Answer:
The only true statement is:
"∠3 + ∠4 = ∠1 + ∠2"
Step-by-step explanation:
When we have two lines intersect each other, there are 4 angles generated in the intersection.
Two vertical angles are angles whose only "contact point" is the intersection itself. The vertical angles are supplementary only if the two lines are perpendicular.
While a linear pair are two angles that are separated by one of the lines, and linear pairs are always supplementary.
Now let's look at the image and see which statements are true and which ones aren't.
"∠2 and ∠3 form a linear pair"
If we look at the image, we can see that ∠2 and ∠3 are not separated by one of the lines, ∠2 and ∠3 are vertical angles, then this statement is false.
"∠1 and ∠4 are supplementary angles"
Two angles are supplementary if the sum of the measures is equal to 180°, such that, as stated before, a linear pair of angles are supplementary angles.
In the case of ∠1 and ∠4, we can see that these angles are vertical angles, and we also can see that the lines are not perpendicular, then angles ∠1 and ∠4 are not supplementary.
"∠2 and ∠3 = 180°"
This is the same as saying that "∠2 and ∠3 are supplementary"
By the exact same reasoning than in the previous statement, we can find that this one is false.
"∠3 + ∠4 = ∠1 + ∠2"
We know that ∠1 and ∠2 are a linear pair, then ∠1 and ∠2 are supplementary angles, then:
∠1 + ∠2 = 180°
And we know that ∠3 and ∠4 also are a linear pair, and then also are supplementary angles, then we also have:
∠3 + ∠4 = 180°
Then is true that:
∠3 + ∠4 = ∠1 + ∠2
This statement is true.
"∠2 and ∠4 are vertical angles"
We can see that ∠2 and ∠4 are separated by one of the lines, then these angles form a linear pair.
The statement is false.
