Consider the figure, then complete the following using less than, equals, or greater than. 𝑋𝑌 what 𝑋𝑊.
So, let’s have a look at the shape and see what we can tell about it. To start with, it’s a quadrilateral, a four-sided shape. And within our quadrilateral, there are two triangles. This small line on line 𝑊𝑍 and line 𝑌𝑍 means that these two lines are equal in length. And we’re given two angles that are both equal to 39 degrees. So, we know that the measure of angle 𝑊𝑍𝑋 is equal to the measure of 𝑋𝑍𝑌. And there’s one other thing we can say about this shape; the line 𝑋𝑍 appears in both triangles, so that means that it’s common to both triangles.
So, if we were to consider the attributes of both these triangles, triangle 𝑊𝑋𝑍 and triangle 𝑋𝑌𝑍, then we can say that they each have a side length that it’s the same length. Line 𝑊𝑍 is equal to line 𝑌𝑍. They also both have an angle which we know is the same size; the measure of 𝑊𝑍𝑋 is equal to the measure of 𝑋𝑍𝑌. And finally, we know that they have yet another side that’s the same length; the line 𝑋𝑍 appears in both triangles.
Since the angle that we’re given is the included angle between the two sides, that means it’s in between them, this means that we can use the SAS or side-angle-side congruence rule. This means that we can say that triangle 𝑊𝑋𝑍 and triangle 𝑋𝑌𝑍 are congruent using the side-angle-side rule. If two triangles are congruent, that means they are exactly the same shape and size. So, corresponding angles and sides will be the same in each triangle. So, for our angles, that means the measure of angle 𝑋𝑊𝑍 will be equal to the measure of 𝑋𝑌𝑍. And the measure of angle 𝑊𝑋𝑍 will be equal to the measure of 𝑌𝑋𝑍.
And for our lines, we were already given that the line 𝑌𝑍 is equal to the line 𝑊𝑍. And we know that the line 𝑋𝑍 in both triangles is equal because it’s a common side. And finally, since we’ve established that our triangles are congruent, this means that our third sides, 𝑋𝑊 and 𝑋𝑌, must also be the same length. So, we can write 𝑋𝑌 equals 𝑋𝑊, which means that the missing symbol in our question is equals.