b How can you describe ∠2 and ∠3 as a pair of angles? Does this change when we open the box?
C
c Notice that ∠1 is measured using the box's vertical side.
A
a m∠2=70^(∘)
m∠3=110^(∘)
B
b See solution
C
c ∠ABC is still a straight angle.
Opening will be steeper.
Practice makes perfect
a Seen from the side, the box will look like below. Notice that the vertical sides of the box are parallel.
Since, the vertical sides are parallel, we can, by the Alternate Interior Angles Theorem, say that ∠1≅ ∠2. Also, ∠1 and ∠3 are consecutive interior angles, so by the Consecutive Interior Angles Theorem, we know that m∠1 and 70^(∘) are sum to 180^(∘). Using these theorems, we can write the following equations:
m∠2 =70^(∘) and m∠3+70^(∘)=180^(∘).
Let's solve the second equation.
b Examining the closed box, we see that ∠2 and ∠3 form a linear pair. According to the Linear Pair Postulate, these angles are supplementary which means their measures add up to 180^(∘). m∠2+m∠3=180^(∘)
When the box is opened, ∠ABC is made up of ∠2 and ∠3 as we now have classified as supplementary. Therefore, ∠ABC has to be a straight angle.
c If m∠1=60^(∘), the measure of ∠2 and ∠3 will change but still remain a linear pair. Let's illustrate the new box.
Again, using the Alternate Interior Angles Theorem and the Consecutive Interior Angles Theorem, we can write the following equations:
m∠2 =60^(∘) and m∠3+60^(∘)=180^(∘).
Let's solve the second equation.
Thus, ∠ABC is still a straight angle since ∠2 and ∠3 still form a linear pair. However, the opening of the box, as seen in the diagram, will be steeper.
