In the figure above, lines k, l, and m intersect at a point. If x + y = u + w , which of the following must be true?

From the given equation we have x=z and t=z. Therefore, option B is the correct answer.

Given that, lines k, l, and m intersect at a point.

What is vertical angle theorem?

Vertical angles theorem or verticallyopposite angles theorem states that two opposite vertical angles formed when two lines intersect each other are always equal (congruent) to each other.

From the given figure,

Using vertical angle theorem, we have

x°=t°

y°=u°

w°=z°

If x + y = u + w, then

x+u=u+z

⇒ x=z

If x + y = u + w, then

t+u=u + z

⇒ t=z

So, x=z and t=z

From the given equation we have x=z and t=z. Therefore, option B is the correct answer.

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Answer:

B)

Step-by-step explanation:

First, let's write down what we know based on the Vertical Angle Theorem and the info given by the question:

1. Vertical angle theorem claims that opposite angles are the same, therefore:

• y=u
• x=t
• w=z

2. We know from the problem that:

• x+y = u+w

3. Now for the proofing:

Since  x+y = u+w, let us try to cross off the variable u and see if we can get x equaling to w.

We know that y=u, so let us switch out y for another u, allowing the same variable to cross out

x+y = u+w --> x+u = u+w --> x=w

Ok now we know x=w, and recall that w=z. So x would also equal to z!

Let us try to proof z=t:

We know that x=t

But now we also know that x=z (from previous)

Therefore we can substitute the z in for x in x=t and voila z=t!

Try finding out if y=w. No matter how many equations you create, they are not the same. It would not make sense either if you use this equation: x+y = u+w. If w and x are the same then y cannot be anything except being equal to "u". We cannot prove that u is equal to anything else either.

Therefore the answer is B)