Question

given the following information, determine which lines, if any, are parallel. state the postulate or theorem that justifies your answer.

1. (mangle bcg + mangle fgc=180) 2. (angle cbfcongangle gfh) 3. (angle efbcongangle fbc) 4. (angle acdcongangle kbf)

find (x) so that (lparallel m). identify the postulate or theorem you used.

1. ((4x - 6)^{circ}) ((3x + 6)^{circ}) 6. ((7x - 24)^{circ}) ((5x + 18)^{circ}) ((2x + 12)^{circ}) ((5x - 15)^{circ})

Explanation:

1.

Step1: Analyze angle - sum condition

Given \(m\angle BCG + m\angle FGC=180^{\circ}\). By the Consecutive Interior Angles Theorem (if two lines are cut by a transversal and a pair of consecutive - interior angles are supplementary, then the two lines are parallel), lines \(AD\) and \(EH\) are parallel.

Step2: Analyze congruent - angle conditions

For \(\angle CBF\cong\angle GFH\), by the Corresponding Angles Postulate (if two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel), lines \(AD\) and \(EH\) are parallel.
For \(\angle EFB\cong\angle FBC\), by the Alternate Interior Angles Theorem (if two lines are cut by a transversal and alternate - interior angles are congruent, then the lines are parallel), lines \(KB\) and \(FC\) are parallel.
For \(\angle ACD\cong\angle KBF\), by the Corresponding Angles Postulate, lines \(KB\) and \(FC\) are parallel.

Step1: Set up equation based on corresponding - angles

If \(l\parallel m\), then the corresponding angles \((4x - 6)^{\circ}\) and \((3x + 6)^{\circ}\) are congruent. So we set up the equation \(4x-6 = 3x + 6\).

Step2: Solve the equation

Subtract \(3x\) from both sides: \(4x-3x-6=3x - 3x+6\), which gives \(x-6 = 6\). Then add 6 to both sides: \(x=6 + 6=12\). We used the Corresponding Angles Postulate.

Step1: Set up equation based on vertical - angles and parallel - lines relationship

The angles \((7x - 24)^{\circ}\) and \((5x + 18)^{\circ}\) are vertical angles. Since \(l\parallel m\), vertical angles are congruent. So we set up the equation \(7x-24=5x + 18\).

Step2: Solve the equation

Subtract \(5x\) from both sides: \(7x-5x-24=5x - 5x+18\), which gives \(2x-24 = 18\). Then add 24 to both sides: \(2x=18 + 24=42\). Divide both sides by 2: \(x = 21\). We used the Vertical Angles Theorem and the properties of parallel lines.

Answer:

1. Lines \(AD\) and \(EH\) are parallel by the Consecutive Interior Angles Theorem.
2. Lines \(AD\) and \(EH\) are parallel by the Corresponding Angles Postulate.
3. Lines \(KB\) and \(FC\) are parallel by the Alternate Interior Angles Theorem.
4. Lines \(KB\) and \(FC\) are parallel by the Corresponding Angles Postulate.

5.