In the square 𝐴𝐵𝐶𝐷, the points 𝐵, 𝐶, and 𝑂 are collinear. Given that the measure of angle 𝑂𝐴𝐶 equals 28 degrees, find the measure of angle 𝐴𝑂𝐶.
In this question, we have a square 𝐴𝐵𝐶𝐷. And we recall that a square has four equal sides and four equal angles that are 90 degrees each. On our diagram, we can see that there’s a point 𝑂 also labelled. And we’re told that 𝐵𝐶 and 𝑂 are collinear. The word “collinear” means lying on a straight line. So what we’re really being told here is that point 𝑂 is on the line 𝐵𝐶. We’re told in this question that the measure of angle 𝑂𝐴𝐶 equals 28 degrees. So let’s mark that on our diagram.
We’re asked to calculate the measure of angle 𝐴𝑂𝐶, which is this angle marked in orange. We can see that these angles are part of a triangle 𝐴𝑂𝐶. Let’s see if we can calculate the other missing angle in this triangle. We can see that this angle is formed from the diagonal of our square, the line 𝐴𝐶. Recall that, in a square, the diagonals of a square bisect the angles. The word “bisect” means to cut exactly into two pieces. Since the angles in our square are 90 degrees, then the diagonal would cut that angle exactly into two pieces, giving two angles of 45 degrees. This means that the measure of angle 𝐴𝐶𝑂 must be equal to 45 degrees.
So now we know two of the angles in our triangle, we can use that to help us find the measure of angle 𝐴𝑂𝐶. We can remember that the angles in a triangle add up to 180 degrees. So we can write the measure of angle 𝐴𝑂𝐶 equals 180 degrees take away the sum of 28 degrees and 45 degrees since they are our two other angles. This is the same as working out 180 degrees take away 28 degrees take away 45 degrees. Simplifying this would give us 180 degrees take away 73 degrees, which will give us 107 degrees.
So the measure of angle 𝐴𝑂𝐶 is 107 degrees.