Which of the following angle pairs are alternate interior angles? Angle 𝑎 and angle 𝑔; angle 𝑑 and angle 𝑒; angle 𝑐 and angle ℎ; angle 𝑐 and angle 𝑒; or angle 𝑓 and angle 𝑗.
Looking at the diagram, we can see that the setup in this question is one where when we have two lines and then a transversal — that is a line that crosses both of the other lines. And we have 12 angles formed, where these lines intersect. We’re asked to determine which of the angle pairs are alternate interior angles.
The definition of alternate interior angles is nonadjacent interior angles on opposite sides of the transversal. Now, most of the angle pairs that we’ve been asked about are in the bottom part of the diagram. So we’ll consider the horizontal line shaded in green as the transversal. The part of the diagram now shaded in orange is the interior of the two black lines. So let’s consider the positioning of each of these angle pairs in turn.
Firstly, angles 𝑎 and 𝑔: well, we can see that they are indeed on opposite sides of the transversal. But they’re on the exterior of the black lines. Therefore, 𝑎 and 𝑔 are in fact an example of alternate exterior angles and so not the right type of angles that we’re looking for.
Next, let’s consider the angles pair 𝑑 and 𝑒. We can see that they are on the interior of the diagram. But they’re on the same side of the transversal. Therefore, angles 𝑑 and 𝑒 are what’s known as consecutive interior angles. So again, they’re not the right type of angles.
The same is true of angles 𝑐 and ℎ. They are on the interior of the diagram, but on the same side of the transversal. They’re also consecutive interior angles. Next, let’s consider the angle pair 𝑐 and 𝑒. Now, looking at these, we can see that they are on the interior of the diagram and this time they are on opposite sides of the transversal. They’re also not adjacent. And therefore, they are an example of alternate interior angles. So we have found one pair of angles that are alternate interior the angles.
Now, to consider angles 𝑓 and 𝑗, we actually need to change the setup slightly in terms of which line we’re considering to be the transversal. The transversal in relation to this pair of angles is the line that I’ve now marked in green. The interior of the diagram is now the part inside the two black lines.
Looking at angles 𝑓 and 𝑗, we can see that 𝑓 is an interior angle, but 𝑗 is an exterior angle. They both lie on the same side of the transversal and they both lie on the same side of their black line. Therefore, angles 𝑓 and 𝑗 are what’s known as corresponding angles.
So of the five options given, the only pair of angles that are alternate interior angles are angle 𝑐 and angle 𝑒.