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

# Question Video: Finding the Measure of Two Angles Inside a Square Using the Properties of a Square Mathematics • First Year of Preparatory School

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Given that π΄π΅πΆπ· is a square, find, in degrees, the values of π₯ and π¦.

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### Video Transcript

Given that π΄π΅πΆπ· is a square, find, in degrees, the values of π₯ and π¦.

So we have a diagram of π΄π΅πΆπ·, which is a square. We also have one of the diagonals of the square, π΅π·, drawn in. And a line, π΄πΉ, which isnβt a diagonal of the square. π₯ and π¦ represent angles within this diagram. Now we donβt yet have enough information to be able to calculate π₯ or π¦ directly. So letβs think about any other angles in the diagram that we could find first.

Thereβs an angle of a 100 degrees marked in the diagram. And in fact this angle is vertically opposite angle π΄πΈπ΅, 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 π΄πΈπ΅ is also equal to 100 degrees. Now letβs consider the triangle π΄πΈπ΅, in which angle π₯ is one of the three angles and another is this angle weβve just calculated of 100 degrees.

If we could work out the third angle in this triangle, angle π΄π΅πΈ, then weβd be able to use the fact that the angle sum in a triangle is always 180 degrees to find the value of π₯. So letβs look at this angle more closely. This angle is in the corner of the square. And itβ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 π΄π΅πΆ and π΄π·πΆ, which means that the two portions of this angle are equal. So Iβve marked them both as green arcs.

All of the interior angles in a square are 90 degrees. And therefore the angle π΄π΅πΈ, which weβ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 π΄π΅πΈ. The third angle π₯ is the one we want to calculate. So we can find the value of π₯ by subtracting the other two angles in this triangle from 180 degrees. We have 180 degrees minus 100 degrees minus 45 degrees, π₯ is 35 degrees. So now that we found the value of π₯, we can think about calculating the value of π¦.

In order to do so, letβs think about the triangle π΄π΅πΉ, now marked in pink. This triangle includes the sides π΄π΅ and π΅πΉ. π΄π΅ is the side of the square. And π΅πΉ is part of a side. And so these two sides are at right angles with each other. It also includes angle π₯, which weβve just shown to be 35 degrees. Our method for calculating π¦ then is going to be to calculate the third angle in this triangle, angle π΅πΉπ΄, and then use the fact that itβs on a straight line with π¦ in order to find the value of π¦.

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 π¦ is an exterior angle. And in order to find it, we need to sum the other two interior angles. So thatβs the right angle and the angle of 35 degrees. This gives the value of π¦, 125 degrees. Donβt worry if you couldnβ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 π΅πΉπ΄, by using the fact that the angle sum in a triangle is 180 degrees. And then calculate π¦ as itβ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, π₯ is equal to 35 degrees. π¦ 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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