Question

finding angle measures use the figure to find the measures of the numbered angles. explain your reasoning. 8. 9. 10. 11. you be the teacher your friend describes a relationship between the angles shown. is your friend correct? explain your reasoning. ∠5 is congruent to ∠6.

Explanation:

Step1: Identify angle - pair relationships for problem 8

Since the two horizontal lines are parallel and \(t\) is a transversal, \(\angle1\) and the \(107^{\circ}\) angle are corresponding angles. So \(\angle1 = 107^{\circ}\). Also, \(\angle1\) and \(\angle2\) are a linear - pair. Since \(\angle1+\angle2 = 180^{\circ}\), then \(\angle2=180 - 107=73^{\circ}\).

Step2: Identify angle - pair relationships for problem 9

The \(95^{\circ}\) angle and \(\angle3\) are vertical angles. So \(\angle3 = 95^{\circ}\). \(\angle3\) and \(\angle4\) are a linear - pair. Since \(\angle3+\angle4 = 180^{\circ}\), then \(\angle4=180 - 95 = 85^{\circ}\).

Step3: Identify angle - pair relationships for problem 10

The \(49^{\circ}\) angle and \(\angle5\) are corresponding angles. So \(\angle5 = 49^{\circ}\). \(\angle5\) and \(\angle6\) are a linear - pair. Since \(\angle5+\angle6 = 180^{\circ}\), then \(\angle6=180 - 49=131^{\circ}\).

Step4: Analyze the statement in problem 11

\(\angle5\) and \(\angle6\) are a linear - pair. Linear - pair angles are supplementary (\(\angle5+\angle6 = 180^{\circ}\)), not congruent. So the friend is incorrect.

Answer:

1. \(\angle1 = 107^{\circ}\), \(\angle2 = 73^{\circ}\)
2. \(\angle3 = 95^{\circ}\), \(\angle4 = 85^{\circ}\)
3. \(\angle5 = 49^{\circ}\), \(\angle6 = 131^{\circ}\)
4. Incorrect. Linear - pair angles are supplementary, not congruent.