Find all the missing angles in intersecting lines worksheet.

I'll solve this geometry problem by finding all the missing angles. The image shows 6 diagrams, each with two parallel lines cut by a transversal, and I need to find the measures of angles 1 through 8 in each diagram.

Let me start by analyzing the first diagram (labeled "1)") which has ∠5 = 133.9° given.

First, look closely:
- Diagram 1 shows two horizontal parallel lines intersected by a diagonal transversal.
- Angles are labeled 1-8 around the two intersection points.
- ∠5 is given as 133.9°.
- I need to use angle relationships: vertical angles are equal, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary (add up to 180°).

Since I have the measure of ∠5, I can find other angles using these relationships:
- ∠5 and ∠8 are vertical angles, so ∠8 = ∠5 = 133.9°
- ∠5 and ∠3 are corresponding angles, so ∠3 = ∠5 = 133.9°
- ∠3 and ∠4 are adjacent angles on a straight line, so they're supplementary: ∠4 = 180° - 133.9° = 46.1°
- ∠4 and ∠1 are vertical angles, so ∠1 = ∠4 = 46.1°
- ∠1 and ∠2 are adjacent angles on a straight line, so they're supplementary: ∠2 = 180° - 46.1° = 133.9°
- ∠2 and ∠6 are corresponding angles, so ∠6 = ∠2 = 133.9°
- ∠6 and ∠7 are adjacent angles on a straight line, so they're supplementary: ∠7 = 180° - 133.9° = 46.1°

Let me verify this pattern works for the other diagrams as well.

For diagram 2, ∠8 = 106.8° is given.
- ∠8 and ∠5 are vertical angles, so ∠5 = ∠8 = 106.8°
- ∠5 and ∠3 are corresponding angles, so ∠3 = ∠5 = 106.8°
- ∠3 and ∠4 are adjacent angles on a straight line, so they're supplementary: ∠4 = 180° - 106.8° = 73.2°
- ∠4 and ∠1 are vertical angles, so ∠1 = ∠4 = 73.2°
- ∠1 and ∠2 are adjacent angles on a straight line, so they're supplementary: ∠2 = 180° - 73.2° = 106.8°
- ∠2 and ∠6 are corresponding angles, so ∠6 = ∠2 = 106.8°
- ∠6 and ∠7 are adjacent angles on a straight line, so they're supplementary: ∠7 = 180° - 106.8° = 73.2°

This pattern confirms my approach is correct.

Let me now solve for all 6 diagrams:

Diagram 1 (∠5 = 133.9°):
- ∠1 = 46.1° (vertical to ∠4)
- ∠2 = 133.9° (supplementary to ∠1)
- ∠3 = 133.9° (corresponding to ∠5)
- ∠4 = 46.1° (supplementary to ∠3)
- ∠5 = 133.9° (given)
- ∠6 = 133.9° (corresponding to ∠2)
- ∠7 = 46.1° (supplementary to ∠6)
- ∠8 = 133.9° (vertical to ∠5)

Diagram 2 (∠8 = 106.8°):
- ∠1 = 73.2° (vertical to ∠4)
- ∠2 = 106.8° (supplementary to ∠1)
- ∠3 = 106.8° (corresponding to ∠5)
- ∠4 = 73.2° (supplementary to ∠3)
- ∠5 = 106.8° (corresponding to ∠3)
- ∠6 = 106.8° (corresponding to ∠2)
- ∠7 = 73.2° (supplementary to ∠6)
- ∠8 = 106.8° (given)

Diagram 3 (no angles given):
Looking at the image, I see that diagram 3 has no given angle values. This might be an error or perhaps it's meant to be solved using the same pattern as the others, but without a starting value, I cannot determine specific angle measures. I'll need to check if there's any information I missed.

Actually, looking more carefully at the image, I notice that in diagram 3, there are no given angle values, which means this diagram cannot be solved without additional information. However, since this appears to be a worksheet, it's possible that the intention was to apply the same geometric principles, but without a starting value, we can only express relationships between angles, not their specific measures.

Wait, let me check diagram 4, which has ∠3 = 133.8° given.

