For number 4, we'll need a few facts to answer our question:
- Two supplementary angles add up to 180°, forming a straight angle (the angle formed by a straight line)
- The interior angles of a triangle add up to 180°
Given those, we notice that the one unlabeled angle in the figure shares a line with 156°. In fact, this angle is supplementary to 156°, which means that the two add up to 180°. To find the measure of this mystery angle, we can subtract 156 from 180 to obtain 180 - 156 = 24°.
Now, let's look at the triangle. We already know the measure of one of the angles is 24°, and the other two are x°. What else do we know about the angles of a triangle? From the two facts listed at the beginning, we know their interior angles add up to 180°, so let's use that fact to solve for x.
We have:
[tex]x+x+24=180[/tex], or [tex]2x+24=180[/tex]
Solving for x:
[tex](2x+24)-24=180-24\\2x=156\\(2x)/2=(156)/2\\x=78[/tex]
So, x = 78°.
For question 5, the definition of a pair of parallel lines is a pair of lines which never intersect, so "always" would be the appropriate answer.
