Video Transcript
Given that ๐ด๐ต๐ถ๐ท is a square, find, in degrees, the values of ๐ฅ and ๐ฆ.
So we have a diagram of ๐ด๐ต๐ถ๐ท, which is a square. We also have one of the diagonals of the square, ๐ต๐ท, drawn in. And a line, ๐ด๐น, which isnโt a diagonal of the square. ๐ฅ and ๐ฆ represent angles within this diagram. Now we donโt yet have enough information to be able to calculate ๐ฅ or ๐ฆ directly. So letโs think about any other angles in the diagram that we could find first.
Thereโs an angle of a 100 degrees marked in the diagram. And in fact this angle is vertically opposite angle ๐ด๐ธ๐ต, 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 ๐ด๐ธ๐ต is also equal to 100 degrees. Now letโs consider the triangle ๐ด๐ธ๐ต, in which angle ๐ฅ is one of the three angles and another is this angle weโve just calculated of 100 degrees.
If we could work out the third angle in this triangle, angle ๐ด๐ต๐ธ, then weโd be able to use the fact that the angle sum in a triangle is always 180 degrees to find the value of ๐ฅ. So letโs look at this angle more closely. This angle is in the corner of the square. And itโs 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 ๐ด๐ต๐ถ and ๐ด๐ท๐ถ, which means that the two portions of this angle are equal. So Iโve marked them both as green arcs.
All of the interior angles in a square are 90 degrees. And therefore the angle ๐ด๐ต๐ธ, which weโve just shown is half of this, is equal to 45 degrees. So now as we wanted, we know two of the angles in triangle ๐ด๐ต๐ธ. The third angle ๐ฅ is the one we want to calculate. So we can find the value of ๐ฅ by subtracting the other two angles in this triangle from 180 degrees. We have 180 degrees minus 100 degrees minus 45 degrees, ๐ฅ is 35 degrees. So now that we found the value of ๐ฅ, we can think about calculating the value of ๐ฆ.
In order to do so, letโs think about the triangle ๐ด๐ต๐น, now marked in pink. This triangle includes the sides ๐ด๐ต and ๐ต๐น. ๐ด๐ต is the side of the square. And ๐ต๐น is part of a side. And so these two sides are at right angles with each other. It also includes angle ๐ฅ, which weโve just shown to be 35 degrees. Our method for calculating ๐ฆ then is going to be to calculate the third angle in this triangle, angle ๐ต๐น๐ด, and then use the fact that itโs on a straight line with ๐ฆ in order to find the value of ๐ฆ.
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 ๐ฆ is an exterior angle. And in order to find it, we need to sum the other two interior angles. So thatโs the right angle and the angle of 35 degrees. This gives the value of ๐ฆ, 125 degrees. Donโt worry if you couldnโt 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 ๐ต๐น๐ด, by using the fact that the angle sum in a triangle is 180 degrees. And then calculate ๐ฆ as itโs 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, ๐ฅ is equal to 35 degrees. ๐ฆ 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.
