Video: Finding the Ratio between Two Corresponding Sides in Two Congruent Triangles

Consider the figure, then complete the following using <, =, or >: π‘‹π‘Œ οΌΏ π‘‹π‘Š.

02:38

Video Transcript

Consider the figure, then complete the following using less than, equals, or greater than. π‘‹π‘Œ what π‘‹π‘Š.

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 π‘Šπ‘ and line π‘Œπ‘ 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 π‘Šπ‘π‘‹ is equal to the measure of π‘‹π‘π‘Œ. 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 π‘Šπ‘‹π‘ and triangle π‘‹π‘Œπ‘, then we can say that they each have a side length that it’s the same length. Line π‘Šπ‘ is equal to line π‘Œπ‘. They also both have an angle which we know is the same size; the measure of π‘Šπ‘π‘‹ is equal to the measure of π‘‹π‘π‘Œ. 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 π‘Šπ‘‹π‘ and triangle π‘‹π‘Œπ‘ 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 π‘‹π‘Šπ‘ will be equal to the measure of π‘‹π‘Œπ‘. And the measure of angle π‘Šπ‘‹π‘ will be equal to the measure of π‘Œπ‘‹π‘.

And for our lines, we were already given that the line π‘Œπ‘ is equal to the line π‘Šπ‘. 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, π‘‹π‘Š and π‘‹π‘Œ, must also be the same length. So, we can write π‘‹π‘Œ equals π‘‹π‘Š, which means that the missing symbol in our question is equals.

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