Question

question 1
find the unknown angles ∠a and ∠b, if the four lines cut by the transversal are parallel.

○ 115° and 65°

○ 65° and 65°

○ 115° and 115°

○ 105° and 65°

Explanation:

Step1: Find ∠a

∠a and the 115° angle are supplementary (linear pair), so \( \angle a = 180^\circ - 115^\circ = 65^\circ \)? Wait, no—wait, actually, if the lines are parallel, ∠a and the 115° angle: wait, no, let's recheck. Wait, the 115° and ∠a: are they adjacent? Wait, the transversal cuts the parallel lines. Wait, maybe ∠a and 115° are supplementary? Wait, no, maybe vertical angles or corresponding? Wait, no, first, ∠a and the 115° angle: since they form a linear pair (adjacent angles on a straight line), so \( \angle a + 115^\circ = 180^\circ \), so \( \angle a = 180 - 115 = 65^\circ \)? Wait, no, wait, maybe I made a mistake. Wait, no, the four lines are parallel, cut by a transversal. Then, ∠a and the angle below (for ∠b) – wait, no, let's look at ∠b. Since the lines are parallel, ∠b should be equal to ∠a? Wait, no, let's do step by step.

First, find ∠a: the angle of 115° and ∠a are adjacent and form a linear pair, so they are supplementary. So \( \angle a = 180^\circ - 115^\circ = 65^\circ \)? Wait, no, wait, maybe the 115° and ∠a are vertical angles? No, vertical angles are equal. Wait, no, the diagram: the 115° is above the top parallel line, and ∠a is below that intersection. So they are adjacent, forming a linear pair. So \( \angle a = 180 - 115 = 65^\circ \). Then, for ∠b: since all four lines are parallel, the transversal cuts them, so ∠b should be equal to ∠a (corresponding angles, since the lines are parallel). So ∠b = ∠a = 65°? Wait, no, wait, maybe ∠a is equal to 115°? Wait, no, let's re-express.

Wait, maybe I messed up. Let's start over. The four lines are parallel, cut by a transversal. The angle given is 115°, and ∠a is at the same intersection, below the 115°. So 115° and ∠a are supplementary (linear pair), so \( \angle a = 180 - 115 = 65^\circ \). Then, ∠b: since the lines are parallel, the angle ∠b and ∠a are corresponding angles (because the transversal cuts the parallel lines, so the angle at the bottom line should be equal to ∠a, because they are in the same position relative to the transversal and the parallel lines). So ∠b = ∠a = 65°? Wait, but the options: 65° and 65° is an option (second option: 65° and 65°). Wait, but wait, maybe ∠a is 115°? Wait, no, let's check the linear pair. If two angles are adjacent and form a straight line, they sum to 180. So 115 + a = 180 → a = 65. Then, since the lines are parallel, the angle b is equal to a (corresponding angles), so b = 65. So the angles are 65° and 65°, which is the second option. Wait, but wait, maybe I made a mistake. Wait, the first option is 115 and 65, but that would mean a is 115, but then 115 + 115 = 230 ≠ 180. So no. So step 1: find ∠a: linear pair with 115°, so a = 65. Step 2: find ∠b: since lines are parallel, ∠b is equal to ∠a (corresponding angles), so b = 65. So the answer is 65° and 65°, which is the second option.

Wait, but let's confirm. The four lines are parallel, so the transversal creates equal corresponding angles. So ∠a and ∠b: since all lines are parallel, the angle at the bottom (∠b) should be equal to ∠a, because they are in the same relative position (corresponding angles). So yes, ∠a = 65°, ∠b = 65°, so the option is 65° and 65°.

Step1: Calculate ∠a

∠a and \( 115^\circ \) form a linear pair (supplementary), so:
\( \angle a = 180^\circ - 115^\circ = 65^\circ \).

Step2: Determine ∠b

Since the four lines are parallel, \( \angle b \) and \( \angle a \) are corresponding angles (equal). Thus:
\( \angle b = \angle a = 65^\circ \).

Answer:

65° and 65° (the second option, e.g., "65° and 65°" or the option labeled with 65° and 65°)