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Transcript
Name:____________________ Geometry Theorems Chapters 1-4 KEEP THIS IN YOUR NOTES FOR EVER!!!!! IT IS SUPER USEFUL Postulate 1: If there are two points, then there is_____________________________ that contains them. Postulate 3: If there are three points, then there is_____________________________ that contains them. The Ruler Postulate: ο· Points on a line can be numbered so that:_____________________________________________ ο· Distance between points can be calculated by:__________________________________________ Betweeness of Points Theorem: If B is between points A and C then we can write the equation: π΄πΆ =_____+______ Linear Pairs Theorem: If two angles are a linear pair then they are ____________________________ Vertical Angles Theorem: If two angles are vertical angles then they are________________________ Pythagorean Theorem:__________________________________________________________________ Substitution Property of Equality: If two things are equal, then they can be____________________ for each other. Transitive Property of Equality:______________________________________________________ Addition Property of Equality:_______________________________________________________ Subtraction Property of Equality:_____________________________________________________ Multiplication Property of Equality:___________________________________________________ Division Property of Equality:________________________________________________________ Square Root Property of Equality:____________________________________________________ Reflexive Property of Equality:_______________________________________________________ Theorem 2.2: If two angles are congruent and supplementary, then each is a _______________________ Theorem 2.3: If two angles are both supplementary to a third angle, then theyβre___________________ to each other. Theorem 2.4: If two angles are both complementary to a third angle, then theyβre ___________________ to each other Alternate Interior Angles Theorem: Two lines are _______________ if and only if their alternate interior angles are_____________________. Alternate Exterior Angles Theorem: Two lines are _______________ if and only if their alternate exterior angles are_____________________. Corresponding Angles Theorem: Two lines are _______________ if and only if their corresponding angles are_____________________. Same Side Interior Angles Theorem: Two lines are _______________ if and only if their same side interior angles are_____________________. Same Side Exterior Angles Theorem: Two lines are _______________ if and only if their same side exterior angles are_____________________. Transitivity of Parallel Lines Theorem: If π β₯ π and π β₯ π then ________________ Theorem 3.8: If π β₯ π‘ and π β₯ π‘ then _________________________ Triangle Sum Theorem: The sum of the three interior angles of a triangle is always________________ Exterior Angle Theorem: An exterior angle equals the ______________ of the remote interior angles. Corresponding Parts of Congruent Triangles are Congruent Theorem (CPCTC Thm): If two triangles are congruent, then all their corresponding parts are___________________________ Third Angle Theorem: If two triangles have two angles that are congruent, then their third angles are also __________________ Reflexive Property of Congruence: Any figure is always __________________ to itself. Transitive Property of Congruence: If shape A is congruent to shape B and shape B is congruent to shape C then________________________________________________________________________ SSS Postulate: If two triangles have all three corresponding sides congruent, then the triangles themselves are________________________________ ASA Postulate: If two triangles have a corresponding ____________, _____________ and __________ congruent, then the triangles are congruent. AAS Postulate: If two triangles have a corresponding ____________, _____________ and __________ congruent, then the triangles are congruent. SAS Postulate: If two triangles have a corresponding ____________, _____________ and __________ congruent, then the triangles are congruent. Some Useful Vocabulary that is also used in proofs Bisect: Something is cut into two _________________ pieces Midpoint: The point that divides a line segment into two ____________________ pieces Right Angle: An angle that measures_________ degrees. NOTE: ALL RIGHT ANGLES ARE CONGRUENT TO EACH OTHER!!! Perpendicular Bisector: A line segment that cuts another line into two congruent pieces and does it at a 90 degree angle. Name:_____________________ Date:__________________ Classwork 4-3 Recognizing Theorems and using them in Proofs (1) Look through your list of theorems and decide which theorem justifies each conclusion below. Given: π₯ + 3 = 7 Conclusion π₯ = 4 Given: β 1 = 25° Conclusion: β 2 = 25° Reason:________________ 2 Reason:_______________ Given: Given: Given Line π΄π΅ β₯ πΆπ· and line πΆπ· β₯ πΈπΉ Conclusion: π΄π΅ β₯ πΈπΉ 1 Reason:__________________ π 40° Conclusion: β π = 110° Given: β 1 is supplementary to β 2 and β 2 is supplementary to β 3 Conclusion: πβ 1 = πβ 3 Reason:__________________ Reason:__________________ Given: Line π΄π΅ β₯ πΆπ· and line πΆπ· β₯ πΈπΉ Conclusion: π΄π΅ β₯ πΈπΉ Given: Point π΅ is between points π· and π½ Conclusion: π·π΅ + π΅π½ = π·π½ Given: Ray βββββ π΄π΅ bisects β πΆπ΄π Conclusion: πβ πΆπ΄π΅ = πβ π΅π΄π Reason:__________________ Reason:__________________ Reason:__________________ Given: β π΄π΅πΆ is complementary to β π·πΈπΉ and β π·πΈπΉ is complementary to β πΊπ»πΌ Conclusion: πβ π΄π΅πΆ = πβ πΊπ»πΌ Given: β 1 = 45° Conclusion: β 2 = 135° Given: the diagram Conclusion: The triangles are congruent Reason:__________________ Reason:______________ Reason:________ Given: πβ 1 = πβ 2 Conclusion: π β₯ π Given: The triangles are congruent Conclusion: 2 πβ 3 = πβ 2 3 Reason:_______ Given: the diagram Conclusion: β π = 60 Conclusion: The triangles are congruent. Reason:_________________ π 1 π 2 Reason:_________________ Given: A is the midpoint of CR C A Conclusion: πΆπ΄ = π΄π R Reason:_________________ Given: π β₯ π Conclusion: πβ 1 = πβ 2 70° 1 2 π 1 45° 75° π Reason:__________________ π 2 Reason:_________________ Given: The diagram Conclusion: πβ π = πβ π 33° 67° π π 67° 33° Reason:________________ (2) Come up with your own conclusions AND the reasons. Given Line πΈπ΄ β₯ πΊπ· and line πΊπ· β₯ π΅πΆ Conclusion:______________ Given: π β₯ π Conclusion: __________ __________ 1 π Given: πβ 1 = πβ 2 Conclusion: __________ π 2 1 π π 2 Reason:__________________ Reason:_________________ Reason:_________________ Given: diagram Conclusion __________ __________________ Given: Given: Line πΊπ β₯ π»πΌand line π΅π β₯ π΄πΆ Conclusion: Reason:___________ Reason:__________________ Reason:__________________ Given: the diagram Conclusion: ______________ Given: β 1 = 32° Conclusion:_________ Given: E is the midpoint of BT 82° π 34° Conclusion:______________ 1 52° 68° π B E T Conclusion:______________ 2 Reason:_______________ Reason:__________________ Reason:_________________ Given: β 1 = 36° Given: Ray ββββ πΌπΊ bisects β π πΌπ Conclusion:_______________ Conclusion:__________ 1 ________________________ 2 Given: Conclusion: _____________ _______________________ Reason:__________________ Reason:______________ Reason:_________________ Given: The diagram Given: The triangles are congruent Given: π₯ 2 = 49 Conclusion: ____________ π 60° 80° 60° 80° π Reason:________________ Given: β 1 is supplementary to β 2 and β 2 is supplementary to β 3 C Conclusion_______________ Conclusion: ______________ A Reason:_________ D B Given: Point π΅ is between points π΄ and π Conclusion:_______________ Conclusion:_______________ Reason:__________________ Reason:________________ Given: β 1 is complementary to β 2 and β 2 is complementary to β 3 Conclusion:_______________ Reason:__________________ Reason:__________________