Find the values of 𝑥 and 𝑦.
So we have a diagram of a triangle 𝐴𝐵𝐶 in which we’re told one of the angles is 96 degrees. And the other angles are expressed in terms of these variables, 𝑥 and 𝑦, whose values we wish to calculate.
In order to do this, we’ll need to solve some equations. The first fact that we know about the angles in any triangle is their sum is 180 degrees. We can therefore form an equation involving the sizes of the three angles. Nine 𝑦 minus three plus 𝑥 plus one plus 96 is equal to 180.
This equation can be simplified slightly. On the left-hand side, we have negative three plus one plus 96. So overall, this simplifies to plus 94. We therefore have nine 𝑦 plus 𝑥 plus 94 is equal to 180. Subtracting 94 from both sides of the equation simplifies it further, giving nine 𝑦 plus 𝑥 is equal to 86.
Now we want to calculate the values of 𝑥 and 𝑦. But we aren’t yet in a position to do so as we have just one equation with two unknowns. We need another equation in order to be able to find the values of 𝑥 and 𝑦. Let’s consider what else we know about this triangle.
We’re told in the diagram that two of the sides of this triangle are of the same length, 𝐴𝐵 and 𝐴𝐶. This means that triangle 𝐴𝐵𝐶 is an isosceles triangle. And in terms of the angles, it means that the two base angles, those currently shaded in orange, must be equal to each other.
Therefore, we can form a second equation involving the measures of these two angles. Nine 𝑦 minus three is equal to 𝑥 plus one. Adding three to both sides of this equation simplifies it slightly, to give nine 𝑦 is equal to 𝑥 plus four. So now we have two equations with two unknowns. The first equation: nine 𝑦 plus 𝑥 equals 86. And the second: nine 𝑦 is equal to 𝑥 plus four.
In order to find the values of 𝑥 and 𝑦, we need to solve these two equations simultaneously. Both equations involve nine 𝑦. And therefore, the most straightforward method of solution is going to be to substitute the expression for nine 𝑦 from the second equation into the first equation.
So substituting 𝑥 plus four in place of nine 𝑦 in the first equation gives 𝑥 plus four plus 𝑥 is equal to 86. Combining the like terms, the two 𝑥s gives two 𝑥 plus four is equal to 86. Next, we subtract four from both sides, giving two 𝑥 is equal to 82.
The final step is to divide both sides of the equation by two. And so we have that 𝑥 is equal to 41. So we found the value of 𝑥. And now we need to find the value of 𝑦. To do this, I’m going to choose to substitute 𝑥 equals 41 into equation two. This will give nine 𝑦 is equal to 41 plus four. 41 plus four is 45. And so we have nine 𝑦 is equal to 45. To find the value of 𝑦, we need to divide both sides by nine. This gives 𝑦 is equal to five.
So we’ve found the values of 𝑥 and 𝑦. 𝑥 is equal to 41. 𝑦 is equal to five. The two key facts we used in this question were, firstly, that the angle sum in any triangle is 180 degrees and, secondly, that in an isosceles triangle the two base angles are equal.