Answer: In diagram (11) X = 30 degrees and Y = 60 degrees
In diagram (13) X = 20.55 degrees and Y = 69.45 degrees
Step-by-step explanation: What we have are two right angled triangles with two sides given in each case, therefore we can calculate the two unknown angles (since the third angle equals 90 degrees).
For the second triangle (number 13) we shall use Y as the reference angle. That means we have an opposite (the side facing the reference angle) which is 24 and we have an adjacent (the side that lies between the right angle and the reference angle) which is 9. Hence,
Tan Y = opposite/adjacent
Tan Y = 24/9
Tan Y = 2.6667
Checking with your calculator or table of values, 2.6667 = 69.45
Having two angles, which are 90 and 69.45, the third angle X can be computed as
X = 180 - (90 + 69.45)
Sum of angles in a triangle equals 180
X = 180 - 159.45
X = 20.55 and Y = 69.45
For the first triangle (number 11) we shall use X as the reference angle. We have two sides, the opposite (the side facing the reference angle) which is 3, and the hypotenuse (side facing the right angle) which is 6. Hence,
Sin X = opposite/hypotenuse
Sin X = 3/6
Sin X = 0.5000
Checking with your calculator or table of values,
X = 30 degrees.
Also, now that we have two angles we can easily compute the third angle as follows
Y = 180 - (90 + 30)
Sum of angles in a triangle equals 180
Y = 180 - 120
Y = 60
X = 30 and Y = 60
