

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

The acute angles in a right triangle are always complementary.

## Exterior Angles

Exterior Angle: The angle formed by one side of a polygon and the extension of the adjacent side.

In all polygons, there are two sets of exterior angles, one that goes around clockwise and the other goes around counterclockwise.

Notice that the interior angle and its adjacent exterior angle form a linear pair and add up to 180\begin{align*}180^\circ\end{align*}.

m1+m2=180

Example 4: Find the measure of RQS\begin{align*}\angle{RQS}\end{align*}.

Solution: 112\begin{align*}112^\circ\end{align*} is an exterior angle of RQS\begin{align*}\triangle{RQS}\end{align*} and is supplementary to RQS\begin{align*}\angle{RQS}\end{align*}.

112+mRQSmRQS=180=68

Example 5: Find the measure of the numbered interior and exterior angles in the triangle.

Solution: m1+92=180\begin{align*}m\angle{1} + 92^\circ = 180^\circ\end{align*} by the Linear Pair Postulate. m1=88\begin{align*}m\angle{1} = 88^\circ\end{align*}

m2+123=180\begin{align*}m\angle{2} + 123^\circ = 180^\circ\end{align*} by the Linear Pair Postulate. m2=57\begin{align*}m\angle{2} = 57^\circ\end{align*}

m1+m2+m388+57+m3m3=180by the Triangle Sum Theorem.=180=35

Lastly, m3+m435+m4m4=180by the Linear Pair Postulate.=180=145

In Example 5, the exterior angles are 92, 123\begin{align*}92^\circ, \ 123^\circ\end{align*}, and 145\begin{align*}145^\circ\end{align*}. Adding these angles together, we get 92+123+145=360\begin{align*}92^\circ + 123^\circ + 145^\circ = 360^\circ\end{align*}. This is true for any set of exterior angles for any polygon.

Exterior Angle Sum Theorem: The exterior angles of a polygon add up to 360\begin{align*}360^\circ\end{align*}.

m1+m2+m3m4+m5+m6=360=360

Example 6: What is the value of p\begin{align*}p\end{align*} in the triangle below?

Solution: First, we need to find the missing exterior angle, let’s call it x\begin{align*}x\end{align*}. Set up an equation using the Exterior Angle Sum Theorem.

130+110+xxx=360=360130110=120

x\begin{align*}x\end{align*} and p\begin{align*}p\end{align*} add up to 180\begin{align*}180^\circ\end{align*} because they are a linear pair.

x+p120+pp=180=180=60

Example 7: Find mA\begin{align*}m\angle{A}\end{align*}.

Solution:

mACB+115=180 because they are a linear pairmACB=65mA+65+79=180by the Triangle Sum TheoremmA=36

Remote Interior Angles: The two angles in a triangle that are not adjacent to the indicated exterior angle.

In Example 7 above, A\begin{align*}\angle{A}\end{align*} and 79\begin{align*}79^\circ\end{align*} are the remote interior angles relative to 115\begin{align*}115^\circ\end{align*}.

Exterior Angle Theorem From Example 7, we can find the sum of mA\begin{align*}m\angle{A}\end{align*} and mB\begin{align*}m\angle{B}\end{align*}, which is 36+79=115\begin{align*}36^\circ + 79^\circ = 115^\circ\end{align*}. This is equal to the exterior angle at C\begin{align*}C\end{align*}.

Exterior Angle Theorem: The sum of the remote interior angles is equal to the non-adjacent exterior angle.

mA+mB=mACD

## Proof of the Exterior Angle Theorem

Given: Triangle with exterior 4\begin{align*}\angle{4}\end{align*}

Prove: m1+m2=m4\begin{align*}m\angle{1} + m\angle{2}=m\angle{4}\end{align*}

Statement Reason
1. Triangle with exterior 4\begin{align*}\angle{4}\end{align*} Given
2. m1+m2+m3=180\begin{align*}m\angle{1}+m\angle{2}+m\angle{3}=180^\circ\end{align*} Triangle Sum Theorem
3. m3+m4=180\begin{align*}m\angle{3}+m\angle{4}=180^\circ\end{align*} Linear Pair Postulate
4. m1+m2+m3=m3+m4\begin{align*}m\angle{1}+m\angle{2}+m\angle{3}=m\angle{3}+m\angle{4}\end{align*} Transitive PoE
5. m1+m2=m4\begin{align*}m\angle{1}+m\angle{2}=m\angle{4}\end{align*} Subtraction PoE

Example 8: Find mC\begin{align*}m\angle{C}\end{align*}.

