Question

1. lines m and q are parallel. find the other missing angle measures.

1.
2.
3.
4.
5.
6.

Explanation:

Step1: Find ∠1

∠1 and 110° are supplementary (linear pair), so ∠1 = 180° - 110° = 70°.

Step2: Find ∠2

∠2 and 115° are supplementary (linear pair), so ∠2 = 180° - 115° = 65°.

Step3: Find ∠3

Lines m and q are parallel, and the transversal creates alternate interior angles with ∠2. So ∠3 = ∠2 = 65° (alternate interior angles).

Step4: Find ∠4

∠4 and ∠3 are supplementary (linear pair), so ∠4 = 180° - 65° = 115°.

Step5: Find ∠5

∠5 and 110°: Let's see, ∠1 is 70°, and using triangle or parallel lines. Wait, ∠5 and the angle adjacent to 110° (∠1) and ∠2: Wait, maybe better to use vertical angles or corresponding. Wait, ∠5 and ∠1: Wait, no. Wait, ∠5 and the angle with 110°: Let's use the triangle. Wait, ∠1 is 70°, ∠2 is 65°, so the third angle in the triangle (∠7) would be 180° - 70° - 65° = 45°? Wait, no, maybe I made a mistake. Wait, no, lines m and q are parallel, so ∠5 and ∠1: Wait, ∠5 and ∠1: Wait, ∠1 is 70°, and ∠5: Wait, maybe ∠5 is equal to ∠1? No, wait, let's check again. Wait, the transversal for ∠5: the other transversal (the one with 110°) intersects m and q. So ∠5 and ∠1: Wait, ∠1 is 70°, and ∠5: Let's use vertical angles or corresponding. Wait, ∠5 and the angle opposite to ∠1? No, maybe ∠5 is equal to ∠1? Wait, no, let's do it step by step. Wait, ∠7: ∠7 is equal to ∠5 (alternate interior angles)? Wait, no, ∠7 and ∠5: Wait, lines m and q are parallel, transversal is the one with 110°, so ∠5 and ∠7 are alternate interior? Wait, no, the two transversals: one is the line with 115° and 110°, the other is the other line. Wait, maybe I messed up. Wait, let's start over.

Wait, ∠1 is 70° (from 180 - 110). ∠2 is 65° (180 - 115). Then, in the triangle formed by the two transversals and line q, the angles are ∠1 (70°), ∠2 (65°), so ∠7 (the angle at line q) is 180 - 70 - 65 = 45°? No, that can't be. Wait, no, the triangle is not a triangle, it's a straight line? Wait, no, the two transversals intersect at a point, forming a triangle with line q. So ∠1, ∠2, and ∠7 are the angles of a triangle? Wait, no, ∠1, ∠2, and ∠7 are on a straight line? No, line q is straight, so ∠2 + ∠7 + (angle adjacent to ∠1) = 180°? Wait, I think I made a mistake here. Let's correct.

Wait, ∠1 is 70°, ∠2 is 65°, so ∠7 (the angle between the two transversals on line q) is 180° - 70° - 65° = 45°? No, that's not right. Wait, no, ∠1 and ∠2 are not in a triangle, but on a transversal. Wait, the two transversals intersect at a point, so ∠1, ∠2, and the angle between them (∠7) are on a straight line? No, line q is straight, so ∠2 + ∠7 + (the angle equal to ∠1) = 180°? No, I'm confused. Let's use the other transversal (the one with 110°) for ∠5.

The transversal with 110° intersects m and q, so ∠5 and the angle adjacent to 110° (which is ∠1 = 70°) and the other angle. Wait, ∠5 and ∠1: since lines m and q are parallel, ∠5 should be equal to ∠1? No, ∠1 is 70°, so ∠5 = 70°? Wait, let's check with vertical angles. ∠5 and ∠6: vertical angles? No, ∠6 and ∠1: maybe. Wait, ∠6 is vertical to the angle adjacent to ∠1? No, let's do it properly.

