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2. Parallel & Perpendicular Lines
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Geometry

2.03 Parallel lines and triangles

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Practice
Lesson

Concept summary

Triangle sum theorem

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

A triangle with an auxillary line is drawn on one of the vertices and parallel to the opposite side of the triangle. Three angles were formed from the auxillary 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 auxilary 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 auxilary 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.

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, we get that:

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

Worked examples

Example 1

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

A triangle with two of its interior angles are labelled 72 degrees and 58 degrees respectively.

Approach

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.

Solution

By the triangle sum theorem, we have that:

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

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

Reflection

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 2

Determine whether or not the diagram is valid.

A triangle with an exterior angle measuring 73 degrees and remote interior angles measuring 36 degrees and 39 degrees is drawn.

Approach

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.

Solution

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, it is not valid.

Reflection

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.

Outcomes

G.CO.C.8

Use definitions and theorems about lines and angles to solve problems and to justify relationships in geometric figures.

G.CO.C.9

Use definitions and theorems about triangles to solve problems and to justify relationships in geometric figures.

G.MP1

Make sense of problems and persevere in solving them.

G.MP3

Construct viable arguments and critique the reasoning of others.

G.MP4

Model with mathematics.

G.MP6

Attend to precision.

G.MP7

Look for and make use of structure.

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