Question

1. 18) 19) find the value of x. 20) 21) 22)

Explanation:

Step1: Use the exterior angle theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. For problem 19, we have the equation \(5x + 12+ \text{(interior angle)}=\text{(exterior angle)}\). But if we assume the given \(93^{\circ}\) is the exterior angle and \(5x + 12\) is one of the non - adjacent interior angles (assuming the other non - adjacent interior angle is \(0\) which is wrong. Wait, no, actually, if we consider the linear pair and triangle angle relations wrong. Wait, correct approach: The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. If we assume the given figure for 19: Let's re - check. The exterior angle property: \(5x+12 + y=93\) (where \(y\) is wrong. No, wait, the correct formula for the exterior angle of a triangle: If we have a triangle and an exterior angle formed by one side and the extension of another side, then the measure of the exterior angle is equal to the sum of the measures of the two non - adjacent interior angles. So for problem 19, we have \(5x + 12=93\) (assuming the other non - adjacent interior angle is \(0\) which is wrong. Wait, no, looking at the figure (assuming standard textbook problem), the exterior angle is \(93^{\circ}\) and one non - adjacent interior angle is \(5x + 12\). So \(5x+12=93\).

$$5x+12 = 93$$

Step2: Solve the linear equation

Subtract \(12\) from both sides of the equation \(5x+12 = 93\).

$$5x=93 - 12$$

$$5x=81$$

Divide both sides by \(5\).

$$x=\frac{81}{5}=16.2$$

For problem 20: Using the vertical angle (or adjacent angle in a straight line). The sum of angles on a straight line is \(180^{\circ}\). If we assume the two angles \(51^{\circ}\) and \(4x + 3\) are supplementary (if they form a linear pair).

$$51+(4x + 3)=180$$

$$4x+54 = 180$$

Subtract \(54\) from both sides:

$$4x=180 - 54$$

$$4x=126$$

Divide by \(4\):

$$x = 31.5$$

For problem 21: Using the fact that the sum of angles in a right - angle related (assuming the right - angle symbol). The sum of \(2x\) and \(30^{\circ}\) is \(90^{\circ}\) (if they are complementary).

$$2x+30 = 90$$

Subtract \(30\) from both sides:

$$2x=90 - 30$$

$$2x=60$$

Divide by \(2\):

$$x = 30$$

For problem 22: Using the sum of angles around a point (but if we assume a linear pair). If \(114+(x + 9)=180\) (sum of angles in a linear pair is \(180^{\circ}\)).

$$x+9+114=180$$

$$x + 123=180$$

Subtract \(123\) from both sides:

$$x=180 - 123$$

$$x = 57$$

Answer:

1. \(x = 16.2\)
2. \(x = 31.5\)
3. \(x = 30\)
4. \(x = 57\)