Question

directions: if l || m, classify the marked angle - pair and give their relationship, then solve for x.
1.
2.
3.
4.
5.
6.

Explanation:

Step1: Identify angle - pair relationship for problem 1

Since \(l\parallel m\), the marked angles are corresponding angles. Corresponding angles are equal. So, \(58 = 4x - 2\).

Step2: Solve the equation for \(x\) in problem 1

Add 2 to both sides: \(58+2=4x - 2+2\), which gives \(60 = 4x\). Then divide both sides by 4: \(x=\frac{60}{4}=15\).

Step3: Identify angle - pair relationship for problem 2

Since \(l\parallel m\), the marked angles are alternate - exterior angles. Alternate - exterior angles are equal. So, \(16x + 20=134\).

Step4: Solve the equation for \(x\) in problem 2

Subtract 20 from both sides: \(16x+20 - 20=134 - 20\), which gives \(16x = 114\). Then \(x=\frac{114}{16}=\frac{57}{8}=7.125\).

Step5: Identify angle - pair relationship for problem 3

Since \(l\parallel m\), the marked angles are same - side interior angles. Same - side interior angles are supplementary (\(\text{sum}=180^{\circ}\)). So, \((7x - 1)+125 = 180\).

Step6: Solve the equation for \(x\) in problem 3

First, simplify the left - hand side: \(7x+124 = 180\). Then subtract 124 from both sides: \(7x=180 - 124 = 56\). Divide both sides by 7: \(x = 8\).

Step7: Identify angle - pair relationship for problem 4

Since \(l\parallel m\), the marked angles are alternate - interior angles. Alternate - interior angles are equal. So, \(9x + 2=133\).

Step8: Solve the equation for \(x\) in problem 4

Subtract 2 from both sides: \(9x+2 - 2=133 - 2\), which gives \(9x = 131\). Then \(x=\frac{131}{9}\approx14.56\).

Step9: Identify angle - pair relationship for problem 5

Since \(l\parallel m\), the marked angles are vertical angles (which are equal regardless of parallel lines, but still valid here). So, \(x - 77=3x + 38\).

Step10: Solve the equation for \(x\) in problem 5

Subtract \(x\) from both sides: \(-77 = 2x+38\). Then subtract 38 from both sides: \(-77 - 38=2x\), which gives \(-115 = 2x\). So, \(x=-\frac{115}{2}=-57.5\).

Step11: Identify angle - pair relationship for problem 6

Since \(l\parallel m\), the marked angles are corresponding angles. So, \(11x - 47=6x - 2\).

Step12: Solve the equation for \(x\) in problem 6

Subtract \(6x\) from both sides: \(11x-6x - 47=6x-6x - 2\), which gives \(5x - 47=-2\). Add 47 to both sides: \(5x=45\). Divide both sides by 5: \(x = 9\).

Answer:

1. \(x = 15\)
2. \(x = 7.125\)
3. \(x = 8\)
4. \(x=\frac{131}{9}\approx14.56\)
5. \(x=-57.5\)
6. \(x = 9\)