In the figure, lines 𝑘, 𝑙, and 𝑚 intersect at a point. If 𝑥 plus 𝑧 equals 𝑤 plus 𝑢, which of the following must be true? 1) 𝑧 equals 𝑢, 2) 𝑥 equals 𝑦, or 3) 𝑦 equals 𝑡. A) 2 and 3 only, B) 1 and 2 only, C) 1 and 3 only, or D) 1, 2, and 3.
We’re going to consider which of these angles are equal to each other. Before we do that, let’s go ahead and highlight angles that we already know are equal to one another. When two lines intersect, opposite angles are equal to one another. We know that 𝑥 must be equal to 𝑡 and that 𝑦 is equal to 𝑢 and 𝑧 is equal to 𝑤. Our question told us that 𝑥 plus 𝑧 equals 𝑤 plus 𝑢. 𝑥 plus 𝑧 equals 𝑤 plus 𝑢. We know that 𝑤 and 𝑧 are equal. And that means we could substitute 𝑤 for 𝑧.
If we have a 𝑤 on both sides of our equation, it cancels out. And that tells us that 𝑥 is equal to 𝑢. And we know that 𝑥 is equal to 𝑡 and that 𝑢 is equal to 𝑦. Our yellow angles are the same size as the pink angles. If we make 𝑦 and 𝑢 yellow, it’s easier to identify which of our three statements must be true.
Is 𝑧 equal to 𝑢? We can’t say that that’s true based on the given information. Is 𝑥 equal to 𝑦? It is. And is 𝑦 equal to 𝑡? That also must be true based on our given information. 2 and 3 are true. And so, we select option A, 2 and 3 only.