# Question Video: Determining the Correct Inequality of a Triangle given Its Angles Mathematics

Which of the following is correct? [A] 𝑚∠𝑋𝑍𝑇 > 𝑚∠𝑋 > 𝑚∠𝑋𝑍𝑌 [B] 𝑚∠𝑋𝑌𝑆 > 𝑚∠𝑋𝑍𝑌 > 𝑚∠𝑋𝑌𝑍 [C] 𝑚∠𝑋 > 𝑚∠𝑋𝑍𝑌 > 𝑚∠𝑋𝑌𝑍 [D] 𝑚∠𝑋𝑍𝑌 > 𝑚∠𝑋𝑌𝑍 > 𝑚∠𝑋 [E] 𝑚∠𝑋𝑌𝑍 > 𝑚∠𝑋𝑌𝑆 > 𝑚∠𝑋𝑍𝑌

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### Video Transcript

Which of the following is correct? Is it option (A) the measure of angle 𝑋𝑍𝑇 is greater than angle 𝑋 which is greater than angle 𝑋𝑍𝑌? Option (B) angle 𝑋𝑌𝑆 is greater than angle 𝑋𝑍𝑌 which is greater than angle 𝑋𝑌𝑍. Option (C) angle 𝑋 is greater than angle 𝑋𝑍𝑌 which is greater than angle 𝑋𝑌𝑍. Option (D) angle 𝑋𝑍𝑌 is greater than angle 𝑋𝑌𝑍 which is greater than angle 𝑋. Or finally, option (E) angle 𝑋𝑌𝑍 is greater than angle 𝑋𝑌𝑆 which is greater than angle 𝑋𝑍𝑌.

As many of the angles in our options are the same, it makes sense to fill in all the missing angles on the diagram first. We can do this using our properties of interior and exterior angles. We know that the angles on a straight line sum to 180 degrees. We could use this to calculate the interior angle at vertex 𝑌 and vertex 𝑍. 180 minus 137 is equal to 43. This means that the interior angle at vertex 𝑍 is 43 degrees. 180 minus 115 is equal to 65. So the interior angle at vertex 𝑌 is 65 degrees.

To calculate the third interior angle, the one at vertex 𝑋, we could use the fact that angles in a triangle sum to 180 degrees. We could add 43 and 65 and then subtract this from 180. Alternatively, we could use the fact that any exterior angle of a triangle is equal to the sum of the nonadjacent interior angles. 137 is therefore equal to 65 plus 𝑥. Subtracting 65 from both sides of this equation gives us 𝑥 is equal to 72. We could also have calculated this by subtracting 43 degrees from 115 degrees. The third interior angle is 72 degrees.

We now have the five angles required. Angle 𝑋 is equal to 72 degrees. Angle 𝑋𝑌𝑍 is equal to 65 degrees. Angle 𝑋𝑌𝑆 is equal to 115 degrees. Angle 𝑋𝑍𝑇 is equal to 137 degrees. And finally, angle 𝑋𝑍𝑌 is equal to 43 degrees. We can now substitute these in to our five options. All five of our options want the numbers to be in descending order. This is because the first angle needs to be greater than the second angle which needs to be greater than the third angle.

In option (E), 65 is not greater than 115. In option (D), 43 is not greater than 65. In option (C), while 72 is greater than 43, this is not greater than 65. The same problem occurs with option (B). 115 is greater than 43, but 43 is not greater than 65. In option (A), our numbers are in descending order. 137 is greater than 72 which is greater than 43. This means that the correct answer is angle 𝑋𝑍𝑇 is greater than angle 𝑋 which is greater than angle 𝑋𝑍𝑌.