Question

ng lines parallel
in the diagram, ( g parallel h ), ( mangle1=(4x + 36)^{circ} ) and ( mangle2=(3x - 3)^{circ} ).
what is the measure of ( angle3 )?
( 21^{circ} )
( 159^{circ} )
( 120^{circ} )
( 60^{circ} )

Explanation:

Step1: Use the property of parallel lines

Since \(g\parallel h\), \(\angle1\) and \(\angle2\) are alternate - interior angles. So \(m\angle1 = m\angle2\).

$$4x + 36=3x - 3$$

Step2: Solve the equation for \(x\)

Subtract \(3x\) from both sides: \(4x-3x + 36=3x-3x - 3\), which gives \(x+36=-3\).
Then subtract 36 from both sides: \(x=-3 - 36=-39\). (This is wrong. Wait, no! Wait, actually, since \(g\parallel h\), \(\angle1\) and \(\angle2\) are same - side interior angles. So \(m\angle1+m\angle2 = 180^{\circ}\))

$$4x + 36+3x - 3=180$$

$$7x+33 = 180$$

Subtract 33 from both sides: \(7x=180 - 33 = 147\)
Divide both sides by 7: \(x = 21\)

Step3: Find \(m\angle2\)

Substitute \(x = 21\) into \(m\angle2=(3x - 3)^{\circ}\). So \(m\angle2=(3\times21-3)^{\circ}=(63 - 3)^{\circ}=60^{\circ}\)

Step4: Use the linear - pair relationship

Since \(\angle2\) and \(\angle3\) form a linear pair (\(\angle2+\angle3 = 180^{\circ}\)), then \(m\angle3=180^{\circ}-m\angle2\)

$$m\angle3=180 - 60=120^{\circ}$$

Answer:

\(120^{\circ}\)