Question

1. find the value of x. (9x - 8)° (18x - 15)° (8x - 2)°
2. find the value of x. (7x + 10)° (8x + 7)° (18x - 4)°
3. find the value of x. (8x + 1)° (18x - 18)°
4. find the value of x. (5x + 1)° (12x - 7)°

Explanation:

Step1: Recall triangle - exterior - angle property

The exterior - angle of a triangle is equal to the sum of the two non - adjacent interior angles.

For the first triangle (top - left):

Let's use the exterior - angle property. The exterior angle is \(18x - 18\) and the two non - adjacent interior angles are \(8x + 1\) and the third angle in the triangle.
\[18x-18=(8x + 1)+ \text{third angle}\]
Since the sum of angles in a triangle is \(180^{\circ}\), we can also use the exterior - angle property: \(18x-18=(8x + 1)+(180-(8x + 1-(18x - 18)))\)
\[18x-18=8x + 1+180-(8x + 1-18x + 18)\]
\[18x-18=8x + 1+180-( - 10x+19)\]
\[18x-18=8x + 1+180 + 10x-19\]
\[18x-18=18x+162\]
This is incorrect. Using the exterior - angle property directly: \(18x-18=(8x + 1)+ \text{third angle}\), and since the sum of angles in a triangle is \(180^{\circ}\), we know that \(18x-18=(8x + 1)+(180-(8x + 1+(18x - 18)))\) simplifies to \(18x-18 = 8x+1+(180 - 26x + 17)\)
\[18x-18=8x + 1+197-26x\]
\[18x-18=-18x + 198\]
\[18x+18x=198 + 18\]
\[36x=216\]
\[x = 6\]

For the second triangle (top - right):

Using the exterior - angle property, the exterior angle \(18x-4\) is equal to the sum of the two non - adjacent interior angles \(8x + 7\) and \(7x+10\)
\[18x-4=(8x + 7)+(7x + 10)\]
\[18x-4=8x+7+7x + 10\]
\[18x-4=15x + 17\]
\[18x-15x=17 + 4\]
\[3x=21\]
\[x = 7\]

For the third triangle (bottom - left):

Using the exterior - angle property, the exterior angle \(18x-15\) is equal to the sum of the two non - adjacent interior angles \(9x-8\) and \(8x-2\)
\[18x-15=(9x-8)+(8x - 2)\]
\[18x-15=9x-8+8x - 2\]
\[18x-15=17x-10\]
\[18x-17x=-10 + 15\]
\[x = 5\]

For the fourth triangle (bottom - right):

Since the figure is a right - triangle, and using the exterior - angle property. The exterior angle \(12x-7\) is equal to the sum of the non - adjacent interior angles \(5x + 1\) and \(90\)
\[12x-7=(5x + 1)+90\]
\[12x-7=5x+91\]
\[12x-5x=91 + 7\]
\[7x=98\]
\[x = 14\]

Answer:

For the first triangle: \(x = 6\)
For the second triangle: \(x = 7\)
For the third triangle: \(x = 5\)
For the fourth triangle: \(x = 14\)