Question

2 list the side lengths from shortest to longest for each diagram. a b 3 determine the value of x in each diagram. a b c d

Explanation:

3a

Step1: Use the property of vertical angles

Vertical angles are equal. So the angle adjacent to \(81^{\circ}\) in \(\triangle HIJ\) is \(81^{\circ}\).

Step2: Apply the angle - sum property of a triangle

The sum of interior angles of a triangle is \(180^{\circ}\). In \(\triangle HIJ\), we have \(x + 2x+81 = 180\).
Combine like terms: \(3x+81 = 180\).
Subtract \(81\) from both sides: \(3x=180 - 81=99\).
Divide both sides by \(3\): \(x = 33\).

Step1: Use the exterior - angle property of a triangle

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.
The exterior angle at \(V\) is \(x + 8\), and the two non - adjacent interior angles are \(90^{\circ}\) and \(64^{\circ}\).
So \(x+8=90 + 64\).

Step2: Solve for \(x\)

Simplify the right - hand side: \(x+8=154\).
Subtract \(8\) from both sides: \(x=154 - 8=146\).

Step1: Find the interior angles adjacent to the exterior angles

The interior angle at \(J\) is \(180 - 132=48^{\circ}\), and the interior angle at \(L\) is \(180 - 112 = 68^{\circ}\).

Step2: Apply the angle - sum property of a triangle

In \(\triangle JKL\), the sum of interior angles is \(180^{\circ}\). So \(48+(2x + 4)+68=180\).
Combine like terms: \(2x+(48 + 4+68)=180\), \(2x + 120=180\).
Subtract \(120\) from both sides: \(2x=180 - 120 = 60\).
Divide both sides by \(2\): \(x = 30\).

Answer:

\(x = 33\)

3b