Question

interior angles
find the value of x.

1. 2) 3) 4) 5) 6) 7) 8)

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Explanation:

1. For the first - type of angle - pair (vertically opposite angles are equal):

• Vertically opposite angles are equal. If two angles are vertically opposite, we can set up an equation to solve for \(x\).
• For example, if we have two vertically opposite angles \(3x\) and \(45^{\circ}\), the equation is \(3x = 45\).
• Step1: Set up the equation
• Based on the property of vertically - opposite angles being equal. Let the two vertically opposite angles be \(ax + b\) and \(c\). The equation is \(ax + b=c\).
• Step2: Solve for \(x\)
• First, isolate \(x\). If \(ax + b=c\), then \(ax=c - b\), and \(x=\frac{c - b}{a}\) (assuming \(a

eq0\)).

1. For the second - type of angle - pair (adjacent angles on a straight - line or supplementary angles):

• Adjacent angles on a straight - line add up to \(180^{\circ}\). If we have two adjacent angles \((x + 20)^{\circ}\) and \(100^{\circ}\) on a straight - line, we set up the equation \((x + 20)+100 = 180\).
• Step1: Set up the equation
• Using the property that the sum of adjacent angles on a straight - line is \(180^{\circ}\). Let the two adjacent angles be \(ax + b\) and \(c\). The equation is \((ax + b)+c = 180\).
• Step2: Simplify the equation
• \(ax+(b + c)=180\), then \(ax=180-(b + c)\).
• Step3: Solve for \(x\)
• \(x=\frac{180-(b + c)}{a}\) (assuming \(a

eq0\)).

Since we don't have specific values for each of the 16 cases in the picture, we can't give exact numerical answers. But the general method to solve for \(x\) in each case is as described above. If we take a sample case:
Let's say we have two vertically opposite angles: one is \(5x\) and the other is \(75^{\circ}\).

Step1: Set up the equation

Since vertically opposite angles are equal, we have \(5x=75\).

Step2: Solve for \(x\)

Divide both sides of the equation by 5. \(x = \frac{75}{5}=15\).

Step1: Set up the equation

Since the sum of adjacent angles on a straight - line is \(180^{\circ}\), we have \((3x + 15)+105=180\).

Step2: Simplify the left - hand side

\(3x+(15 + 105)=180\), so \(3x+120 = 180\).

Step3: Isolate \(3x\)

Subtract 120 from both sides: \(3x=180 - 120=60\).

Step4: Solve for \(x\)

Divide both sides by 3: \(x=\frac{60}{3}=20\).

Answer:

\(x = 15\)

If we have two adjacent angles on a straight - line, say \((3x+15)^{\circ}\) and \(105^{\circ}\):