Question

1. if \\( \angle k \\) and \\( \angle b \\) are supplementary and \\( m \angle k = 73 ^ { \circ } \\), solve for \\( m \angle b \\).
2. if \\( \angle x \\) and \\( \angle c \\) are supplementary and \\( m \angle c = 95 ^ { \circ } \\), solve for \\( m \angle x \\).
3. angles \\( \angle a \\) and \\( \angle b \\) are supplementary angles. if \\( m \angle b = 88 ^ { \circ } \\), find \\( m \angle a \\).
4. find the value of \\( b \\).
5. find the value of \\( n \\).
6. find the value of \\( y \\).

Explanation:

Step1: Recall the definition of supplementary angles

Supplementary angles add up to \(180^{\circ}\).

Step2: Solve for \(m\angle h\) (Problem 1)

Since \(\angle k\) and \(\angle h\) are supplementary, \(m\angle k + m\angle h=180^{\circ}\). Given \(m\angle k = 73^{\circ}\), then \(m\angle h=180^{\circ}-73^{\circ}=107^{\circ}\).

Step3: Solve for \(m\angle x\) (Problem 2)

Since \(\angle x\) and \(\angle c\) are supplementary, \(m\angle x + m\angle c=180^{\circ}\). Given \(m\angle c = 95^{\circ}\), then \(m\angle x=180^{\circ}-95^{\circ}=85^{\circ}\).

Step4: Solve for \(m\angle a\) (Problem 3)

Since \(\angle a\) and \(\angle b\) are supplementary, \(m\angle a + m\angle b=180^{\circ}\). Given \(m\angle b = 88^{\circ}\), then \(m\angle a=180^{\circ}-88^{\circ}=92^{\circ}\).

Step5: Solve for \(b\) (Problem 4)

The two angles \((b + 128)^{\circ}\) and \((-2b+58)^{\circ}\) are supplementary. So \((b + 128)+(-2b + 58)=180\).
Simplify the left - hand side: \(b+128-2b + 58=-b+186\).
Set \(-b + 186=180\). Subtract 186 from both sides: \(-b=180 - 186=-6\). Multiply both sides by \(- 1\) to get \(b = 6\).

Step6: Solve for \(n\) (Problem 5)

The two angles \((4n + 112)^{\circ}\) and \((3n+38)^{\circ}\) are supplementary. So \((4n + 112)+(3n + 38)=180\).
Simplify the left - hand side: \(4n+112+3n + 38=7n+150\).
Set \(7n+150 = 180\). Subtract 150 from both sides: \(7n=180 - 150 = 30\). Then \(n=\frac{30}{7}\approx4.29\).

Step7: Solve for \(y\) (Problem 6)

The two angles \((-8y + 85)^{\circ}\) and \((-3y+29)^{\circ}\) are supplementary. So \((-8y + 85)+(-3y + 29)=180\).
Simplify the left - hand side: \(-8y+85-3y + 29=-11y+114\).
Set \(-11y + 114=180\). Subtract 114 from both sides: \(-11y=180 - 114 = 66\). Divide both sides by \(-11\) to get \(y=-6\).

Answer:

1. \(107^{\circ}\)
2. \(85^{\circ}\)
3. \(92^{\circ}\)
4. \(6\)
5. \(\frac{30}{7}\)
6. \(-6\)