Angles of Parallel Lines Cut by Transversals 06:41 minutes
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Transcript Angles of Parallel Lines Cut by Transversals
On their nightly food run, the three raccoons crashed their shopping cart... AGAIN. It's time to go back to the drawing stump. They decide to practice going around the sharp corners and tight angles during the day, before they get their loot. To put this surefire plan into action they'll have to use their knowledge of parallel lines and transversals. Let's look at this map of their city. All the HORIZONTAL roads are parallel lines. They DON'T intersect. But there are several roads which CROSS the parallel ones. These lines are called TRANSVERSALS. The raccoons crashed HERE at angle 1. The measure of angle 1 is 60 degrees. Can you see any other angles that are also 60 degrees? There are a few such angles, and one of them is angle 3. That's because angle 1 and angle 3 are vertical angles, and vertical angles are always equal in measure. Do we have enough information to determine the measure of angle 2? Since angles 1 and 2 are angles on a line, they sum to 180 degrees. That means angle 2 is 120 degrees. And since angles 2 and 4 are vertical, angle 4 must also be 120 degrees. Now we know all of the angles around this intersection, but what about the angles at the other intersection? Let's take a look at angle 5. If we translate angle 1 along the transversal until it overlaps angle 5, it looks like they are congruent. And they are! That means angle 5 is also 60 degrees. Angle 1 and angle 5 are examples of CORRESPONDING angles. Corresponding angles are pairs of angles that are in the SAME location around their respective vertices. And whenever two PARALLEL lines are cut by a transversal, pairs of corresponding angles are CONGRUENT. That means you only have to know the measure of one angle from the pair, and you automatically know the measure of the other! Can you see other pairs of corresponding angles here? Angles 2 and 6 are also corresponding angles. So are angles 3 and 7 and angles 4 and 8. That means the measure of angle 2 equals the measure of angle 6, the measure of angle 3 equals the measure of angle 7, and the measure of angle 4 equals the measure of angle 8. We already know that angles 4 and 6 are both 120 degrees, but is it ALWAYS the case that such angles are congruent? It is! Let's show this visually. Now, let's use our knowledge of vertical and corresponding angles to prove it. We are going to use angle 2 to help us compare the two angles. Angle 4 must be equal to angle 2 because they are vertical angles. And angle 6 must be equal to angle 2 because they are corresponding angles. Since angle 6 and angle 4 are both equal to the same angle, they also must be equal to each other! We call angle pairs like angle 6 and angle 4 alternate interior angles because they are found on ALTERNATE sides of the transversal and they are both INTERIOR to the two parallel lines. Can you see another pair of alternate interior angles? 3 and 5 are ALSO alternate interior. If two parallel lines are cut by a transversal, alternate interior angles are always congruent. We just looked at alternate interior angles, but we also have pairs of angles that are called alternate EXTERIOR angles. Based on the name, which angle pairs do you think would be called alternate exterior angles? Well, they need to be EXTERIOR to the parallel lines and on ALTERNATE sides of the transversal. 1 and 7 are a pair of alternate exterior angles and so are 2 and 8. Notice that the measure of angle 1 equals the measure of angle 7 and the same is true for angles 2 and 8. If two parallel lines are cut by a transversal, alternate exterior angles are always congruent. In fact, when parallel lines are cut by a transversal, there are a lot of congruent angles. Look at what happens when this same transversal intersects additional parallel lines. We can use congruent angle pairs to fill in the measures for THESE angles as well. The raccoons only need to practice driving their shopping cart around ONE corner to be ready for ALL the intersections along this transversal. For each transversal, the raccoons only have to measure ONE angle. They can then use their knowledge of corresponding angles, alternate interior angles, and alternate exterior angles to find the measures for ALL the angles along that transversal. Now it's time for some practice before they do a little...um... shopping. While they are riding around, let's review what we've learned. When parallel lines are cut by a transversal, congruent angle pairs are created. Corresponding angles are in the SAME position around their respective vertices and there are FOUR such pairs. Alternate interior angles are on ALTERNATE sides of the transversal and INTERIOR to the parallel lines and there are two such pairs. Alternate EXTERIOR angles are on alternate sides of the transversal and EXTERIOR to the parallel lines and there are also two such pairs. The raccoons are trying to corner the market on food scraps, angling for a night-time feast! Well, THAT was definitely a TURN for the worse!