Angle Relationships
Learning Target
DOK 3: I can create an argument comparing the relationships between the angles created when a transversal intersects two parallel lines.
Intersecting Lines
Any time two lines intersect there are four angles created by this intersection.
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Linear Pairs
Because a line is defined to be 180 degrees, any adjacent angles in this intersection must add up to 180 degrees (they are supplementary angles).
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Angles ∠1 and ∠2 are a linear pair (and therefore supplementary), because they build a straight line across the bottom. Angles ∠3 and ∠4 are also supplementary because they build a straight line across the top.
This fact is true for any adjacent angles in the intersection. Angles ∠1 and ∠4 are supplementary for the same reason.
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Vertical Angles
Angles across from each other in an intersection of lines are called vertical angles.
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Angles ∠1 and ∠3 are vertical angles. So are angles ∠2 and ∠4.
Because both angles ∠1 and ∠3 are supplementary to angles ∠2 and ∠4, it follows that these angles must have equal measurements. If angle ∠1 is 110 degrees, angle ∠3 is also 110 degrees.
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To summarize: A linear pair of angles are always supplementary, while vertical angles are always congruent.
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Transversals
A line that crosses two other lines is called a transversal. The red line below is a transversal.
Transversal Angles
A transversal creates eight angles. If the two lines it crosses (green) are parallel, there are many important connections between the angles.
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Corresponding Angles
Angles on the same corner of the intersections are called corresponding angles.
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Angles ∠1 and ∠5 are corresponding angles, because they are both at the top of the intersections. Angles ∠2 and ∠6 are also corresponding angles, as well as angles ∠3 and ∠7 and angles ∠4 and ∠8.
Alternate Angles
Angles on the opposite side of the transversal are called alternate angles.
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Angles ∠2 and ∠8 are alternateangles, because they are both on different sides of the red transversal line.. Angles ∠1 and ∠7 are also alternate angles, as well as many other combinations.
Interior and Exterior Angles
Angles between the parallel lines are called interior angles, while angles outside the parallel lines are called exterior angles.
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Interior
Exterior
Exterior
Alternate Interior/Exterior
We combine the terms alternate and interior/exterior. Alternate interior angles are angles on the opposite sides of the transversal that are between the parallel lines.
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Angles ∠2 and ∠8 are alternate interior angles, So are angles ∠3 and ∠5.
Explore!
Use the lab at http://tube.geogebra.org/student/m1237025 to explore the relationships between the angles created by a transversal and parallel lines. What relationship do you observe between corresponding angles? How about alternate interior angles? Alternate exterior angles?