### Video Transcript

π΄π΅πΆ is an isosceles triangle, with π΄π΅ equal to π΄πΆ. πΆπ·πΈπΉ is a rectangle. Angle πΆπ΄π΅ is equal to 42 degrees. Angle πΉπΆπ΄ is equal to 63 degrees. Angle π΄π΅π· is equal to 90 degrees. Show that triangle πΆπ΅π· 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 mustnβt, however, forget to give a reason each time we work out an angle.

Firstly, weβre given that π΄π΅ is equal to π΄πΆ in an isosceles triangle π΄π΅πΆ. This means that the angles π΄π΅πΆ and π΄πΆπ΅ 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 π΄πΆπ΅ and π΄π΅πΆ are both 69 degrees. There are two angles we can now work out. The first is this angle down here. This is called angle πΆπ΅π·.

We are told that angle π΄π΅π· is equal to 90 degrees. We can therefore find the measure of angle πΆπ΅π· by subtracting what we just worked out from 90. 90 minus 69 is 21, so πΆπ΅π· is equal to 21 degrees.

And we can use a similar process here for angle π΅πΆπ·. 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. Thatβs 138, so angle π΅πΆπ· is 138 degrees.

Notice we now have two angles out of a possible three in triangle πΆπ΅π·. We can calculate the measure of angle πΆπ·π΅ by subtracting the two that we know from 180 degrees. 180 minus 138 plus 21 is 21 degrees.

Now letβs refer back to the question. We were trying to prove that the triangle πΆπ΅π· is isosceles; that is to say, two of its angles are equal. We have just shown that angles πΆπ·π΅ and πΆπ΅π· are both 21 degrees. So we have shown that triangle πΆπ΅π· must be isosceles as required.