Find the measure of angle 𝐸𝐶𝐷.
First, let’s look carefully at the diagram and identify exactly which angle we’re being asked to find. Angle 𝐸𝐶𝐷 is the angle formed when you travel from 𝐸 to 𝐶 to 𝐷, so it’s this angle here that I’ve marked in orange. Now looking at the diagram carefully, we can see that we have three parallel lines. We know that they’re parallel because of the arrows on them in blue to indicate this. We also have two angles marked, the angles of 103 degrees and 42 degrees. So within this question, we’re going to need to use angle rules about angles in parallel lines, in order to take a step-by-step approach to calculating angle 𝐸𝐶𝐷.
We can’t calculate angle 𝐸𝐶𝐷 immediately from the information we’ve been given. We need to calculate some other angles first. So we look carefully at the diagram and we see what other angles it’s possible for us to calculate.
Let’s look first of all at the larger pair of parallel lines, so that’s lines 𝐴𝐵 and 𝐸𝐹. We’ve been told that the angle at the top of these parallel lines is 103 degrees. This means that we can also calculate the angle I’ve marked in green, angle 𝐴𝐸𝐹. The angle of 103 degrees and angle 𝐴𝐸𝐹 are an example of cointerior angles in parallel lines; they sit within the parallel lines.
There’s a key fact about this type of angles that will enable us to calculate angle 𝐴𝐸𝐹. It’s this: cointerior angles are supplementary, which means they sum to 180 degrees. So if we know that one of these two co-interior angles is 103 degrees, we can calculate the other by subtracting it from 180 degrees. So we have that angle 𝐴𝐸𝐹 is equal to 180 minus 103; it’s 77 degrees. So how does this help?
Well, let’s look back at the original diagram. We’ve now found that the angle marked in green is 77 degrees. We also know that part of that angle is 42 degrees. This means of course that we can work out what the other part of the angle is, the angle formed when go from 𝐹 to 𝐸 to 𝐶. So angle 𝐹𝐸𝐶 is equal to 77 minus 42, it’s 35 degrees. So by calculating this angle of 35 degrees, we’ve now found an angle in the bottom pair of parallel lines, that’s the parallel lines 𝐶𝐷 and 𝐸𝐹. And remember, it’s an angle within this pair of parallel lines that we’re ultimately asked to calculate, angle 𝐸𝐶𝐷.
So now looking at this pair of parallel lines, we can see that again, we have a pair of cointerior angles: the angle of 35 degrees and angle 𝐸𝐶𝐷. We already have our key fact about cointerior angles which is that they’re supplementary. So we can work out angle 𝐸𝐶𝐷 by subtracting 35 degrees from 180 degrees. This tells us that angle 𝐸𝐶𝐷 is equal to 145 degrees.
So within this question, we couldn’t calculate the required angle immediately from the information given. We had to look at other parts of the diagram first and calculate those angles that it was possible to find, using a step-by-step approach. The key angle rule that we used within this question was that cointerior angles in parallel lines are supplementary, which means that they sum to 180 degrees.