Lesson 16Parallel Lines and the Angles in a Triangle
Let’s see why the angles in a triangle add to 180 degrees.
Learning Targets:
 I can explain using pictures why the sum of the angles in any triangle is 180 degrees.
16.1 True or False: Computational Relationships
Is each equation true or false?
16.2 Angle Plus Two
Here is triangle .
Select the Midpoint tool
and click on two points or a segment to find the midpoint.
Rotate triangle around the midpoint of side . Right click on the point and select Rename to label the new vertex .

Rotate triangle around the midpoint of side . Right click on the point and select Rename to label the new vertex .

Look at angles , , and . Without measuring, write what you think is the sum of the measures of these angles. Explain or show your reasoning.

Is the measure of angle equal to the measure of any angle in triangle ? If so, which one? If not, how do you know?

Is the measure of angle equal to the measure of any angle in triangle ? If so, which one? If not, how do you know?

What is the sum of the measures of angles , , and ?
16.3 Every Triangle in the World
Here is . Line is parallel to line .

What is ? Explain how you know.
 Use your answer to explain why .

Explain why your argument will work for any triangle: that is, explain why the sum of the angle measures in any triangle is .
Are you ready for more?

Using a ruler, create a few quadrilaterals. Use a protractor to measure the four angles inside the quadrilateral. What is the sum of these four angle measures?

Come up with an explanation for why anything you notice must be true (hint: draw one diagonal in each quadrilateral).
16.4 Four Triangles Revisited
This diagram shows a square that has been made by images of triangle under rigid transformations.
Lesson 16 Summary
Using parallel lines and rotations, we can understand why the angles in a triangle always add to . Here is triangle . Line is parallel to and contains .
A 180 degree rotation of triangle around the midpoint of interchanges angles and so they have the same measure: in the picture these angles are marked as . A 180 degree rotation of triangle around the midpoint of interchanges angles and so they have the same measure: in the picture, these angles are marked as . Also, is a straight line because 180 degree rotations take lines to parallel lines. So the three angles with vertex make a line and they add up to (). But are the measures of the three angles in so the sum of the angles in a triangle is always !
Lesson 16 Practice Problems

For each triangle, find the measure of the missing angle.

Is there a triangle with two right angles? Explain your reasoning.

In this diagram, lines and are parallel.
Angle measures and angle measures .
 What is ?
 What is ?
 What is ?

The two figures are congruent.
 Label the points , and that correspond to , , and in the figure on the right.
 If segment measures 2 cm, how long is segment ? Explain.
 The point is shown in addition to and . How can you find the point that corresponds to ? Explain your reasoning.