Topics
3. Triangles
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United States of America
Grades 9-12

3.01 Triangles and angles

Lesson
Practice

Introduction

We will apply what we know about parallel lines and transversals from lesson  2.01 Parallel lines and transversals  and physical manipulation of angles to solve problems involving triangles and prove the triangle angle sum theorem and the exterior angle theorem.

Interior angles of triangles

Exploration

Drag the vertices of the triangle to change the size of each angle. Check the box to explore.

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  1. What do you notice about the relationship between the angles in the triangle?
Triangle sum theorem

The sum of the measures of the interior angles of a triangle is 180\degree

A triangle with an auxiliary line is drawn on one of the vertices and parallel to the opposite side of the triangle. Three angles were formed from the auxiliary line and the vertex. Each non-parallel side of the triangle acts like a transversal of the two parallel lines. 2 Pairs of alternate interior angles are marked as equal.

Using the parallel postulate we know that we can construct an auxiliary line through one of the vertices of a triangle that is parallel to the opposite side.

The three marked angles that have the shared vertex on the auxiliary line can be used to help us prove that the sum of the measures of the interior angles of a triangle must be 180 \degree.

Examples

Example 1

Prove the triangle sum theorem.

Worked Solution
Create a strategy

To prove this theorem, construct a line \overleftrightarrow{QR} through point B that is parallel to \overline{AC}.

Triangle A B C with auxiliary line Q R drawn at vertex B.

We are given \triangle ABC with \overline{QR} \parallel \overline{AC} and need to prove m \angle BAC + m \angle ABC + m \angle BCA = 180.

Apply the idea
To prove: m \angle BAC + m \angle ABC + m \angle BCA = 180
StatementsReasons
1. \overline{QR} \parallel \overline{AC}Given
2.\angle QBC and \angle RBC are supplementaryLinear angle pair
3.m \angle QBC + m \angle RBC = 180Definition of supplementary angles
4.m \angle QBC = m \angle QBA + m \angle ABCAngle addition postulate
5.m \angle QBA + m\angle ABC + m \angle RBC = 180Substitution property
6.\begin{aligned} \angle QBA \cong \angle BAC \\ \angle RBC \cong \angle BCA \end{aligned}Alternate interior angle theorem
7.\begin{aligned} m\angle QBA = m\angle BAC\\ m\angle RBC = m\angle BCA \end{aligned}Congruent angles have the same angle measure
8.m \angle BAC + m \angle ABC + m \angle BCA =180Substitution property of equality

Example 2

Determine the measure of the third interior angle of the triangle.

A triangle with two of its interior angles are labeled 72 degrees and 58 degrees respectively.
Worked Solution
Create a strategy

The triangle sum theorem tells us that the sum of the measures of the triangle will be 180 \degree. Let the measure of the third interior angle be x\degree and solve for it.

Apply the idea

By the triangle sum theorem, we have that:

\displaystyle x+72+58\displaystyle =\displaystyle 180Triangle sum theorem
\displaystyle x+130\displaystyle =\displaystyle 180Evaluate the addition
\displaystyle x\displaystyle =\displaystyle 50Subtract 130 from both sides

So the measure of the third interior angle of the triangle is 50 \degree.

Reflect and check

Since any interior angle of a triangle is supplementary to the sum of the other two interior angles, we could also use the calculation x=180-(72+58) to reach the same result.

Example 3

Consider the diagram shown below:

A triangle with the first angle measuring x degrees and the second angle measuring 33 degrees. An internal line segment is drawn from the vertex of the third angle to the side opposite it. This internal segment divides the third angle into two angles measuring y degrees and 77 degrees, and also makes an angle of 65 degrees with the side opposite the third angle. The angles x degrees, 77 degrees and 65 degrees are on the same side if the internal line segment.

Solve for x and y.

Worked Solution
Create a strategy

Calculate the measure of x using the triangle sum theorem, then calculate the measure of y using the triangle sum theorem.

Apply the idea

The sum of the angles in a triangle is equal to 180 \degree, so we have:

\displaystyle x+65+77\displaystyle =\displaystyle 180Triangle sum theorem
\displaystyle x +142\displaystyle =\displaystyle 180Combine like terms
\displaystyle x\displaystyle =\displaystyle 38Subtract 142 from both sides

Now, we can see that the large triangle made up of both smaller triangles will also have an interior angle sum of 180 \degree. Note that one angle of the larger triangle is (y+77) \degree. So, we have:

\displaystyle y + 77 + 38+ 33\displaystyle =\displaystyle 180Triangle sum theorem
\displaystyle y+148\displaystyle =\displaystyle 180Combine like terms
\displaystyle y\displaystyle =\displaystyle 32Subtract 148 from both sides
Reflect and check

We could also solve for y after finding the unknown angle in the smaller bottom triangle. We know that a straight line is 180 \degree, so we know that the 65 \degree angle and the unknown angle that is supplementary to it must have a sum of 180 \degree: 180 \degree - 65 \degree= 115 \degree Then, we have:

\displaystyle y+115+33\displaystyle =\displaystyle 180Triangle sum theorem
\displaystyle y+148\displaystyle =\displaystyle 180Combine like terms
\displaystyle y\displaystyle =\displaystyle 32Subtract 148 on both sides
Idea summary

The triangle sum theorem states that the sum of the measures of the interior angles of a triangle is 180 \degree.

Exterior angles of triangles

Exploration

Drag the vertices of the triangle to change the size of each angle. Check the boxes to explore.

Loading interactive...
  1. What do you notice about the relationship between the interior and exterior angles?

Using the triangle sum theorem, we can also relate the measures of exterior angles and remote interior angles of a triangle.

Exterior angle of a polygon

The angle outside of a polygon, between one side of the polygon and the extension of an adjacent side. This angle forms a linear pair with the interior angle it is adjacent to.