Wait, ∠5: the transversal (the one with 110°) cuts m and q, so ∠5 and the angle above m (let's say ∠x) are alternate interior angles. ∠x is equal to ∠1 (70°), so ∠5 = 70° (alternate interior angles). Yes, that makes sense. So ∠5 = 70°.

Step6: Find ∠6

∠6 and ∠1 are vertical angles? No, ∠6 and ∠5: ∠6 and ∠5 are supplementary? Wait, no, ∠6 and ∠5: lines m and q are parallel, so ∠6 and ∠1: Wait, ∠6 is vertical to the angle that's equal to ∠1? Wait, ∠6 and ∠1: ∠1 is 70°, so ∠6 = 70°? No, wait, ∠6 and ∠5: ∠5 is 70°, so ∠6 = 180° - 70°? No, that's not. Wait, ∠6 and ∠5 are adjacent? No, ∠6 is on line q, ∠5 is on line m. Wait, ∠6 and ∠1: ∠1 is 70°, and ∠6 is equal to ∠1 (corresponding angles). So ∠6 = 70°.

Step7: Find ∠7

∠7 and ∠2: Wait, ∠7 is on line q, between the two transversals. ∠2 is 65°, ∠1 is 70°, so ∠7 = 180° - 65° - 70° = 45°? No, that can't be. Wait, no, ∠7 and ∠3: ∠3 is 65°, and lines m and q are parallel, so ∠7 = ∠3? No, ∠3 is 65°, ∠7: Wait, ∠7 and ∠5: ∠5 is 70°, so ∠7 = 180° - 65° - 70° = 45°? I think I made a mistake here. Wait, maybe the initial approach was wrong. Let's list all angles:

∠1: 70° (180 - 110)

∠2: 65° (180 - 115)

∠3: 65° (alternate interior with ∠2)

∠4: 115° (supplementary to ∠3)

∠5: 70° (alternate interior with ∠1, since lines m and q are parallel, transversal is the one with 110°)

∠6: 70° (vertical to ∠5? No, ∠6 and ∠5: ∠6 is equal to ∠1, so 70°)

∠7: 180° - ∠2 - ∠1? No, ∠7 is on line q, so ∠2 + ∠7 + ∠6 = 180°? Wait, ∠2 is 65°, ∠6 is 70°, so ∠7 = 180 - 65 - 70 = 45°? That seems odd, but maybe.

Wait, maybe the problem has a typo, but let's proceed with the correct angle chasing:

Wait, let's re-express:

- ∠1: 180 - 110 = 70° ✔️

- ∠2: 180 - 115 = 65° ✔️

- ∠3: alternate interior with ∠2, so 65° ✔️

- ∠4: supplementary to ∠3, 180 - 65 = 115° ✔️

- ∠5: alternate interior with ∠1 (since lines m || q, transversal is the one with 110°), so 70° ✔️

- ∠6: vertical to ∠1? No, ∠6 is equal to ∠1 (corresponding angles), so 70° ✔️

- ∠7: on line q, so ∠2 + ∠7 + ∠6 = 180°? 65 + ∠7 + 70 = 180 → ∠7 = 45°? Maybe.

But the problem asks for the other missing angles, so let's list the ones asked (∠1, ∠2, ∠3, ∠4, ∠5, ∠6):

∠1: 70°

∠2: 65°

∠3: 65°

∠4: 115°

∠5: 70°

∠6: 70° (or maybe ∠6 is equal to ∠1, 70°)

Wait, maybe I messed up ∠5 and ∠6. Let's check with vertical angles. The angle with 110° and ∠1 are vertical? No, ∠1 is 70°, and the angle opposite to ∠1 (vertical angle) would be equal to ∠1, which is 70°, and that angle is equal to ∠5 (corresponding angles), so ∠5 = 70°, and ∠6 is vertical to that angle, so ∠6 = 70°. Yes, that makes sense.

So summarizing:

∠1: 70°

∠2: 65°

∠3: 65°

∠4: 115°

∠5: 70°

∠6: 70°

(∠7: 45°, but it's not asked)

Answer:

∠1 = 70°, ∠2 = 65°, ∠3 = 65°, ∠4 = 115°, ∠5 = 70°, ∠6 = 70°