triangle: Find the value of x and y

Question

In adjoining figure AE || BC,      angle DAE = x °,              angle ACB = (x - 15)°,           angle BA =( x/2 + y)°,  

and           angle ABC = (y+15)°             find the value of x and y.                

Hints:    angle CAE = angle ACB  = (x - 15°),  angle DAB is striaght angle , 

            sum of the angles of triangle ABC = 180°

Answer 1

The measures of the angles of triangle ABC add up to 180 degrees:

(x-15) + (x/2 +y) + (y+15) = 180

(x + x/2) + (y + y) + (-15 + 15) = 180            Like terms grouped together

3x/2 + 2y = 180

We also know that ∠DAB = x + (x - 15) + (x/2 + y) and that ∠DAB is a straight angle, so:

x + (x - 15) + (x/2 + y) = 180

5x/2 + y -15 = 180

5x/2 + y = 195

We now have a system of equations we can use to solve for x and y:

5x/2 + y = 195
3x/2 + 2y = 180

There are multiple ways to solve a system of equations; I chose to multiply the top equation by -2 and use elimination:

-5x - 2y = -390
3x/2 + 2y = 180
-7x/2     = -210

Multiply both sides by -2/7 and we get:

x = 60

You can plug that in to the equation of your choice and find that y = 45.

Just to check, we can find that the angles of the triangle are 45, 75 and 60, which add to 180.