b The angle measuring (3y-17)^(∘) and the one immediately to its left form a linear pair.
C
c Replace the expressions of the angles with the values that make the lines parallel. Is this figure possible given the angles?
A
a x=54
B
b y=47.5
C
c No
Practice makes perfect
a If lines p and q are parallel, the corresponding angles that measure (x+56)^(∘) and (2x+2)^(∘) should be congruent. Therefore, by equating the measures of these angles and solving for x, we can determine the value of x that makes p ∥ q.
b To determine the value of y that makes r and s parallel, we have to relate the two angles that contain y. The angle that measures (3y-17)^(∘) and the one immediately to its left form a linear pair. By the Linear Pair Postulate, we know they are supplementary. Therefore, we can express its measure as 180^(∘)-(3y-17)^(∘).
If r and s are parallel, the corresponding angles whose measures are (y+7)^(∘) and 180^(∘)-(3y-17)^(∘) should be congruent. Therefore, by equating the measures of these angles and solving for y, we can determine the value of y that makes r ∥ s.
c Let's calculate the measures of the different angles that make it so that r and s are parallel as well as p and q. We have already established that x= 54 and y= 47.5
Let's replace the expressions in the figure with these values.
We can see that this is an impossible figure. The angles whose measures are (x+56)^(∘) and (y+7)^(∘) form a linear pair and are therefore supplementary. However, the calculated measures of these angles will not add up to 180^(∘).
110^(∘)+54.5^(∘)=164.5^(∘)
Therefore, given the expressions, the lines cannot both be parallel at the same time. We could also use the fact that (x+56)^(∘) and (3y-17)^(∘) are alternate exterior angles and should be equal if r and s are parallel.