Diagram 4 (∠3 = 133.8°):
- ∠1 = 46.2° (vertical to ∠4)
- ∠2 = 133.8° (supplementary to ∠1)
- ∠3 = 133.8° (given)
- ∠4 = 46.2° (supplementary to ∠3)
- ∠5 = 133.8° (corresponding to ∠3)
- ∠6 = 133.8° (corresponding to ∠2)
- ∠7 = 46.2° (supplementary to ∠6)
- ∠8 = 133.8° (vertical to ∠5)

Diagram 5 (∠2 = 65.2°):
- ∠1 = 114.8° (supplementary to ∠2)
- ∠2 = 65.2° (given)
- ∠3 = 65.2° (corresponding to ∠5, and also vertical to ∠2? Wait, let me reconsider...)
Actually, ∠2 and ∠3 are adjacent angles on a straight line, so they're supplementary: ∠3 = 180° - 65.2° = 114.8°
- ∠4 = 65.2° (vertical to ∠2, or supplementary to ∠3)
- ∠5 = 114.8° (corresponding to ∠3)
- ∠6 = 65.2° (corresponding to ∠2)
- ∠7 = 114.8° (supplementary to ∠6)
- ∠8 = 65.2° (vertical to ∠6)

Diagram 6 (∠7 = 103.7°):
- ∠1 = 76.3° (vertical to ∠4)
- ∠2 = 103.7° (supplementary to ∠1)
- ∠3 = 103.7° (corresponding to ∠5)
- ∠4 = 76.3° (supplementary to ∠3)
- ∠5 = 103.7° (corresponding to ∠3)
- ∠6 = 103.7° (corresponding to ∠2)
- ∠7 = 103.7° (given)
- ∠8 = 76.3° (vertical to ∠5? Wait, ∠8 should be vertical to ∠5, so ∠8 = 103.7°, but that contradicts the supplementary relationship with ∠7. Let me recalculate.)

Actually, ∠7 and ∠8 are adjacent angles on a straight line, so they're supplementary: ∠8 = 180° - 103.7° = 76.3°
Then ∠5 = ∠8 = 76.3° (vertical angles)
∠3 = ∠5 = 76.3° (corresponding angles)
∠4 = 180° - 76.3° = 103.7° (supplementary to ∠3)
∠1 = ∠4 = 103.7° (vertical angles)
∠2 = 180° - 103.7° = 76.3° (supplementary to ∠1)
∠6 = ∠2 = 76.3° (corresponding angles)

So for Diagram 6:
- ∠1 = 103.7°
- ∠2 = 76.3°
- ∠3 = 76.3°
- ∠4 = 103.7°
- ∠5 = 76.3°
- ∠6 = 76.3°
- ∠7 = 103.7°
- ∠8 = 76.3°

Now, going back to Diagram 3, since no angle is given, I cannot provide specific numerical values. However, I can state the relationships:
- ∠1 = ∠4 = ∠6 = ∠7 (all equal due to vertical and corresponding angle relationships)
- ∠2 = ∠3 = ∠5 = ∠8 (all equal due to vertical and corresponding angle relationships)
- ∠1 + ∠2 = 180° (supplementary angles)

But since the worksheet asks for "all of the missing angles" and provides spaces for numerical answers, it's likely that Diagram 3 was intended to have a given angle, or perhaps it's a mistake in the worksheet.

Given the information available, I'll provide solutions for diagrams 1, 2, 4, 5, and 6, and note that Diagram 3 cannot be solved without additional information.

Here are the complete solutions:

Diagram 1 (∠5 = 133.9°):
- ∠1 = 46.1°
- ∠2 = 133.9°
- ∠3 = 133.9°
- ∠4 = 46.1°
- ∠5 = 133.9°
- ∠6 = 133.9°
- ∠7 = 46.1°
- ∠8 = 133.9°

Diagram 2 (∠8 = 106.8°):
- ∠1 = 73.2°
- ∠2 = 106.8°
- ∠3 = 106.8°
- ∠4 = 73.2°
- ∠5 = 106.8°
- ∠6 = 106.8°
- ∠7 = 73.2°
- ∠8 = 106.8°

Diagram 3 (No given angle):
Cannot be solved without additional information.

Diagram 4 (∠3 = 133.8°):
- ∠1 = 46.2°
- ∠2 = 133.8°
- ∠3 = 133.8°
- ∠4 = 46.2°
- ∠5 = 133.8°
- ∠6 = 133.8°
- ∠7 = 46.2°
- ∠8 = 133.8°

Diagram 5 (∠2 = 65.2°):
- ∠1 = 114.8°
- ∠2 = 65.2°
- ∠3 = 114.8°
- ∠4 = 65.2°
- ∠5 = 114.8°
- ∠6 = 65.2°
- ∠7 = 114.8°
- ∠8 = 65.2°

Diagram 6 (∠7 = 103.7°):
- ∠1 = 103.7°
- ∠2 = 76.3°
- ∠3 = 76.3°
- ∠4 = 103.7°
- ∠5 = 76.3°
- ∠6 = 76.3°
- ∠7 = 103.7°
- ∠8 = 76.3°

Note: In Diagram 3, since no angle is provided, the specific measures cannot be determined. The relationships between angles would be: ∠1 = ∠4 = ∠6 = ∠7, ∠2 = ∠3 = ∠5 = ∠8, and ∠1 + ∠2 = 180°.