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In the figure, 𝐴𝐵 is parallel to 𝐷𝐸.
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Which congruence criterion could be used to prove the two triangles are congruent?
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So, in this question we have two triangles, triangle 𝐴𝐵𝐶 and triangle 𝐶𝐷𝐸.
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We need to check if they are congruent, which means they’re exactly the same shape and the same size.
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Let’s remind ourselves of the criteria which must be met if we want to show that two triangles are congruent.
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The first option is SSS which stands for side, side, side.
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In this case, we would need to show that there are corresponding sides which are equal in each triangle.
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The second option is SAS, which is side, angle, side, where the angle has to be the included angle between the two sides.
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The third possibility is angle, side, angle where the side is the included side between the two given angles.
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Alternatively, we could have two angles and a side.
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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.
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So, let’s take a look at our diagram and see what we could establish about the properties of these two triangles.
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We could begin by marking on our two parallel lines 𝐴𝐵 and 𝐷𝐸.
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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.
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We also have another set of alternate interior angles.
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So, we could say that the measure of angle 𝐴𝐵𝐶 is equal to the measure of angle 𝐶𝐷𝐸.
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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.
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So if we take a look at our diagram, we can’t say anything for definite about the lengths of the sides.
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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.
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So, if we look at our congruency criteria, we can’t use the first four options because they involve the side lengths.
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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.
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The only thing we have been able to show in our diagram is that we have three corresponding angles equal to each other.
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But this is not sufficient to show congruence.
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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.
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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.