Question

1. find x. (15x + 7)°/(4x + 3)° 3. find w. (8w - 9)° (w + 7)° 4. find the m∠lmp and m∠pmo. (6y + 2)° (2y + 11)° 5. ∠1 is a complement of ∠2, and m∠1 = 23°. find m∠2. 6. ∠5 is a supplement of ∠6 and m∠5 = 78°. find m∠6.

Explanation:

2. Find x

Step1: Set up equation

Since the two angles are vertical - angles and vertical angles are equal, we have \(15x + 7=4x + 3\).

Step2: Isolate x - terms

Subtract \(4x\) from both sides: \(15x-4x + 7=4x-4x + 3\), which simplifies to \(11x+7 = 3\).

Step3: Isolate the constant

Subtract 7 from both sides: \(11x+7 - 7=3 - 7\), getting \(11x=-4\).

Step4: Solve for x

Divide both sides by 11: \(x=-\frac{4}{11}\).

Step1: Set up equation

Since the two angles are vertical - angles and vertical angles are equal, we have \(8w - 9=w + 7\).

Step2: Isolate w - terms

Subtract \(w\) from both sides: \(8w-w - 9=w - w+7\), which simplifies to \(7w-9 = 7\).

Step3: Isolate the constant

Add 9 to both sides: \(7w-9 + 9=7 + 9\), getting \(7w = 16\).

Step4: Solve for w

Divide both sides by 7: \(w=\frac{16}{7}\).

Step1: Set up equation

Since \(\angle LMP\) and \(\angle PMO\) are complementary (the angle at \(M\) is a right - angle, \(90^{\circ}\)), we have \((6y + 2)+(2y+11)=90\).

Step2: Combine like terms

\(6y+2y+2 + 11=90\), which simplifies to \(8y+13 = 90\).

Step3: Isolate the y - term

Subtract 13 from both sides: \(8y+13 - 13=90 - 13\), getting \(8y = 77\).

Step4: Solve for y

Divide both sides by 8: \(y=\frac{77}{8}\).

Step5: Find \(m\angle LMP\)

\(m\angle LMP=6y + 2=6\times\frac{77}{8}+2=\frac{462}{8}+2=\frac{462 + 16}{8}=\frac{478}{8}=\frac{239}{4}=59.75^{\circ}\).

Step6: Find \(m\angle PMO\)

\(m\angle PMO=2y + 11=2\times\frac{77}{8}+11=\frac{77}{4}+11=\frac{77+44}{4}=\frac{121}{4}=30.25^{\circ}\).

Answer:

\(x =-\frac{4}{11}\)

3. Find w