Solution: Using the Exterior Angle Theorem

mC+16mTCA=121=105

If you forget the Exterior Angle Theorem, you can do this problem just like Example 7.

Example 9: Algebra Connection Find the value of x\begin{align*}x\end{align*} and the measure of each angle.

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

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

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

3(12)+9=453(12)+4=408(12)1=95

Example 10: Algebra Connection Find the value of x\begin{align*}x\end{align*} and the measure of each angle.

Solution: Set up an equation using the Exterior Angle Theorem.

(4x+2)+(2x9)=(5x+13)interior anglesexterior angle (6x7)=(5x+13)  x=20

Substitute in 20\begin{align*}20^\circ\end{align*} for x\begin{align*}x\end{align*} to find each angle.

4(20)+2=822(20)9=31Exterior angle: 5(20)+13=113

Know What? 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 180\begin{align*}180^\circ\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.

## Review Questions

• Questions 1-16 are similar to Examples 1-8.
• Questions 17 and 18 use the definition of an Exterior Angle and the Exterior Angle Sum Theorem.
• Question 19 is similar to Example 3.
• Questions 20-27 are similar to Examples 9 and 10.

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

1. Find the lettered angles, af\begin{align*}a - f\end{align*}, in the picture to the right. Note that the two lines are parallel.
2. Draw both sets of exterior angles on the same triangle.
1. What is m1+m2+m3\begin{align*}m\angle{1}+m\angle{2}+m\angle{3}\end{align*}?
2. What is m4+m5+m6\begin{align*}m\angle{4}+m\angle{5}+m\angle{6}\end{align*}?
3. What is m7+m8+m9\begin{align*}m\angle{7}+m\angle{8}+m\angle{9}\end{align*}?
4. List all pairs of congruent angles.
3. Fill in the blanks in the proof below. Given: The triangle to the right with interior angles and exterior angles. Prove: m4+m5+m6=360\begin{align*}m\angle{4}+m\angle{5}+m\angle{6}=360^\circ\end{align*}
Statement Reason
1. Triangle with interior and exterior angles. Given
2. m1+m2+m3=180\begin{align*}m\angle{1}+m\angle{2}+m\angle{3}=180^\circ\end{align*}
3. 3\begin{align*}\angle{3}\end{align*} and 4\begin{align*}\angle{4}\end{align*} are a linear pair, 2\begin{align*}\angle{2}\end{align*} and 5\begin{align*}\angle{5}\end{align*} are a linear pair, and 1\begin{align*}\angle{1}\end{align*} and 6\begin{align*}\angle{6}\end{align*} are a linear pair
4. Linear Pair Postulate (do all 3)
5. m1+m6=180m2+m5=180m3+m4=180\begin{align*}m\angle{1}+m\angle{6}=180^\circ\!\\ m\angle{2}+m\angle{5}=180^\circ\!\\ m\angle{3}+m\angle{4}=180^\circ\end{align*}
6. m1+m6+m2+m5+m3+m4=540\begin{align*}m\angle{1}+m\angle{6}+m\angle{2}+m\angle{5}+m\angle{3}+m\angle{4}=540^\circ\end{align*}
7. m4+m5+m6=360\begin{align*}m\angle{4}+m\angle{5}+m\angle{6}=360^\circ\end{align*}
1. Fill in the blanks in the proof below. Given: ABC\begin{align*}\triangle{ABC}\end{align*} with right angle B\begin{align*}B\end{align*}. Prove: A\begin{align*}\angle{A}\end{align*} and C\begin{align*}\angle{C}\end{align*} are complementary.
Statement Reason
1. ABC\begin{align*}\triangle ABC\end{align*} with right angle B\begin{align*}B\end{align*}. Given
2. Definition of a right angle
3. mA+mB+mC=180\begin{align*}m\angle{A} + m\angle{B}+m\angle{C}=180^\circ\end{align*}
4. mA+90+mC=180\begin{align*}m\angle{A}+90^\circ+m\angle{C}=180^\circ\end{align*}
5.
6. A\begin{align*}\angle{A}\end{align*} and C\begin{align*}\angle{C}\end{align*} are complementary

Algebra Connection Solve for x\begin{align*}x\end{align*}.

## Review Queue Answers

1. acute isosceles
2. obtuse scalene
3. right scalene
4. B\begin{align*}B\end{align*} is the vertex, 180\begin{align*}180^\circ\end{align*},

8 , 9 , 10

Feb 22, 2012

## Last Modified:

Jun 11, 2015
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CK.MAT.ENG.SE.1.Geometry-Basic.4.1