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Determine the values of 𝑥 and 𝑦.

So in this question, we’ve been given one big triangle that’s been split into two separate right triangles. And we’ve got to find some of the missing angles. Firstly, because the base of the big triangle is aligned and this angle here is ninety degrees, then this angle here must be ninety degrees as well because the measure of the angles on a line sum to one hundred and eighty degrees. And the other rule that we’re gonna use is that the sum of the measures of the interior angles in a triangle is a hundred and eighty degrees. In other words, the angles in a triangle sum to a hundred and eighty degrees.

So let’s look at that left-hand triangle first. Forty-seven degrees plus ninety degrees plus 𝑦 degrees: using that second rule there, the sum of the measures of the interior angles in a triangle is a hundred and eighty degrees. They must sum to a hundred and eighty degrees. So forty-seven plus ninety plus 𝑦 equals a hundred and eighty. And that means that a hundred and thirty-seven plus 𝑦 is a hundred and eighty. Then if we take away a hundred and thirty-seven from both sides of that equation, we end up with 𝑦 is equal to forty-three. Now let’s consider the other triangle. Again the sum of the measures of the interior angles in a triangle is a hundred and eighty degrees. So forty-three plus ninety plus 𝑥 is equal to a hundred and eighty and forty-three plus ninety is a hundred and thirty-three. Then if we take away a hundred and thirty-three from both sides of that equation, we get 𝑥 is equal to forty-seven.

Now one final comment before we finish. Remember 𝑦 was the number of degrees and 𝑥 was the number of degrees. So 𝑦 and 𝑥 are just numbers. If we’d have given our answer as 𝑥 equals forty-seven degrees and 𝑦 equals forty-three degrees, then technically we would have been wrong because 𝑥 is the number of degrees. So really we’d been saying that this angle here in this corner down here would be forty-seven degrees degrees, and that would be wrong. So our answer is 𝑥 is forty-seven and 𝑦 is forty-three.