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Parallel Lines Cut By A Transversal is a line that goes through two parallel lines. They form many different types of angles. They form angles called vertical angles, linear pairs, corresponding angles, interior angles, exterior angles, alternative exterior angles, alternative interior angles, and consecutive interior angles.
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Vertical Angles on parallel line cut by a transversal are angles that are on opposite sides from each other that equal the same thing. In the diagram on the left, angles a and d , b and c, e and h, and f and g are all vertical angles. A and C can't be a vertical angle because both of them don't equal to each other, and that they are not the opposite from each other.
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A and C isn't a vertical angle, but it can be a linear pair, which is two angles that create a supplementary angle. B and D, B and A, C and D, E and B, C and E, A and F, D and F, E and F, G and A, G and D, B and H, C and H, E and G, F and H, and G and H. A and D are not linear pairs because they equal less than 180. This diagram shows some of the linear pair that can be made a parallel lines cut by a transversal.
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Corresponding Angles are two angles who have the same position. 1 and 5, 2 and 6, 3, and 7, and 4 and 8. They all have to equal the same, but they cannot be on opposite from each other, unlike vertical angles. It can't be 1 and 7 either because no angle can be outside and be the same, it has to be in the same place as the other angle, like 1 and 5.
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Alternative Interior Angles are two angles that are opposites angles from the inside of the parallel lines cut by a transversal. This is somehow similar to the vertical angle, but it happens in the inside of it. Both angles must equal the same, and not different. In the diagram on the left, Angles C and F and Angles E and D are the only two pairs that are alternative interior angles. A and H nor B and G can be alternative interior because they are in the inside of the angles.
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Alternative Exterior Angles is the opposite of what alternative interior is asking. Alternative Exterior Angles are asking for angles that are opposite angles that are outside of the parallel lines cut by a transversal. Like the alternative interior, both angles must equal the same thing as well. The alternative exterior angles from the left diagram is Angles A and H and Angles B and G. It cannot be C and F nor D and F, because they are inside, like what was covered before.
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Consecutive Interior is almost like alternative interior angles, but it looks for something else in the inside of the picture. It looks for inside angles, but that lie on the same side of the transversal. They cannot equal the same, or outside the lines. Angles 3 and 5 and Angles 4 and 6 are both consecutive interior angles.
Relationship between the angles:
- Vertical Angles: ∠ 1 ≅ ∠ 2
- Linear Pairs: ∠ 1 + ∠ 2 = 180
- Corresponding Angles: ∠ 1 ≅ ∠ 2
- Interior Angles: It Depends (alternative or corresponding?)
- Exterior Angles: It Depends (alternative or corresponding?)
- Alternative Interior Angles: ∠ 1 ≅ ∠ 2
- Alternative Exterior Angles: ∠ 1 ≅ ∠ 2
In solving problems on Parallel Lines Cut by a Transversal, we use only the two formula*:
- Congruent: ∠ 1 ≅ ∠ 2
- Supplementary: ∠ 1 + ∠ 2 = 180
How to solve using both formulas*:
- Locate the special angle
- Identify the relationship and write the appropriate equation (either ∠ 1 ≅ ∠ 2 or ∠ 1 + ∠ 2 = 180)
- Substitute
- Solve for x
- Solve each angles and make sure they hold up the relationship you identify in step 2.
*Notes from my teacher, Ms. Koltunova & Mr. Perry