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Question Video: Finding the Measure of an Angle Using the Congruence of Triangles Mathematics • 8th Grade

Find π‘šβˆ π΅π΄πΈ.

02:48

Video Transcript

Find the measure of angle 𝐡𝐴𝐸.

In the diagram below, we have the large triangle 𝐡𝐢𝐷 on the outside. And there’s three different smaller triangles inside it. The angle that we need to find is this angle 𝐡𝐴𝐸. We could find the measure of this angle if we knew the angle 𝐷𝐴𝐢 and if we knew the angle 𝐢𝐴𝐸 as we have a straight line. Or, if we looked at the smaller triangle 𝐡𝐴𝐸, we could work out our missing angle if we knew this other angle 𝐡𝐸𝐴. Now, it might be tempting to say that angle 𝐡𝐸𝐴 looks like a 90-degree angle, but we don’t know this for sure, so we couldn’t use it in a calculation.

Instead, let’s look at the diagram and notice that we have some notation on the line sections to indicate that we have pairs of corresponding sides the same length. Let’s see if we can find if we have a pair of congruent triangles. We can remember that congruent triangles will be the same shape and size. The line 𝐷𝐴 on triangle 𝐷𝐴𝐢 is marked as being the same length as line 𝐸𝐴 on triangle 𝐸𝐴𝐢. We have another pair of sides marked as the same length. That’s side 𝐷𝐢 and side 𝐸𝐢. We’re not given any information about corresponding angles being the same, but we do actually have another side to consider: the line 𝐴𝐢 is common to both triangles. And therefore, we can say that this is a corresponding congruent side in both triangles.

So now that we’ve shown that we have three pairs of corresponding sides congruent, we can say that triangle 𝐷𝐴𝐢 is congruent to triangle 𝐸𝐴𝐢 by the SSS or side-side-side congruency criterion. So how does this help us with the original question to find the missing angle of 𝐡𝐴𝐸? Well, we were given that this angle at 𝐢𝐷𝐴 is a right angle of 90 degrees, and so the corresponding angle at 𝐢𝐸𝐴 would also be 90 degrees. We then use the fact that the angles on a straight line sum to 180 degrees, and our straight line 𝐢𝐡 means that the angle 𝐴𝐸𝐡 will also be 90 degrees.

We can use our final angle fact to help us with this triangle 𝐡𝐸𝐴. We can recall that the angles in a triangle add up to 180 degrees. To find our angle 𝐡𝐴𝐸 in this triangle 𝐡𝐸𝐴, we calculate 180 degrees subtract 90 degrees subtract 46 degrees, which gives us the answer that the measure of angle 𝐡𝐴𝐸 is 44 degrees.

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