One of the angles in a trapezium is 50 degrees. Tick one box for each statement. For each statement, it must be true, it might be true, or it cannot be true. There is at least one right angle in the trapezium. There are three acute angles in the trapezium. At least one angle in the trapezium is 130 degrees. And there are two 50-degree angles in the trapezium.
Before we began, let’s consider the definition of a trapezium. It’s a four-sided polygon with a pair of opposite sides that are parallel. And in our case, we know one angle measures 50 degrees. If we sketch a trapezoid 𝐴𝐵𝐶𝐷, where 𝐵𝐶 and 𝐴𝐷 are parallel, we can use this sketched image to try and answer each of these statements. Let’s consider if angle 𝐴 measured 50 degrees.
Under these conditions, it cannot be true that there is at least one right angle in the trapezium. Because angle 𝐴 and 𝐵 fall on the same line, we know that angle 𝐴 and 𝐵 together must be equal to 180 degrees. So we can say that angle 𝐵 must be equal to 180 degrees minus 50 degrees which is 130 degrees. But we want to consider if there could be at least one right angle in this trapezium. If we turned angle 𝐶 into a right angle and made it measure 90 degrees, then the measure of angle 𝐷 would also have to be 90 degrees because remember angle 𝐶 and must add up to 180 degrees.
Does the image we’ve sketched meet the definition for a trapezium? It is four-sided. It still has one pair of opposite sides that are parallel. This tells us that it is possible to have at least one right angle in the trapezium. It might be true. But because we weren’t given more information, we can’t say for certain what the measure of 𝐶 and 𝐷 are.
Now, we consider if it is possible that there would be three acute angles in this trapezium. Remember that based on the information we’re given, we know for certain that one angle measures 50 degrees and one angle measures 130 degrees. We remember that an acute angle is an angle that is less than 90 degrees. 50 degrees is less than 90 degrees. So we do have one acute angle. We’ve already said that angle 𝐶 plus angle 𝐷 must equal 180 degrees. If we plugged in an acute angle for angle 𝐶, for example, 60 degrees, then angle 𝐷 would have to be equal to 120 degrees because together these two angles measure 180 degrees. It is never possible for two acute angles to add up to 180 degrees. And since it’s not possible for both angle 𝐶 and angle 𝐷 to be acute angles, in this trapezium it cannot be true that there are three acute angles.
The third statement says at least one angle in a trapezium is 130 degrees. We’ve already shown that this must be true. And our fourth statement says that there are two 50-degree angles in the trapezium. We can sketch a smaller angle for angle 𝐷 and say that it is 50 degrees. If we do that, we can calculate the measure of angle 𝐶, which will be 130 degrees. Is this shape a trapezium? It does have four sides and it still has a pair of opposite parallel sides. It is a possible trapezium in which at least one angle measures 50 degrees. So we say that it might be true. This is not the only option for the trapezium, but it is one possible option.
The best strategy to solve something like this is to sketch out what you already know, as we’ve done here.