𝐴𝐵𝐶 is an isosceles triangle, with 𝐴𝐵 equal to 𝐴𝐶. 𝐶𝐷𝐸𝐹 is a rectangle. Angle 𝐶𝐴𝐵 is equal to 42 degrees. Angle 𝐹𝐶𝐴 is equal to 63 degrees. Angle 𝐴𝐵𝐷 is equal to 90 degrees. Show that triangle 𝐶𝐵𝐷 is isosceles. You must give a reason for each stage of your working.
These questions can look really scary, but I like to go through and add the angles I do know or can work out to the diagram. The answer will usually follow fairly quickly. We mustn’t, however, forget to give a reason each time we work out an angle.
Firstly, we’re given that 𝐴𝐵 is equal to 𝐴𝐶 in an isosceles triangle 𝐴𝐵𝐶. This means that the angles 𝐴𝐵𝐶 and 𝐴𝐶𝐵 as marked must be equal. We can work out the value of these two angles by subtracting 42 from 180, since angles in a triangle sum to 180 degrees. 180 minus 42 is 138, so the two angles have a total measure of 138 degrees.
Since these two angles are equal, we can divide 138 by two to work out the measure of each of the angles. 138 divided by two is 69, so angles 𝐴𝐶𝐵 and 𝐴𝐵𝐶 are both 69 degrees. There are two angles we can now work out. The first is this angle down here. This is called angle 𝐶𝐵𝐷.
We are told that angle 𝐴𝐵𝐷 is equal to 90 degrees. We can therefore find the measure of angle 𝐶𝐵𝐷 by subtracting what we just worked out from 90. 90 minus 69 is 21, so 𝐶𝐵𝐷 is equal to 21 degrees.
And we can use a similar process here for angle 𝐵𝐶𝐷. This time, we use the fact that angles around a point add to 360 degrees. So we can subtract the given angles of 63 and 90 plus the one we just worked out, 69, from 360. That’s 138, so angle 𝐵𝐶𝐷 is 138 degrees.
Notice we now have two angles out of a possible three in triangle 𝐶𝐵𝐷. We can calculate the measure of angle 𝐶𝐷𝐵 by subtracting the two that we know from 180 degrees. 180 minus 138 plus 21 is 21 degrees.
Now let’s refer back to the question. We were trying to prove that the triangle 𝐶𝐵𝐷 is isosceles; that is to say, two of its angles are equal. We have just shown that angles 𝐶𝐷𝐵 and 𝐶𝐵𝐷 are both 21 degrees. So we have shown that triangle 𝐶𝐵𝐷 must be isosceles as required.