Video: Using Triangle Congruence Criteria to Establish Congruence

In the figure, 𝐴𝐵 ∥ 𝐷𝐸. Which congruence criterion could be used to prove the two triangles are congruent?

02:55

Video Transcript

In the figure, 𝐴𝐵 is parallel to 𝐷𝐸. Which congruence criterion could be used to prove the two triangles are congruent?

So, in this question we have two triangles, triangle 𝐴𝐵𝐶 and triangle 𝐶𝐷𝐸. We need to check if they are congruent, which means they’re exactly the same shape and the same size. Let’s remind ourselves of the criteria which must be met if we want to show that two triangles are congruent. The first option is SSS which stands for side, side, side. In this case, we would need to show that there are corresponding sides which are equal in each triangle.

The second option is SAS, which is side, angle, side, where the angle has to be the included angle between the two sides. The third possibility is angle, side, angle where the side is the included side between the two given angles. Alternatively, we could have two angles and a side. Or finally, in our option for right triangles only, if we have the hypotenuse and a leg, that’s a side of the triangle corresponding, then that would show congruence.

So, let’s take a look at our diagram and see what we could establish about the properties of these two triangles. We could begin by marking on our two parallel lines 𝐴𝐵 and 𝐷𝐸. Since we have two parallel lines, this means that the measure of angle 𝐵𝐴𝐶 is equal to the measure of angle 𝐶𝐸𝐷 since these are alternate interior angles. We also have another set of alternate interior angles. So, we could say that the measure of angle 𝐴𝐵𝐶 is equal to the measure of angle 𝐶𝐷𝐸. And for the third angle, we can say that the measure of angle 𝐴𝐶𝐵 is equal to the measure of angle 𝐷𝐶𝐸 because these are vertical or opposite angles.

So if we take a look at our diagram, we can’t say anything for definite about the lengths of the sides. We weren’t given any measurements and we can’t measure with a ruler because we weren’t told that the diagram was drawn to scale. So, if we look at our congruency criteria, we can’t use the first four options because they involve the side lengths. We also can’t use the final congruency criterion because we don’t know the length of the sides or hypotenuse, and we don’t know if any of the angles is a right angle.

The only thing we have been able to show in our diagram is that we have three corresponding angles equal to each other. But this is not sufficient to show congruence. If we have two triangles, which have three corresponding angles equal to each other, this would only show that they’re similar but not necessarily congruent. So, since we haven’t been able to demonstrate any of the congruency criteria have been fulfilled, then the answer is there’s not enough information to prove congruence.

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