Question

1. write an equation using the angles in this diagram. then solve your equation to find the value of x.
2. write and solve an equation to find the value of y in the diagram in problem 1.
3. find the area of the unshaded regions. show all your work.

a.
8 cm
15 cm
4 cm
30 cm
b.

Explanation:

Step1: Solve for x in angle - related problem

Since the sum of angles around a point is 360 degrees and we have a right - angle (90 degrees) and a 45 - degree angle, and other angles involving x. Assuming the relevant angles sum to 180 degrees (if they are on a straight - line or in a related angle - set), we have the equation \(x + 45+90 + 2x=180\).
Combining like terms: \(3x+135 = 180\).
Subtract 135 from both sides: \(3x=180 - 135\), so \(3x = 45\).
Divide both sides by 3: \(x = 15\).

Step2: Solve for y in angle - related problem

If y is related to the same set of angles, and assuming it is supplementary to one of the known or x - related angles. Let's say if y is supplementary to an angle formed by the combination of x and other known angles. If we assume a straight - line situation where one side of the line has angles that we have already considered for x, and the other side is y. Since the sum of angles on a straight - line is 180 degrees, and we know from step 1 that the sum of the non - y angles on that straight - line is \(45 + 90+2x\). Substituting \(x = 15\), we get \(45+90 + 2\times15=45+90 + 30=165\). Then \(y=180 - 165=15\).

Step3: Find area of unshaded region in part a

The area of the large rectangle is \(A_{1}=30\times15 = 450\) \(cm^{2}\).
The area of the shaded part: The vertical part of the shaded region has dimensions \(4\times30\) and the horizontal part (excluding the overlapping part) has dimensions \((15 - 4)\times8\). The area of the vertical shaded part is \(A_{s1}=4\times30 = 120\) \(cm^{2}\), and the area of the horizontal shaded part is \(A_{s2}=(15 - 4)\times8=11\times8 = 88\) \(cm^{2}\). The total shaded area \(A_{s}=120+88 = 208\) \(cm^{2}\). The unshaded area \(A_{u}=450-208 = 242\) \(cm^{2}\).

Step4: Find area of unshaded region in part b

The area of the large triangle is \(A_{t}=\frac{1}{2}(10 + 6+9)\times(9 + 5+8)=\frac{1}{2}\times25\times22 = 275\).
The area of the shaded rectangle has dimensions \(6\times5 = 30\). The area of the unshaded region is \(275-30=245\).

Answer:

1. \(x = 15\)
2. \(y = 15\)
3. a. \(242\) \(cm^{2}\)

b. \(245\)