Question Video: The Definition of a Vertex Angle Mathematics

Video Transcript

If two angles are vertically opposite, are they equal in measure? Consider two lines 𝐴𝐵 and 𝐶𝐷 which intersect one another at the point 𝑚. We now bring in an axiom. That’s just a statement that’s used in proof, which we take to be true. The axiom is that angles on a straight line sum to 180 degrees.

So, let’s take line 𝐴𝐵. We can say that angle 𝐴𝑚𝐷 and angle 𝐵𝑚𝐷 add to make 180 degrees. Similarly, we can say that angle 𝐴𝑚𝐶 and angle 𝐵𝑚𝐶 also add to make 180 degrees. And, in fact, if we consider the line 𝐶𝐷, we can form two further statements. That is, angle 𝐴𝑚𝐶 and 𝐴𝑚𝐷 add to make 180 as do angles 𝐵𝑚𝐶 and 𝐵𝑚𝐷.

So, we’re going to define one of our angles. Let’s define 𝐴𝑚𝐷 to be equal to 𝑥 degrees. And so, we can say in these two statements that 𝑥 plus 𝐵𝑚𝐷 equals 180 degrees, but also 𝐴𝑚𝐶 plus 𝑥 equals 180. We rearrange both of our equations by subtracting 𝑥 from both sides. Our first equation becomes 𝐵𝑚𝐷 equals 180 minus 𝑥. And our other equation becomes 𝐴𝑚𝐶 equals 180 minus 𝑥.

Now, notice, we’ve shown that both 𝐵𝑚𝐷 is equal to 180 minus 𝑥 and 𝐴𝑚𝐶 is equal to 180 minus 𝑥. These three dots mean “therefore,” and we can say that, therefore, 𝐵𝑚𝐷 must be equal to 𝐴𝑚𝐶. On our diagram, that’s this one and this one. Now, in fact, since this is the case and angles on a straight line sum to 180 degrees, we can show that both 𝐴𝑚𝐷 and 𝐵𝑚𝐶 are also equal. And so, we say that vertically opposite angles are equal. And so, the answer to this question is yes.