Video Transcript
In the given figure, find all the angles that must have measure greater than the measure of angle three.
We’re asked to compare the measures of the various angles in the figure to that of the angle labeled three and to find all those that must have measure greater than angle three.
Now there is a property concerning the exterior and nonadjacent interior angles of a triangle we can recall that can help us make some comparisons with the measure of angle three. This property tells us that the measure of any exterior angle in a triangle is greater than the measure of either nonadjacent interior angle of that triangle.
In the triangle shown with interior angles 𝑎, 𝑏, and 𝑐 and exterior angle 𝑑, this means that the measure of angle 𝑑 is greater than that of either angle 𝑎 or angle 𝑏. So now considering the given figure with angles one to eight, if we look first at the triangle with interior angles three, four, and seven, we see that angle eight is an exterior angle to this triangle and that the nonadjacent interior angles in this case are angles three and seven. Hence, by the exterior angle property, exterior angle eight must have measure greater than both angles three and seven.
Using the same property on the same triangle, but this time with exterior angle six, this time we see that the interior nonadjacent angles to exterior angle six are angles three and four. And so by the exterior angle property, we must have that the measure of angle six is greater than the measures of both angles three and four.
Making a note of this, we see that so far we’ve found that both angles eight and six must have measure greater than that of angle three. In fact, we can show that these are the only two angles in the figure that must have measure greater than that of angle three. Staying with the triangle with interior angles three, four, and seven, we’re given no measures for these angles. But we can see from the figure if we label the vertex where angles one, two, and three meet as 𝐴, since we don’t actually know where the projection from 𝐴 meets the line containing angles six and seven — we can call this point 𝐷 — it’s entirely possible that angle three could have a measure of 90 degrees or more.
Recalling that the measures of the angles in a triangle sum to 180 degrees, so that the measures of angles three, four, and seven sum to 180 degrees, then if the measure of angle three is 90 degrees or more, 180 degrees minus this must be less than or equal to 90 degrees. And so, the sum of the measures of angles four and seven must be less than or equal to 90 degrees. Since neither of angles four and seven can be the zero angle, this means that both have measures less than 90 degrees. And since we’re working from the premise that the measure of angle three is 90 degrees or more, then both of these measures must be less than that of angle three.
Remember, we’re asked for all the angles that must have measure greater than that of angle three. And we’ve shown that there are circumstances where this is not the case for angles four and seven. So we can eliminate these two angles.
Now, under the same circumstances, that’s where the measure of angle three is 90 degrees or more, let’s consider angles one and two. These combine to make a straight angle with angle three at the point we’ve labeled 𝐴. And so, the measures of these three angles must sum to 180 degrees. But by exactly the same argument as before, since 180 degrees minus the measure of angle three is less than 90 degrees, the sum of the measures of angles one and two must be less than 90 degrees. Hence, both the measures of angles one and two must be less than 90 degrees. And again, we’ve found some circumstances where the measures of both angles one and two are less than the measure of angle three. We can therefore eliminate these two angles from the set of angles that must have measure greater than that of angle three.
So now, we’ve compared all angles with angle three except angle five. And we’re again going to consider the possibility that the measure of angle three is 90 degrees or more. In fact, we can eliminate angle five immediately by recalling that we showed by the exterior angle property that angle six must have greater measure than angle three. And since the measure of angle three is 90 degrees or more, the measure of angle six must be greater than 90 degrees.
This being the case, then by our previous arguments, we know that the measures of both angles two and five must be less than 90 degrees and hence must be less than the measure of angle three. In particular, we’ve shown that there are circumstances where the measure of angle five is not greater than that of angle three. So we can eliminate angle five.
We’ve shown therefore that the only two angles in the figure which must have measure greater than that of angle three are angles six and eight.
