In the given figure, Matthew stated
that 𝑥 is an obtuse angle measuring 105 degrees, and Daniel stated that 𝑥 is an
acute angle measuring 75 degrees. Determine which of the two is
correct without using a protractor.
In the diagram, we’re given three
angles: a 63-degree angle, a 42-degree angle, and 𝑥. We’re told here not to use a
protractor, so we shouldn’t try and measure angle 𝑥 directly. When we see this type of
instruction, especially in an exam question, very often the actual angle isn’t drawn
We’ll therefore need to find a way
to work out the value of 𝑥 without measuring. That way, we’ll be able to tell
whether Matthew is correct or Daniel is correct. You may have already noticed that
these three angles sit upon a straight line. And we should remember that the
angles on a straight line add up to 180 degrees. We can also say that these three
angles are supplementary as supplementary angles add up to 180 degrees.
If we add the value of the two
other angles, 63 degrees and 42 degrees, well, 60 and 40 would give us 100, and
three and two would give us five. So we know that these two angles
would add to 105 degrees. But we need to find the value of
the angle 𝑥, which is remaining. Since these three angles add to 180
degrees, we would calculate 180 degrees subtract 105 degrees, which gives us the
value of 75 degrees. As Daniel was the person who
correctly identified that 𝑥 is 75 degrees, then Daniel would be our answer.