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Question Video: Proving the Angle Sum of a Triangle Mathematics

In the given figure, if 𝐴𝐢βˆ₯𝐷𝐸, π‘šβˆ π΄π΅πΈ = 55Β°, and π‘šβˆ πΆ = 75Β°, find the measure of ∠𝐴𝐡𝐢.

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Video Transcript

In the given figure, if 𝐴𝐢 is parallel to 𝐷𝐸, the measure of angle 𝐴𝐡𝐸 equals 55 degrees, and the measure of angle 𝐢 equals 75 degrees, find the measure of angle 𝐴𝐡𝐢.

The first piece of information that we are given here is that 𝐴𝐢 is parallel to 𝐷𝐸. We can also fill in the two angle measures that we are given that the measure of angle 𝐴𝐡𝐸 is 55 degrees and the measure of angle 𝐢 is equal to 75 degrees. We can then establish that the angle that we wish to calculate is here, the measure of angle 𝐴𝐡𝐢.

So there are a couple of ways in which we could find the measure of this unknown angle. But both methods will use the fact that we have these parallel line segments. By using the parallel lines 𝐴𝐢 and 𝐸𝐷 and the transversal 𝐡𝐢, we can identify a pair of alternate interior angles. And since alternate interior angles are congruent, we can say that the measure of angle 𝐷𝐡𝐢 is equal to the measure of angle 𝐢. These will both be 75 degrees.

We can notice then that these three angles made at the vertex 𝐡 all lie on a straight line. We recall that the angle measures on a straight line sum to 180 degrees. Therefore, we can write that the measure of angle 𝐴𝐡𝐸 plus the measure of angle 𝐴𝐡𝐢 plus the measure of angle 𝐷𝐡𝐢 must be equal to 180 degrees. We can then simply fill in the angle information that we know. Then, by adding 55 degrees and 75 degrees, we get 130 degrees. We can then simplify this equation by subtracting 130 degrees from both sides, leaving us with the measure of angle 𝐴𝐡𝐢 is equal to 50 degrees. And so we have found the value of this unknown angle.

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