Note that in the picture we are asked about the angles QRW and RWQ. This two angles are part of the the same triangle. So we can ignore the rest of the image and focus only on the triangle that contains both. We have the following
Using this information, we can see that the measure of the three sides of the triangle are different. This means that the triangle is a scalene triangle. This type of triangles have the property that the measure of each angle is different from the other. So, we know that the measure of angles QRW and RWQ are different.
To dig deeper in the problem, we will use the sines law for triangle, which states the following. If we have a triangle of the form
Then, we have the following
[tex]\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}[/tex]
In this case, we have c=45 and a=47, so we have
[tex]\frac{\sin A}{47}=\frac{\sin C}{45}[/tex]
where A is the angle RWQ and C is the angle QRW, if we multiply both sides by 45, we get
[tex]\sin C=\frac{45}{47}\cdot\sin A[/tex]
In this case, we have that 45/47 < 1. Since angles A and C are less than 180° degrees, the sine of this angles is a positive number, so we have that
[tex]\sin C=\frac{45}{47}\cdot\sin A<\sin A[/tex]
Note also that angles A and C should have a measure less than 90°. Over this interval, the sine function is an increasing function, which means that if sin(C)So we have that the measure of angle QRW is less than the measure of angle RWQ.
