JMC 2026 Q19

Added: 2026-05-07

[00:00:10]We are given various bits of information about this diagram in the question, which I've tried to summarize here by using different colors to mark sides which we know to be of equal lengths, and writing down a fact about one of the angles.

[00:00:28]We're looking to find the value of angle TPS.

[00:00:32]So, let's label that angle X.

[00:00:37]As it stands, it's very hard to extract information about X, and we'll probably want to introduce some more variables.

[00:00:45]Now, bearing in mind we have information that these two side lengths are equal, we know that triangle QSP is isosceles, and we can recall a useful angle fact, which is that the base angles in an isosceles are equal.

[00:01:03]So, if I label this angle here, so that's angle QPT as Y, I know that the base angle in the triangle QSP is X + Y, and the other base angle is also X + Y.

[00:01:27]Now, I can repeat a similar type of logic on triangle TRP, which is also isosceles with the length here being the same as the length here. So, in this case, if I introduce a new label for angle SPR, let's call that one Z, then I'm able to work out the base angle as X + Z, which needs to repeat here.

[00:02:00]Now, looking at triangle TSP, I know all of its interior angles in terms of X, Y, and Z.

[00:02:09]We can recall the angle fact that the sum of interior angles in a triangle is 180°, and write an equation which summarizes that.

[00:02:20]X + X + Z + X + Y needs to equal 180.

[00:02:41]However, I also note from the question that the angle QPR needs to equal 100°, and therefore I can write another equation. Y + X + Z is 100°.

[00:03:07]If I take the difference between these two equations, the top equation has 3X + Y + Z, the bottom equation has just 1X + Y + Z. So, on the left-hand side, if I take the difference, I will be left with 2X.

[00:03:29]On the right-hand side, 180 - 100 is just 80°.

[00:03:39]To find the value of X, I now just need to divide by two.

[00:03:49]So, we get 40°, and the correct answer is C.