Question

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

Explanation:

Part (a)

Step1: Use vertical angles property

Vertical angles are equal. So, the non - labeled angle in the triangle is \(99^{\circ}\) (since \(180 - 81=99\)).

Step2: Apply triangle angle sum property

The sum of angles in a triangle is \(180^{\circ}\). So, \(x + 2x+99 = 180\).
Simplify the left - hand side: \(3x+99 = 180\).
Subtract 99 from both sides: \(3x=180 - 99\), \(3x = 81\).
Divide both sides by 3: \(x=\frac{81}{3}=27\).

Part (b)

Step1: Find the non - labeled angle in the triangle

The angle adjacent to \(90^{\circ}\) is \(90^{\circ}\) (linear pair). Using the exterior angle property of a triangle (exterior angle is equal to the sum of the two non - adjacent interior angles).
The non - labeled interior angle of the triangle (adjacent to \(64^{\circ}\)): Let the angle be \(y\), \(y = 90 - 64=26^{\circ}\).

Step2: Apply exterior angle property

The exterior angle \((x + 8)\) is equal to the sum of the two non - adjacent interior angles (\(90^{\circ}\) and \(26^{\circ}\)). So, \(x + 8=90+26\).
Simplify the right - hand side: \(x + 8=116\).
Subtract 8 from both sides: \(x=116 - 8 = 108\).

Part (c)

Step1: Find the non - labeled angle in the triangle

The angle adjacent to \(132^{\circ}\) is \(180 - 132 = 48^{\circ}\) (linear pair). The angle adjacent to \(112^{\circ}\) is \(180 - 112=68^{\circ}\) (linear pair).

Step2: Apply triangle angle sum property

The sum of angles in a triangle is \(180^{\circ}\). So, \(48+(2x + 4)+68 = 180\).
Simplify the left - hand side: \(2x+(48 + 4+68)=2x + 120\).
Set up the equation: \(2x+120 = 180\).
Subtract 120 from both sides: \(2x=180 - 120\), \(2x = 60\).
Divide both sides by 2: \(x = 30\).

Part (d)

Step1: Find the non - labeled angle in the triangle

The angle adjacent to \(180 - 90=90^{\circ}\) (linear pair). The sum of angles in a triangle is \(180^{\circ}\).
Using the exterior angle property (the exterior angle at \(G\) which is \(180-(180 - 90)=90^{\circ}\) is not needed here. Using the angle sum of a triangle formula: \((3x + 2)+(2x + 18)+90=180\).
Simplify the left - hand side: \(3x+2x+(2 + 18+90)=5x+110\).
Set up the equation: \(5x+110 = 180\).
Subtract 110 from both sides: \(5x=180 - 110\), \(5x = 70\).
Divide both sides by 5: \(x = 14\).

Answer:

a. \(x = 27\)
b. \(x = 108\)
c. \(x = 30\)
d. \(x = 14\)