Question

1. solve for x.

(10x + 1)°
(12x - 5)°

1. solve for x.

(4x + 7)°
(2x + 5)°

1. ∠g and ∠h are complementary angles. if m∠g=(6x - 15)° and m∠h=(3x + 6)°, find m∠h.
2. ∠1 and ∠2 are vertical angles. if m∠1=(5x + 12)° and m∠2=(6x - 11)°, find m∠1.

Explanation:

30.

Step1: Set up equation

Vertical - angles are equal. So, \(10x + 1=12x - 5\).

Step2: Isolate x - terms

Subtract \(10x\) from both sides: \(1 = 12x-10x - 5\), which simplifies to \(1 = 2x - 5\).

Step3: Solve for x

Add 5 to both sides: \(1 + 5=2x\), so \(6 = 2x\). Then divide both sides by 2: \(x=\frac{6}{2}=3\).

Step1: Set up equation

The two angles \((4x + 7)\) and \((2x + 5)\) are complementary (since they form a right - angle). So, \((4x + 7)+(2x + 5)=90\).

Step2: Combine like terms

\(4x+2x+7 + 5=90\), which simplifies to \(6x+12 = 90\).

Step3: Isolate x - term

Subtract 12 from both sides: \(6x=90 - 12=78\).

Step4: Solve for x

Divide both sides by 6: \(x=\frac{78}{6}=13\).

Step1: Set up equation

Complementary angles add up to \(90^{\circ}\). So, \((6x - 15)+(3x + 6)=90\).

Step2: Combine like terms

\(6x+3x-15 + 6=90\), which simplifies to \(9x-9 = 90\).

Step3: Isolate x - term

Add 9 to both sides: \(9x=90 + 9=99\).

Step4: Solve for x

Divide both sides by 9: \(x=\frac{99}{9}=11\).

Step5: Find \(m\angle H\)

Substitute \(x = 11\) into the expression for \(m\angle H\): \(m\angle H=(3x + 6)=(3\times11 + 6)=33 + 6=39^{\circ}\).

Answer:

\(x = 3\)

31.