Question Video: Finding the Measure of Two Angles Inside a Square Using the Properties of a Square

Given that ๐ด๐ต๐ถ๐ท is a square, find, in degrees, the values of ๐‘ฅ and ๐‘ฆ.


Video Transcript

Given that ๐ด๐ต๐ถ๐ท is a square, find, in degrees, the values of ๐‘ฅ and ๐‘ฆ.

So we have a diagram of ๐ด๐ต๐ถ๐ท, which is a square. We also have one of the diagonals of the square, ๐ต๐ท, drawn in. And a line, ๐ด๐น, which isnโ€™t a diagonal of the square. ๐‘ฅ and ๐‘ฆ represent angles within this diagram. Now we donโ€™t yet have enough information to be able to calculate ๐‘ฅ or ๐‘ฆ directly. So letโ€™s think about any other angles in the diagram that we could find first.

Thereโ€™s an angle of a 100 degrees marked in the diagram. And in fact this angle is vertically opposite angle ๐ด๐ธ๐ต, as these two angles are formed when a pair of straight lines intersect. A key fact about vertically opposite angles is that they are equal or congruent. And therefore, angle ๐ด๐ธ๐ต is also equal to 100 degrees. Now letโ€™s consider the triangle ๐ด๐ธ๐ต, in which angle ๐‘ฅ is one of the three angles and another is this angle weโ€™ve just calculated of 100 degrees.

If we could work out the third angle in this triangle, angle ๐ด๐ต๐ธ, then weโ€™d be able to use the fact that the angle sum in a triangle is always 180 degrees to find the value of ๐‘ฅ. So letโ€™s look at this angle more closely. This angle is in the corner of the square. And itโ€™s formed where the diagonal meets the vertex. A key fact about the diagonals of a square is that they bisect, or cut in half, a pair of opposite angles, in this case angles ๐ด๐ต๐ถ and ๐ด๐ท๐ถ, which means that the two portions of this angle are equal. So Iโ€™ve marked them both as green arcs.

All of the interior angles in a square are 90 degrees. And therefore the angle ๐ด๐ต๐ธ, which weโ€™ve just shown is half of this, is equal to 45 degrees. So now as we wanted, we know two of the angles in triangle ๐ด๐ต๐ธ. The third angle ๐‘ฅ is the one we want to calculate. So we can find the value of ๐‘ฅ by subtracting the other two angles in this triangle from 180 degrees. We have 180 degrees minus 100 degrees minus 45 degrees, ๐‘ฅ is 35 degrees. So now that we found the value of ๐‘ฅ, we can think about calculating the value of ๐‘ฆ.

In order to do so, letโ€™s think about the triangle ๐ด๐ต๐น, now marked in pink. This triangle includes the sides ๐ด๐ต and ๐ต๐น. ๐ด๐ต is the side of the square. And ๐ต๐น is part of a side. And so these two sides are at right angles with each other. It also includes angle ๐‘ฅ, which weโ€™ve just shown to be 35 degrees. Our method for calculating ๐‘ฆ then is going to be to calculate the third angle in this triangle, angle ๐ต๐น๐ด, and then use the fact that itโ€™s on a straight line with ๐‘ฆ in order to find the value of ๐‘ฆ.

This is actually equivalent to applying the following fact about triangles. The exterior angle of a triangle is equal to the sum of the other two interior angles. So ๐‘ฆ is an exterior angle. And in order to find it, we need to sum the other two interior angles. So thatโ€™s the right angle and the angle of 35 degrees. This gives the value of ๐‘ฆ, 125 degrees. Donโ€™t worry if you couldnโ€™t remember that fact, you could do it using the step-by-step approach I mentioned earlier.

First calculate the other interior angle in the triangle, angle ๐ต๐น๐ด, by using the fact that the angle sum in a triangle is 180 degrees. And then calculate ๐‘ฆ as itโ€™s on the straight line with this angle. And therefore, the sum of these two angles is also 180 degrees. Our solution to the problem then, ๐‘ฅ is equal to 35 degrees. ๐‘ฆ is equal to 125 degrees. Remember, the key fact that we used early on in the question was that the diagonals of a square bisect a pair of opposite angles.

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