Geometry 3.3 Proofs Notes Name: _______________________________ Some Important Properties to Remember Reflexive Property: For any real number a, _______________________ (This property also works for congruence) Symmetric Property: For any real numbers a and b, if π = π, then _________________ (This property also works for congruence) Transitive Property: For any real numbers a, b, and c, if π = π, and π = π, then __________ (This property also works for congruence) Substitution Property: If π = π, then π can be substituted for π in any equation or expression. (Can ONLY be used when we are talking about things that are equal, NOT congruent.) Parallel Lines and Transversals Corresponding Angles Postulate If two lines are parallel, then corresponding angles are _______________________. Alternate Interior Angles Theorem If two lines are parallel, then alternate interior angles are ________________________. Alternate Exterior Angles Theorem If two lines are parallel, then alternate exterior angles are ________________________. Same- Side Interior Angles Theorem If two lines are parallel, then same side interior angles are ________________________. Vertical Angles Theorem If two angles are vertical angles, then they are _______________. Linear Pair Theorem If two angles are a linear pair, then they are _________________. Converse to Corresponding Angles Postulate If two lines are cut by a transversal and corresponding angles are _________________________, then the two lines are parallel. Converse to Alternate Interior Angles Theorem If two lines are cut by a transversal and alternate interior angles are _________________________, then the two lines are parallel. Converse to Alternate Exterior Angles Theorem If two lines are cut by a transversal and alternate exterior angles are _________________________, then the two lines are parallel. Converse to Same-Side Interior Angles Theorem If two lines are cut by a transversal and same side interior angles are _________________________, then the two lines are parallel. Remember: In order to use any of the theorems or postulates above, we must first define the angles as what type of angle pair they are. Directions: Complete each 2-column proof below. Given: π β₯ β and < 1 β < 2 1. Prove: π β₯ π Statements Reasons 1. π β₯ β 1. _________________________________________________ 2. β 1 and β 3 are corresponding angles. 2. _________________________________________________ 3. β 1 β β 3 3. _________________________________________________ 4. β 1 β β 2 4. _________________________________________________ 5. β 3 β β 2 5. _________________________________________________ 6. β 3 and β 2 are alternate exterior angles. 6. _________________________________________________ 7. π β₯ π 7. _________________________________________________ 2. Given: π β₯ π and β 1 β β 2 Prove: π β₯ π Statements Reasons 1. π β₯ π 1. _________________________________________________ 2. β 1 and β 3 are alternate interior angles. 2. _________________________________________________ 3. β 1 β β 3 3. _________________________________________________ 4. β 1 β β 2 4. _________________________________________________ 5. β 3 β β 2 5. _________________________________________________ 6. β 3 and β 2 are alternate interior angles. 6. _________________________________________________ 7. π β₯ π 7. _________________________________________________ 3. Given: π β₯ π and β 1 β β 3 Prove: π β₯ π Statements Reasons 1. π β₯ π 1. _________________________________________________ 2. β 2 πππ β 1 are corresponding angles. 2. _________________________________________________ 3. β 2 β β 1 3. _________________________________________________ 4. β 1 β β 3 4. _________________________________________________ 5. β 2 β β 3 5. _________________________________________________ 6. β 2 πππ β 1 are alternate interior angles. 6. _________________________________________________ 7. π β₯ π 7. _________________________________________________ 4. Given: β 1 β β 2, and β 3 β β 4 Prove: Μ Μ Μ Μ π΄π΅ β₯ Μ Μ Μ Μ πΆπ· Statements Reasons 1. β 1 β β 2 1. _________________________________________________ 2. β 2 and β 3 are vertical angles. 2. _________________________________________________ 3. β 2 β β 3 3. _________________________________________________ 4. β 1 β β 3 4. _________________________________________________ 5. β 3 β β 4 5. _________________________________________________ 6. β 1 β β 4 6. _________________________________________________ 7. β 1 and β 4 are alternate interior angles. 7. _________________________________________________ 8. Μ Μ Μ Μ π΄π΅ β₯ Μ Μ Μ Μ πΆπ· 8. _________________________________________________ 5. Given: π β₯ π, and β 2 β β 3 Prove: π β₯ π Statements Reasons
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