Question

determine the value of x in each diagram.
(a)
(b)
(c)
(d)

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180°. Also, vertical angles are equal.

a)

The vertical angle to 81° is also 81°. In the triangle with angles \(x^{\circ}\), \(2x^{\circ}\) and 81°, we have the equation \(x + 2x+81 = 180\).
Combining like - terms: \(3x+81 = 180\).
Subtract 81 from both sides: \(3x=180 - 81=99\).
Divide both sides by 3: \(x = 33\).

b)

The sum of the interior angles of the triangle is 180°. The angle adjacent to 90° inside the triangle is 90°. So, \(90+64+(x + 8)=180\).
First, simplify the left - hand side: \(90+64+x + 8=162+x\).
Then, set up the equation \(162+x = 180\).
Subtract 162 from both sides: \(x=180 - 162 = 18\).

c)

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. The exterior angle at \(J\) is 132°. So, \(132=(2x + 4)+(180 - 112)\).
First, simplify the right - hand side: \(132=(2x + 4)+68\).
Then, \(132=2x+72\).
Subtract 72 from both sides: \(2x=132 - 72 = 60\).
Divide both sides by 2: \(x = 30\).

d)

The angle at \(F\) is 90°. In the triangle with angles \((3x + 2)^{\circ}\), \((2x + 18)^{\circ}\) and 90°, we have the equation \((3x + 2)+(2x + 18)+90 = 180\).
Combine like - terms: \(5x+110 = 180\).
Subtract 110 from both sides: \(5x=180 - 110 = 70\).
Divide both sides by 5: \(x = 14\).

Answer:

a. \(x = 33\)
b. \(x = 18\)
c. \(x = 30\)
d. \(x = 14\)