A triangle with one side extended to the left as a line. The exterior angle is the angle between the extended side and the adjacent side of the triangle.
Remote interior angles

The interior angles of a polygon that are not adjacent to a given exterior angle.

A triangle has one of its exterior angles labeled. The remote interior angles are labeled as the two interior angles of the triangle that are not adjacent to the exterior angle.
Triangle exterior angle theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles of the triangle.

Triangle A B C is drawn such that angle B A P is outside the triangle between side B A and side A C forming a linear pair with angle B A C. Angle B A P is the exterior angle marked with an arc. Angle B marked with 2 arcs and angle C marked with 3 arcs are remote interior angles.

For this triangle, the triangle exterior angle theorem tells us that:

m \angle PAB = m \angle B + m \angle C

Examples

Example 4

Prove the triangle exterior angle theorem.

Worked Solution
Create a strategy

To prove this theorem, extend \overline{AC} to \overrightarrow{AD}.

Triangle A B C with an exterior angle B C D forming a linear pair with angle B C D.

We are given \triangle ABC with exterior angle \angle BCD and need to prove m \angle BAC + m \angle ABC = m \angle BCD.

We can use the triangle sum theorem and the definition of supplementary angles to help us prove the statement.

Apply the idea

We can prove the statement by using a two column proof.

To prove: m \angle BAC + m \angle ABC = m \angle BCD
StatementsReasons
1. m \angle BAC + m \angle ABC + m \angle BCA =180Triangle sum theorem
2.m \angle BCA + m \angle BCD = 180Linear pair postulate
3.m \angle BAC + m \angle ABC + m \angle BCA = m \angle BCA + m \angle BCDTransitive property of equality
4.m \angle BAC + m \angle ABC = m \angle BCDSubtraction property of equality
Reflect and check

We can also use a flow chart proof to prove the statement:

A flow chart proof with 4 levels and the reason below each step. The second level have 2 steps. On the first level, the step is labeled triangle A B C with exterior angle B C D, with reason Given. On the second level, the left step is labeled measure of angle B A C plus measure of angle A B C plus measure of angle B C A equals 180, with reason Triangle sum theorem, and the right step is labeled measure of angle B C A plus measure of angle B C D equals 180, with reason Linear pairs. On the third level, the left and right step from the second level now converge into a single step. The step is labeled measure of angle B A C plus measure of angle A B C plus measure of angle B C A equals measure of angle B C A plus measure of angle B C D, with reason Transitive property of equality. On the fourth level, the step is labeled measure of angle B A C plus measure of angle A B C equals measure of angle B C D, with reason Subtraction property of equality.

Example 5

Determine whether or not the angle measures given in the diagram are valid.

A triangle with an exterior angle measuring 73 degrees and remote interior angles measuring 36 degrees and 39 degrees is drawn.
Worked Solution
Create a strategy

The triangle exterior angle theorem tells us that the measure of the exterior angle should be equal to the sum of the measures of the two remote interior angles. If this is not the case, then the diagram cannot be valid.

Apply the idea

The exterior angle of the triangle has a measure of 73 \degree.

The two remote interior angles of the triangle have measures of 36\degree and 39 \degree.

Adding the measures of the two remote angles together gives us:

36+39=75\neq 73

Since the angle measures of the figure do not satisfy the triangle exterior angle theorem, they are not valid.

Reflect and check

Another way to show that the figure is not valid would be to find the measure of the third interior angle of the triangle, using the triangle sum theorem, and then showing that it is not supplementary to the exterior angle:

\displaystyle 36+39+x\displaystyle =\displaystyle 180Triangle sum theorem
\displaystyle 75+x\displaystyle =\displaystyle 180Combine like terms
\displaystyle x\displaystyle =\displaystyle 105Subtract 75 from both sides

Since the third angle in the triangle is 105 \degree, we should expect that it creates a linear pair with the exterior angle:105 + 73 = 178 \neq 180Since the angles are not supplementary, the angle measures are not valid.

Example 6

Solve for x in the diagram shown:

Triangle A B C. Angle B has a measure of 2 x minus 6 degrees. Angle C has a measure of 2 x plus 5 degrees. The exterior angle adjacent to vertex A has a measure of 109 degrees.
Worked Solution
Create a strategy

Relate the known angle in the diagram to the other two angles with measures given in terms of the variable x.

In a triangle, the measure of the exterior angle is equal to the sum of the measures of the two remote interior angles. We will express this relationship using an equation and solve for x.

Apply the idea

We have:

\displaystyle \left(2x-6\right)+\left(2x+5\right)\displaystyle =\displaystyle 109Triangle exterior angle theorem
\displaystyle 4x-1\displaystyle =\displaystyle 109Combine like terms
\displaystyle 4x\displaystyle =\displaystyle 110Add 1 to both sides
\displaystyle x\displaystyle =\displaystyle 27.5Divide both sides by 4
Reflect and check

Confirm that the angles satisfy the exterior angle theorem by substituting 27.5 in each expression for x:

\displaystyle \left(2x-6\right) \degree + \left(2x +5\right) \degree\displaystyle =\displaystyle 190 \degreeTriangle exterior angle theorem
\displaystyle \left[2\left(27.5\right)-6\right] \degree + \left[2\left(27.5\right) +5\right] \degree\displaystyle =\displaystyle 109 \degreeSubstitute x=27.5
\displaystyle \left(55-6\right) \degree + \left(55 + 5\right) \degree\displaystyle =\displaystyle 109 \degreeEvaluate the multiplication
\displaystyle 109 \degree\displaystyle =\displaystyle 109 \degreeEvaluate the subtraction and addition
Idea summary

The triangle exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles of the triangle.

Outcomes

G.CO.C.10

Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

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