Given that triangle 𝑅𝑆𝑉 is congruent to triangle 𝑇𝑉𝑆, find the values of 𝑥 and 𝑦.
When we’re given a congruency relationship between two triangles, we can use the order of the letters to help us identify congruent angles. If we take angle 𝑅 in triangle 𝑅𝑆𝑉, we could see that this is congruent to angle 𝑇 in triangle 𝑇𝑉𝑆. Angle 𝑆 in triangle 𝑅𝑆𝑉, or more specifically angle 𝑅𝑆𝑉, is congruent with angle 𝑉 in triangle 𝑇𝑉𝑆. Our final third angle in each triangle, angle 𝑉 in triangle 𝑅𝑆𝑉 and angle 𝑆 in triangle 𝑇𝑉𝑆, are also congruent.
So to find the angle 𝑥, we know that this equates to angle 𝑇. This will be congruent to angle 𝑅 in triangle 𝑅𝑆𝑉. We don’t know the value of angle 𝑅. But we can work it out using a key fact about the angles in a triangle. And that is that the angles in a triangle add up to 180 degrees.
And therefore, to find angle 𝑅 in triangle 𝑅𝑆𝑉, we have 180 degrees subtract 29 degrees and subtract 90 degrees, which gives us 61 degrees. So now we know that angle 𝑅 is 61 degrees. And therefore, the congruent angle 𝑇 must also be 61 degrees. So 𝑥 must be equal to 61. Notice that we don’t need to include the degree symbol since it was already given to us.
We could also have solved for 𝑥 by recognising that angle 𝑆 is also 29 degrees since it’s congruent to angle 𝑉 in triangle 𝑅𝑆𝑉. And then calculate it using the angle sum in our triangle. To find the value of 𝑦, for the next part of the question, we need to look at the congruent sides in our triangles. We know that since the line 𝑆𝑉 is in both triangles, then this is a congruent side in each triangle. The line 𝑅𝑉 in triangle 𝑅𝑆𝑉 is congruent to the line 𝑆𝑇 in triangle 𝑇𝑉𝑆. We could also see this from the pattern in the letters in each triangle.
And finally, we can say that line 𝑅𝑆 in triangle 𝑅𝑆𝑉 is congruent to line 𝑇𝑉 in triangle 𝑇𝑉𝑆. Since we’re told that the line 𝑇𝑉 is 42 and the line 𝑅𝑆 is two 𝑦 minus four, then we can set these equal and solve for 𝑦. We have two 𝑦 minus four equals 42. So rearranging by adding four will give us two 𝑦 equals 42 plus four, which is 46. We then divide both sides of the equation by two, to give 𝑦 equals 23. Therefore, our final answer is 𝑥 equals 61, 𝑦 equals 23.