In the following figure, find the measure of angle 𝑃𝑄𝑆.
So, let’s have a look at the diagram that we’re given. We could see that there are two lengths, 𝑃𝑆 and 𝑆𝑅, which are marked as the same length. We can see that we have two right angles. And we can see that there’s an angle, 𝑄𝑆𝑅, which is marked as 47 degrees. There are two triangles, 𝑃𝑄𝑆 and 𝑆𝑄𝑅. And there’s a length 𝑄𝑆 which is common to both triangles. Let’s see if we can work out if these two triangles are congruent, which means exactly the same shape and the same size.
So, let’s take a look at the two triangles. In both triangle 𝑃𝑄𝑆 and triangle 𝑆𝑄𝑅, there is the length 𝑄𝑆. So, both of these are the same length. They both have an angle of 90 degrees. And in triangle 𝑃𝑄𝑆, we’re told that the length 𝑃𝑆 is equal to the length of line 𝑆𝑅 in triangle 𝑆𝑄𝑅. So, it might seem that we could apply the congruency rule, side-angle-side. But in this case, we can’t because the angle isn’t the included angle between the two sides that we’re checking. So, we can’t use the side-angle-side rule.
However, since both triangles have a 90-degree angle, this means that they are right triangles. So, we can check if we could apply the congruency rule, HL, for hypotenuse and leg. In this case, the longest side, the hypotenuse of both triangles, is the line 𝑄𝑆 which is common to both. And the line 𝑃𝑆 in triangle 𝑃𝑄𝑆 is equal to the line 𝑆𝑅 in triangle 𝑆𝑄𝑅. So, we have a side which is the same length in both triangles. So, we’ve shown that our two triangles are congruent by using the hypotenuse-leg congruency rule.
So now, let’s look at putting some of the information into our diagram. Since the triangles are congruent, we know that triangle 𝑃𝑄𝑆 must also have a 47-degree angle. But which one of them is equal to 47? In triangle 𝑆𝑄𝑅, our 47-degree angle sits between the hypotenuse and the line 𝑆𝑅. Therefore, in triangle 𝑃𝑄𝑆, the 47-degree angle must also be between the hypotenuse and the congruent side 𝑃𝑆. In the question, we’re asked to calculate the measure of angle 𝑃𝑄𝑆. And we could do this by remembering that the angles in a triangle add up to 180 degrees. This means that the angle 𝑃𝑄𝑆 is equal to 180 take away the sum of 90 and 47.
And so our final answer is the measure of angle 𝑃𝑄𝑆 is equal to 43 degrees.