In problem 9, we are given that \angle 2 : \angle 3 = 2:7. We need to find the measures of \angle 2, \angle 3, \angle 4, and \angle 5. The lines a and b are parallel.

Problem 9

We are given that lines a and b are parallel and the ratio \angle 2 : \angle 3 = 2:7. We need to find the measures of \angle 2, \angle 3, \angle 4, and \angle 5.

Solution:

Since \angle 2 and \angle 3 form a linear pair on line a, they are supplementary:

\angle 2 + \angle 3 = 180^{\circ}

We are given the ratio \angle 2 : \angle 3 = 2:7. Let \angle 2 = 2x and \angle 3 = 7x.

Substituting these into the equation for supplementary angles:

2x + 7x = 180^{\circ}

9x = 180^{\circ}

x = \frac{180^{\circ}}{9}

x = 20^{\circ}

Now we can find the measures of \angle 2 and \angle 3:

• \angle 2 = 2x = 2 \times 20^{\circ} = 40^{\circ}
• \angle 3 = 7x = 7 \times 20^{\circ} = 140^{\circ}

Since lines a and b are parallel:

• \angle 4 and \angle 2 are alternate interior angles, so \angle 4 = \angle 2. Therefore, \angle 4 = 40^{\circ}.
• \angle 5 and \angle 3 are alternate interior angles, so \angle 5 = \angle 3. Therefore, \angle 5 = 140^{\circ}.

Answers:

• \angle 2 = 40^{\circ}
• \angle 3 = 140^{\circ}
• \angle 4 = 40^{\circ}
• \angle 5 = 140^{\circ}