Question

directions: solve each problem. be sure to include a picture, formula, and theorem that you do. remember that all work should be neat and organized in a vertical fashion.

1. find all angles and the values of each variable. provide formulas and theorems for all work that you do.

Explanation:

Step1: Use the vertical - angle theorem

Vertical angles are equal. So, \(2x = 3x + 30\) (this is wrong, actually, we should use the property of linear pairs and vertical angles. Let's start over.

Since \(2x\) and \(4x - 100\) are supplementary (they form a linear pair), we have the equation \(2x+(4x - 100)=180\)

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Step2: Find the measure of each angle

• Angle ①: \(2x=2\times\frac{140}{3}=\frac{280}{3}\approx93.33^{\circ}\)
• Angle ②: \(4x-100 = 4\times\frac{140}{3}-100=\frac{560}{3}-100=\frac{560 - 300}{3}=\frac{260}{3}\approx86.67^{\circ}\)
• Angle ③: Angle ③ is equal to angle ② (vertical angles), so \(\frac{260}{3}\approx86.67^{\circ}\)
• Angle ④: Angle ④ is equal to angle ① (vertical angles), so \(\frac{280}{3}\approx93.33^{\circ}\)

Answer:

Angle ①: \(\frac{280}{3}\approx93.33^{\circ}\), Angle ②: \(\frac{260}{3}\approx86.67^{\circ}\), Angle ③: \(\frac{260}{3}\approx86.67^{\circ}\), Angle ④: \(\frac{280}{3}\approx93.33^{\circ}\)