Question

find the value of x.
1.
2.
3.
4.

Explanation:

Step1: Use the triangle angle - sum property (sum of angles in a triangle is \(180^{\circ}\)) for the first triangle

The first triangle has angles \((5x - 2)^{\circ}\), \((3x + 4)^{\circ}\), and \(90^{\circ}\).
So, \((5x-2)+(3x + 4)+90=180\).
Simplify the left - hand side: \(5x-2+3x + 4+90=(5x+3x)+(-2 + 4+90)=8x+92\).
Then \(8x+92 = 180\).
Subtract 92 from both sides: \(8x=180 - 92=88\).
Divide both sides by 8: \(x=\frac{88}{8}=11\).

Step2: Use the triangle angle - sum property for the second triangle

The sum of angles in a triangle is \(180^{\circ}\). So \((8x - 8)+(x + 10)+(x + 2)=180\).
Simplify the left - hand side: \(8x-8+x + 10+x + 2=(8x+x+x)+(-8 + 10+2)=10x + 4\).
Then \(10x+4 = 180\).
Subtract 4 from both sides: \(10x=180 - 4 = 176\).
Divide both sides by 10: \(x=\frac{176}{10}=17.6\).

Step3: Use the exterior - angle property (exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles)

The exterior angle is \((12x + 3)^{\circ}\), and the two non - adjacent interior angles are \((11x-2)^{\circ}\) and \((8x + 1)^{\circ}\).
So \(12x+3=(11x-2)+(8x + 1)\).
Simplify the right - hand side: \(11x-2+8x + 1=(11x+8x)+(-2 + 1)=19x-1\).
Then \(12x+3=19x-1\).
Subtract \(12x\) from both sides: \(3=19x-12x-1\).
Simplify: \(3 = 7x-1\).
Add 1 to both sides: \(7x=3 + 1=4\).
Divide both sides by 7: \(x=\frac{4}{7}\approx0.57\).

Step4: Use the exterior - angle property

The exterior angle is \((17x-23)^{\circ}\), and the two non - adjacent interior angles are \((7x + 2)^{\circ}\) and the right - angle (assuming it's a right - triangle from the figure, but if not, the sum of non - adjacent interior angles). Let's assume it's a right - triangle (if not, the general formula for exterior angle \(E=a + b\)). If it's a right - triangle, one of the non - adjacent interior angles is \(90^{\circ}\).
So \(17x-23=90+(7x + 2)\).
Simplify the right - hand side: \(90+7x + 2=7x+92\).
Then \(17x-23=7x+92\).
Subtract \(7x\) from both sides: \(17x-7x-23=92\).
Simplify: \(10x-23=92\).
Add 23 to both sides: \(10x=92 + 23=115\).
Divide both sides by 10: \(x = 11.5\).

Answer:

1. \(x = 11\)
2. \(x=17.6\)
3. \(x=\frac{4}{7}\)
4. \(x = 11.5\)