Unit 2A: Angle Pairs and Transversal Notes
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1 Unit 2A: Angle Pairs and Transversal Notes Day 1: Special angle pairs Day 2: Angle pairs formed by transversal through two nonparallel lines Day 3: Angle pairs formed by transversal through parallel lines Day 4: Proving lines are parallel simple Day 5: Proving lines are parallel complex Day 6: Constructions of parallel lines and review Day 7: Test
2 Day 1: Angle pairs Many pairs of angle have special relationships. Some relationships are because of the measurement of the angles while others are because of the position of the angles. Adjacent angles are two angles in the same plane with a common vertex and a common side, but no common interior points. 1 and 2 are adjacent angles. Linear pairs are adjacent angles (the share a leg) and are supplementary (they add up to 180. Supplementary angles are two angles whose measures have a sum of 180. (They do not have to be adjacent) Complementary angles are two angles whose measures have a sum of 90. (They do not have to be adjacent) Vertical angles are two nonadjacent angles formed by two intersecting lines. Vertical angles are congruent. Practice: ABD and m BDC are supplementary. Find the measurement of both angles. 1) m ABD = 5x, m BDC = (17x- 18) 2) m ABD = (3x + 12), m BDC = (7x 32) Practice: ABD and m BDE are complementary. Find the measurement of both angles. 3) m ABD = (5y + 1), m BDE = (3y - 7) 4) m ABD = (4y + 5), m BDE = (4y + 8)
3 Day 2: Non parallel transversal angle relationship A transversal is a line that intersects lines at two different points. The transversal PQ and the lines AB and CD form eight angles. Corresponding angles lie on the side of the transversal PQ, on the same sides of lines AB and CD. Alternate interior angles are angles that lie on sides of the transversal PQ, between lines AB and CD. Alternate exterior angles lie on sides of the transversal PQ, of lines AB and CD. Same- side interior angles or consecutive interior angles lie on the side of the transversal PQ, between lines AB and CD. Name the angles for the two figures. Corresponding angles Figure 1 Figure 2 Figure 1. Alternate interior angles Alternate Exterior angles Same- side interior angles Figure 2.
4 Day 3:When the Transversal cut through 2 parallel lines Corresponding angles postulate: If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 2 5 Name 3 more: Alternating interior angles theorem: If two parallel lines are cut by a transversal, then the alternating interior angles are congruent. Alternating Exterior angles theorem: If two parallel lies are cut by a transversal, then the alternating exterior angles are congruent. Same- side interior angles: If two parallel lines are cut by a transversal, then the same- side interior angles are supplementary. 3 6 Name 3 more: 2 6 Name 3 more: = 180 Name 3 more: Same- side exterior angles: If two parallel lines are cut by a transversal, then the same- side exterior angles are supplementary = 180 Name 3 more:
5 Day 4: Simple proofs Postulates and theorem that proves two lines are parallel Converse of Corresponding angles postulate: If the corresponding angles are congruent, then the two lines are parallel 1 = 110; 5 = 110 Line AB CD Converse of same side interior angles theorem: If the same- side interior angles are supplementary, then the two lines are parallel. 3 = 70; 5 = 110 Line AB CD Converse of same- side exterior angles theorem: If the same- side exterior angles are supplementary, then the two lies are parallel. 1 = 110; 7 = 70 Line AB CD Converse of alternating interior angles theorem: If the alternating interior angles are congruent, then the two lines are parallel. 3 = 70; 6 = 70 Line AB CD Converse of Alternating exterior angles theorem: If the alternating exterior angles are congruent, then the two lines are parallel. 1 = 110; 8 = 110 Line AB CD
6 Proofs: 1. You will be given certain facts and information a. They can provide a picture or not. If not you should always draw an accurate picture for the information given 2. You will be ask to prove a certain conclusion base on the information given and what you can deduce using your geometry and algebra knowledge. 3. Order is very important in proofs. Therefore you must always start with what is given and move to the next idea. a. You must ask yourself what is the next thing that I can now prove from what is given or from what I have already proven. 4. In a two- column proof, the statement is what you want to say and the reason is why you can say what you want to say. a. The statement is often the definition, and the reason is often the postulate / theorem/ word 5. DO NOT ASSUME! Everything must be proven. Lets practice:
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11 Day 6: Construction of Parallel lines How to construct parallel lines P m Create a line parallel to line m that goes through point P. 1. Add a point A on line m. 2. Connect point P to the point A to create an angle 3. Create a circle between the two points with point A as the center. 4. Label the two points created by circle A point B and C 5. Using the same compass setting, go to point P and create the same circle 6. Label one point created by circle P on line AB!!!! point D 7. Measure the distance between the points B and C using the compass 8. Using the same compass measurements, go to point D and make a circle 9. Where the circle D intercept circle P, label point E. 10. Create line DE!!!! You try:
GEOMETRY POSTULATES AND THEOREMS. Postulate 1: Through any two points, there is exactly one line.
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