Question

directions: find the value of x so that l || m. state the converse used.
6.
7.
converse
converse
8.
9.
converse
converse
10.
11.
converse
converse
12.
13.
converse

Explanation:

Step1: Identify angle - relationships

For parallel lines \(l\) and \(m\), we use angle - relationships such as corresponding angles, alternate interior angles, or same - side interior angles. When two parallel lines are cut by a transversal, these angles are equal or supplementary.
Let's take problem 6 for example. If the angles \(52^{\circ}\) and \((3x - 7)^{\circ}\) are corresponding angles (assuming they are in the corresponding - angle position for \(l\parallel m\)), then they are equal.

Step2: Set up an equation

We set up the equation \(3x-7 = 52\).

Step3: Solve the equation for \(x\)

Add 7 to both sides of the equation: \(3x=52 + 7\), so \(3x=59\). Then divide both sides by 3: \(x=\frac{59}{3}\approx19.67\).

For problem 7, if the angles are corresponding (assuming the appropriate angle - relationship for \(l\parallel m\)), we set up an equation based on the equality of the angle measures. Let the two angle measures be \(y_1\) and \(y_2\). If \(y_1=y_2\), we solve for \(x\).

For problem 8, if \((x + 7)^{\circ}\) and \((6x-63)^{\circ}\) are alternate interior angles (assuming the correct position for \(l\parallel m\)), then \(x + 7=6x-63\).

Step4: Solve the new equation

Subtract \(x\) from both sides: \(7 = 5x-63\). Add 63 to both sides: \(70 = 5x\). Divide by 5: \(x = 14\).

We follow similar steps for the remaining problems.

Answer:

1. \(x=\frac{59}{3}\), Corresponding - Angles Converse
2. \(x = 14\), Alternate - Interior - Angles Converse

(Note: Since the full details of all angle - relationships in each problem are not explicitly stated in the question, we assume the most common angle - relationships for parallel lines. You would need to follow the above steps for each individual problem to get all the values of \(x\) and the corresponding converses.)