Question Video: Using Triangle Congruence Criteria to Establish Congruence | Nagwa Question Video: Using Triangle Congruence Criteria to Establish Congruence | Nagwa

Question Video: Using Triangle Congruence Criteria to Establish Congruence Mathematics • First Year of Preparatory School

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Which congruence criterion can be used directly to prove that triangles 𝐴𝐡𝐢 and 𝐴𝐷𝐢 in the given figure are congruent?

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Video Transcript

Which congruence criterion can be used directly to prove that triangles 𝐴𝐡𝐢 and 𝐴𝐷𝐢 in the given figure are congruent? Option (A) ASA, option (B) SAS, option (C) SSS, or option (D) RHS.

Let’s begin this problem by identifying the two triangles. Triangle 𝐴𝐷𝐢 is on the left, and triangle 𝐴𝐡𝐢 is on the right. And we are asked to determine which of the congruency criteria we could use to prove that these two triangles are congruent. The first thing we might observe is that both these triangles have a right angle, which means they are both right triangles.

And there is one congruency criterion which applies in right triangles. It is the RHS criterion, which means right angle-hypotenuse-side. So have we got enough information to apply this criterion here? We can start by noting down the information about the right angles, as we have the two angles 𝐴𝐷𝐢 and 𝐴𝐡𝐢, which both have a measure of 90 degrees.

Next in the criterion would be the hypotenuse of the right triangle. If it’s difficult to work out which side is the hypotenuse, it’s always helpful to remember that in a right triangle, the hypotenuse is always the side opposite the right angle. Since this line segment 𝐴𝐢, which is the hypotenuse of both triangles, is a common side, then we know that this length is congruent in each of the triangles 𝐴𝐡𝐢 and 𝐴𝐷𝐢. Finally, we can note that the line segments 𝐢𝐷 and 𝐢𝐡 are marked as congruent. So, we have another pair of congruent sides in each triangle.

And so, we have demonstrated that each triangle has a right angle, the hypotenuse in each is congruent, and there is another pair of sides congruent. Therefore, we can prove that triangles 𝐴𝐡𝐢 and 𝐴𝐷𝐢 are congruent using the RHS criterion, which was the answer given in option (D).

As an aside, it is common that when we are dealing with congruent triangles, there may be more than one congruence criterion which we could use. So, let’s look at the other options.

We can exclude the angle-side-angle criterion given in option (A), as there is only information about one angle in the diagram, which was the right angle. We can also exclude option (B), the side-angle-side criterion, as the 90-degree angle we were given was not included between the two sides whose congruency we could establish. And finally, the SSS rule cannot be applied directly as we weren’t given any information about the congruency of all three pairs of sides.

Therefore, the RHS criterion is the only one we could directly apply from these options.

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