Question

1.
(5x - 2)°
(3x + 4)°
2.
(9x - 8)°
(x + 2)°
(x + 10)
3.
(11x - 2)°
(19x + 3)°
(9x + 1)°
4.
(7x + 2)°
(17x - 23)°

Explanation:

Step1: Analyze Problem 1 (Right Triangle)

In a right triangle, the two non - right angles sum to \(90^{\circ}\). So, \((5x - 2)+(3x + 4)=90\).
Combine like terms: \(5x+3x-2 + 4=90\), which simplifies to \(8x+2 = 90\).
Subtract 2 from both sides: \(8x=90 - 2=88\).
Divide by 8: \(x=\frac{88}{8}=11\).

Step2: Analyze Problem 2 (Triangle with Angles)

In a triangle, the sum of interior angles is \(180^{\circ}\). So, \((x + 2)+(x + 10)+(9x-8)=180\).
Combine like terms: \(x+x + 9x+2 + 10-8=180\), which simplifies to \(11x + 4=180\).
Subtract 4: \(11x=180 - 4 = 176\).
Divide by 11: \(x=\frac{176}{11}=16\).

Step3: Analyze Problem 3 (Triangle with Angles)

The sum of angles in a triangle is \(180^{\circ}\). So, \((11x-2)+(9x + 1)+(19x+3)=180\).
Combine like terms: \(11x+9x+19x-2 + 1+3=180\), which simplifies to \(39x + 2=180\).
Subtract 2: \(39x=180 - 2=178\). Wait, this seems wrong. Wait, maybe it's an exterior - interior angle relation? Wait, no, the three angles are of a triangle. Wait, \(11x-2+9x + 1+19x+3=(11x+9x+19x)+(-2 + 1+3)=39x + 2\). If \(39x+2 = 180\), \(39x=178\), \(x=\frac{178}{39}\approx4.56\). But maybe I misread the diagram. Alternatively, if it's a triangle with an exterior angle? Wait, no, the diagram shows three angles of a triangle.

Step4: Analyze Problem 4 (Exterior Angle)

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So, \(17x-23=(7x + 2)+(the other non - adjacent angle)\). Wait, from the diagram, the two non - adjacent interior angles are \((7x + 2)\) and the other angle (let's assume the triangle has two angles, and the exterior angle is equal to their sum). Wait, if the two non - adjacent interior angles are \((7x + 2)\) and the angle equal to \((7x + 2)\) (since the triangle has a pair of equal angles? Wait, the diagram shows a triangle with a line extended, so exterior angle \(17x-23\) is equal to the sum of the two remote interior angles. If the two remote interior angles are \((7x + 2)\) and \((7x + 2)\) (isosceles triangle), then \(17x-23=(7x + 2)+(7x + 2)\).
Simplify right side: \(14x+4\).
So, \(17x-23=14x + 4\).
Subtract \(14x\): \(3x-23 = 4\).
Add 23: \(3x=27\), \(x = 9\).

Answer:

Problem 1: \(x = 11\); Problem 2: \(x = 16\); Problem 3: \(x=\frac{178}{39}\) (or re - check diagram); Problem 4: \(x = 9\)