 Hello and welcome to this session. In this session we are going to discuss the following question and the question says that in triangle XYZ Y is equal to 17, measure of angle X is equal to 114 degrees and measure of angle Z is equal to 23 degrees. Solve the triangle. We know that law of signs states that the triangle A, B, C be any triangle with A, B and C representing the measure of sides opposite to angles with measurements A, B and C respectively. Then sin of angle A upon A is equal to sin of angle B upon B is equal to sin of angle C upon C or this can be written as A upon sin of angle A is equal to B upon sin of angle B is equal to C upon sin of angle C. With this key idea let us proceed to the solution to solve a triangle means to find the length of all its sides and measure of all its angles. Let us draw its angle. This is triangle XYZ with angle X is equal to 114 degrees. Angle Z is equal to 23 degrees. Also side opposite to angle Y that is XZ which we denote by Y is equal to 17. Let side opposite to angle X be of length X and side opposite to angle Z be of length Z. We know 3 of the 6 measures that is measure of angle X is given as 114 degrees, measure of angle Z is given as 23 degrees and Y is equal to 17. We need to find measure of angle Y, X and Z. From this figure we see that ASA is given that this angle side angle is given. So we begin by finding measure of angle Y. We know that sum of all angles of a triangle is 180 degrees. So here measure of angle X plus measure of angle Y plus measure of angle Z is equal to 180 degrees which implies that measure of angle X that is 114 degrees plus measure of angle Y plus measure of angle Z that is 23 degrees is equal to 180 degrees which implies that 114 degrees plus 23 degrees is equal to 137 degrees plus measure of angle Y is equal to 180 degrees that is measure of angle Y is equal to 180 degrees minus 137 degrees which implies that measure of angle Y is equal to 43 degrees thus measure of angle Y is equal to 43 degrees. Now we need to find the length X and the length Z from the key idea using law of signs we have sign of angle X upon X is equal to sign of angle Y upon Y and sign of angle Y upon Y is equal to sign of angle Z upon Z. First we take this equation now putting values of angle X, angle Y and Y we get sign of 114 degrees upon X is equal to sign of 43 degrees upon 17. After cross multiplication we get the value of X as 17 into sign of 114 degrees whole upon sign of 43 degrees. Now using calculator we find values of sign of 114 degrees and sign of 43 degrees. This implies that X is approximately equal to 17 into sign of 114 degrees such as 0.91 whole upon sign of 43 degrees such as 0.68 which implies that X is approximately equal to 15.47 upon 0.68 which is approximately equal to 22.75. So X is approximately equal to 22.75 now we use this equation to find the value of Z which implies that sign of angle Y that is sign of 43 degrees upon Y that is 17 is equal to sign of angle Z that is sign of 23 degrees upon Z. After cross multiplication we get the value of Z as 17 into sign of 23 degrees whole upon sign of 43 degrees. Using calculator we find the values of sign of 23 degrees and sign of 43 degrees and this implies that Z is approximately equal to 17 into sign of 23 degrees that is 0.39 whole upon sign of 43 degrees that is 0.68 which implies that Z is approximately equal to 6.63 upon 0.68 which is approximately equal to 9.75. So Z is approximately equal to 9.75 thus we have got the values of X and Z. So we have got measure of angle Y as 43 degrees X is approximately equal to 22.75 and Z is approximately equal to 9.75 which is the required answer. This completes our session. Hope you enjoyed this session.