If the measure of one angle of a triangle is half the measure of the second angle and less than the measure of the third one by 92 degrees, then what are the measures of the three angles?
So we know from the question that we have three angles. And we call the first angle 𝑥. Then if we take a look at the information that we’ve got, we can see the first angle is half the measure of the second angle. So the second angle is gonna be two 𝑥 and that’s because half of two 𝑥 is 𝑥. So now, what we’ve got is a term for our first angle, which is 𝑥, a term for our second angle, which is two 𝑥. Well, the expression for our third angle is gonna be 𝑥 plus 92 degrees. And that’s because we’re told that the first angle is less than the measure of the third angle by 92 degrees. Well, therefore, the angle that is the third is gonna be the original angle one, so 𝑥, plus the 92 degrees.
So great, what do we do next? Well, we use one of the angle facts we know. And that comes from the properties of triangle. And that’s that we know the angles in a triangle sum to 180 degrees. So therefore, what we could do is set up an equation because we can say that, therefore, 𝑥 plus two 𝑥 plus 𝑥 plus 92 equals 180. And that’s because we’ve added together each of the angles. So next, what we can do is collect like terms. So we’ve got 𝑥, then positive two 𝑥, and another positive 𝑥. So therefore, we get four 𝑥 plus 92 equals 180.
So now, we can look at next stage. And next stage would be to subtract 92 from each side of the equation. And when we do that, we get four 𝑥 is equal to 88. Then if we divide by four, we get 𝑥 is equal to 22. So great, have we solved the problem? Well, no, we haven’t because even though we found out what 𝑥 is, we need to find the measures of the three angles. Well, we found the measure of the first angle cause it’s 22 degrees. We’ve measured the second angle. It’s gonna be equal to two multiplied by 22 degrees, which will be equal to 44 degrees. And the measure of the third angle is gonna be equal to 22 degrees plus 92 degrees, which gonna give us 114 degrees. So therefore, we can say the measures of the three angles are 22 degrees, 44 degrees, and 114 degrees.