Which of the following is true? Option (A) 𝐴𝐵 equals 𝐴𝐶, option
(B) 𝐶𝐴 equals 𝐶𝐵, option (C) 𝐵𝐶 equals 𝐵𝐴.
In the diagram, we have a triangle
𝐴𝐵𝐶, we’ve got a line 𝐵𝐶, and we’ve got this ray 𝐵𝐴. In the options that we’re given,
we’re really looking to see if there’s a pair of lines which are equal in
length. We’re not given any hash marks on
any of the lines, which would indicate that two are equal. So let’s have a look at the angles
If we use the fact that the angles
on a straight line add up to 180 degrees, then we should be able to work out this
angle 𝐴𝐶𝐵 and this angle 𝐴𝐵𝐶. Starting with angle 𝐴𝐶𝐵, we can
write that that’s equal to 180 degrees subtract 98 degrees. We can work this out by calculating
180 degrees subtract 100 degrees and then adding on two, which would give us a value
of 82 degrees. We can add this value to the
diagram. The next angle, angle 𝐴𝐵𝐶, must
be equal to 180 degrees subtract 131 degrees. Subtracting 130 degrees and then
another one degree would give us 49 degrees.
Now that we’ve found these two
angles in this diagram, we might think that it’s not very helpful. But let’s see if we can have a look
at calculating the other angle in the triangle 𝐴𝐵𝐶. We should remember that the angles
in a triangle add up to 180 degrees. So we’ll have our unknown angle
𝐶𝐴𝐵 plus angle 𝐴𝐶𝐵, which we worked out as 82 degrees, plus the angle 𝐴𝐵𝐶,
which we worked out as 49 degrees, all must add to give 180 degrees. This means angle 𝐶𝐴𝐵 plus 131
degrees equals 180 degrees. Subtracting 131 degrees from both
sides of this equation gives us that angle 𝐶𝐴𝐵 is equal to 49 degrees. We can then add that onto our
diagram and see if there’s anything to notice.
Well, we should hopefully see that
we have in fact got two angles that are the same size. Both of these are 49 degrees. This means that we’ve got an
isosceles triangle here in triangle 𝐴𝐵𝐶. Any triangle that has two equal
angles must have two equal sides. And therefore, it’s an isosceles
triangle. The two sides that are equal will
be this side 𝐶𝐴 and this side 𝐶𝐵. We can then give our answer that
𝐶𝐴 equals 𝐶𝐵, which was the option given in option (B).