Euclid's Parallel Postulate Examples

According to the parallel postulate, what must the measures of ∠4 and ∠5 add up to?

∠4 and ∠5 are consecutive interior angles. If their sum is less than 180°, then we'll know the two lines are not parallel. Since they are marked in the figure as being parallel (see the arrowheads), they have to equal 180° exactly.

Are lines OE and FG parallel?

According to Euclid's parallel postulate, parallel lines have supplementary consecutive interior angles. Since the consecutive interior angles of OE and FG are given, we can simply check and see whether or not they add up to 180.

142 + 36 = 178
178 ≠ 180

Since the two lines have consecutive interior angles that aren't supplementary, we know that they aren't parallel. Oh well.

What are the values of j and k?

The parallel postulate states that if the measures of the consecutive interior angles of two lines add up to a number other than 180, the lines aren't parallel. Since the lines in the figure are, we know that the consecutive angles must be supplementary to each other. We can set up an algebraic equation to represent this.

j + 35 + 2k + 5 = 180

Now, we can simplify as much as we can.

j + 2k + 40 = 180
j + 2k = 140

Uh oh. We're stuck. Let's back up and look at the picture again. Since the adjacent angles 2k + 5 and 62° make a straight angle, we can set up another equation that sets the sum of these two to 180°.

2k + 5 + 62 = 180

Now, we can solve for k.

2k + 67 = 180
2k = 113
k = 56.5

Substituting that into the other equation, we can solve for j.

j + 2k = 140
j + 2(56.5) = 140
j + 113 = 140
j = 27

It's always a good idea to double-check your work. We can do this by using other characteristics of parallel lines (like congruent corresponding angles).

j + 35 ≟ 62
27 + 35 ≟ 62
62 = 62