Lesson 16: Parallel Lines and the Angles in a Triangle

Is each equation true or false?

$62-28= 60-30$

$3\boldcdot \text{-}8= (2\boldcdot \text{-}8) - 8$

Here is triangle $ABC$.

Select the Midpoint tool  and click on two points or a segment to find the midpoint.

1. Rotate triangle $ABC$ $180^\circ$ around the midpoint of side $AC$. Right click on the point and select Rename to label the new vertex $D$.

2. Rotate triangle $ABC$ $180^\circ$ around the midpoint of side $AB$. Right click on the point and select Rename to label the new vertex $E$.

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

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

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

6. What is the sum of the measures of angles $ABC$, $BAC$, and $ACB$?

Here is $\triangle ABC$. Line $DE$ is parallel to line $AC$.

1. What is $m{\angle DBA} + b + m{\angle CBE}$? Explain how you know.

2. Use your answer to explain why $a + b + c = 180$.

3. Explain why your argument will work for any triangle: that is, explain why the sum of the angle measures in any triangle is $180^\circ$.

This diagram shows a square $BDFH$ that has been made by images of triangle $ABC$ under rigid transformations.