# Question Video: Using Properties of Congruence between Two Triangles to Solve a Problem Mathematics • 8th Grade

Given that β³πππ β β³πππ, find the values of π₯ and π¦.

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### Video Transcript

Given that triangle πππ is congruent to triangle πππ, find the values of π₯ and π¦.

When weβ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 π in triangle πππ, we could see that this is congruent to angle π in triangle πππ. Angle π in triangle πππ, or more specifically angle πππ, is congruent with angle π in triangle πππ. Our final third angle in each triangle, angle π in triangle πππ and angle π in triangle πππ, are also congruent.

So to find the angle π₯, we know that this equates to angle π. This will be congruent to angle π in triangle πππ. We donβt know the value of angle π. 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 π in triangle πππ, we have 180 degrees subtract 29 degrees and subtract 90 degrees, which gives us 61 degrees. So now we know that angle π is 61 degrees. And therefore, the congruent angle π must also be 61 degrees. So π₯ must be equal to 61. Notice that we donβt need to include the degree symbol since it was already given to us.

We could also have solved for π₯ by recognising that angle π is also 29 degrees since itβs congruent to angle π in triangle πππ. And then calculate it using the angle sum in our triangle. To find the value of π¦, for the next part of the question, we need to look at the congruent sides in our triangles. We know that since the line ππ is in both triangles, then this is a congruent side in each triangle. The line ππ in triangle πππ is congruent to the line ππ in triangle πππ. We could also see this from the pattern in the letters in each triangle.

And finally, we can say that line ππ in triangle πππ is congruent to line ππ in triangle πππ. Since weβre told that the line ππ is 42 and the line ππ is two π¦ minus four, then we can set these equal and solve for π¦. We have two π¦ minus four equals 42. So rearranging by adding four will give us two π¦ equals 42 plus four, which is 46. We then divide both sides of the equation by two, to give π¦ equals 23. Therefore, our final answer is π₯ equals 61, π¦ equals 23.