To solve for 'x' and 'y' in an isosceles triangle with angle expressions provided, we need to follow a few steps involving the properties of an isosceles triangle and the sum of angles in a triangle.
Step 1: Recognize the properties of an isosceles triangle.
In an isosceles triangle, the two base angles are equal. So, we have two expressions for the base angles that are given as:
1. The first base angle: 5x - 17 degrees
2. The second base angle: 4y + 3x degrees
Since they are equal, we can equate the two expressions:
5x - 17 = 4y + 3x
Step 2: Solve the above equation for one of the variables.
Let's solve it for 'x':
5x - 3x = 4y + 17
2x = 4y + 17
Now, divide both sides by 2 to solve for 'x':
x = 2y + 8.5
Now we have an expression for 'x' in terms of 'y'.
Step 3: Use the sum of the angles in a triangle.
The sum of the angles in any triangle is 180 degrees. The isosceles triangle has one vertex angle and two base angles which are equal. Using the expressions given for the base angles and the vertex angle, we can form an equation:
Base angle + Base angle + Vertex angle = 180 degrees
(5x - 17) + (4y + 3x) + (2x + 25) = 180
Step 4: Plug in the expression for 'x' from Step 2 into this equation.
Let's first combine like terms in the equation:
5x + 4y + 3x + 2x + 25 - 17 = 180
10x + 4y + 8 = 180
Now substitute our expression for 'x' from Step 2:
10(2y + 8.5) + 4y + 8 = 180
20y + 85 + 4y + 8 = 180
24y + 93 = 180
Step 5: Solve for 'y':
24y = 180 - 93
24y = 87
y = 87 / 24
y = 3.625
Step 6: Substitute the value of 'y' into the expression for 'x' to find 'x':
x = 2y + 8.5
x = 2(3.625) + 8.5
x = 7.25 + 8.5
x = 15.75
So, the value of 'x' is 15.75 and the value of 'y' is 3.625.
