Lesson 16

Is each equation true or false?

\(62-28= 60-30\)

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

Consider 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.

Given that angle \(BAC\) measures 53 degrees, find as many other angle measures as you can.

Using parallel lines and rotations, we can understand why the angles in a triangle always add to \(180^\circ\). Here is triangle \(ABC\). Line \(DE\) is parallel to \(AC\) and contains \(B\).

A 180 degree rotation of triangle \(ABC\) around the midpoint of \(AB\) interchanges angles \(A\) and \(DBA\) so they have the same measure: in the picture these angles are marked as \(x^\circ\). A 180 degree rotation of triangle \(ABC\) around the midpoint of \(BC\) interchanges angles \(C\) and \(CBE\) so they have the same measure: in the picture, these angles are marked as \(z^\circ\). Also, \(DBE\) is a straight line because 180 degree rotations take lines to parallel lines. So the three angles with vertex \(B\) make a line and they add up to \(180^\circ\) (\(x + y + z = 180\)). But \(x, y, z\) are the measures of the three angles in \(\triangle ABC\) so the sum of the angles in a triangle is always \(180^\circ\)!