Common Segments Theorem (Flowchart Proof)
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1 Common Segments Theorem (Flowchart Proof) Below is a flow proof proving the Common Segments Theorem. The Common Segments Theorem states that if a segment is combined with two congruent segments then the resulting segments are congruent. Use the questions on the next page to help complete the flow proof. : QR = ST Prove: QS = RT Q R S T QR = ST #1 QR + RS = RS + ST #4 #2 #3 #5 Segment Addition Postulate #8 #7 RS + ST = RT #6 QS = RT Addition Property of Equality RS = RS Segment Addition Postulate Substitution Property QR + RS = QS
2 Vertical Angles Theorem (Paragraph Proof) Below is a paragraph proof proving the Vertical Angles Theorem. Vertical Angles are two non-adjacent angles formed by two intersecting lines. The Vertical Angles Theorem states that if two angles are vertical angles, then the angles are congruent. Use the questions on the next page to help complete the paragraph proof. W : VXW and ZXY are vertical angles as shown. Prove: VXW ZXY V X Y Z It is (a) that VXW and ZXY are vertical angles. By definition of vertical angles, VXW and ZXY are formed by intersecting lines. By definition of a linear pair, VXZ and VXW form a linear pair and (b). Because of the (c), VXZ and VXW are supplementary and (d). By definition of supplementary, (e) and m VXZ + m ZXY = 180. Because of the (f), m VXZ + m VXW = m VXZ + m ZXY. m VXW = m VXW because of the Reflexive Property of Equality. Because of the (g), m VXW = m ZXY. By definition of congruent angles, (h). Transitive Property of Equality VXZ and ZXY form a linear pair given VXW ZXY m VXZ + m VXW = 180 Linear Pair theorem VXZ and ZXY are supplementary
3 Linear Pair Theorem (Two-Column Proof) Below is a two-column proof proving the Linear Pair Theorem. A linear pair is a pair of adjacent angles whose non-common sides are opposite rays. The Linear Pair Theorem states that if two angles form a linear pair, then the angles are supplementary. Use the questions on the next page to help complete the two-column proof. : JMK and LMK form a linear pair Prove: JMK and LMK are supplementary K J M L Statement JMK and LMK form a linear pair Reason a. b. Definition of a linear pair c. Definition of opposite rays m JML = 180 d. m JMK + m LMK = m JML e. m JMK + m LMK = 180 f. g. h. Definition of a Straight Angle Definition of Supplementary JMK and LMK are supplementary Angle Addition Postulate JML is a straight angle Substitution Property MJ and ML are opposite rays
4 Congruent Supplements Theorem (Two-Column Proof) Below is a two-column proof proving the Congruent Supplements Theorem. The Congruent Supplements Theorem states that if two angles are supplements of the same or congruent angles, then the two angles are congruent. B : AEB and BEC are supplementary DEC and BEC are supplementary A E C Prove: AEB DEC D Statement AEB and BEC are supplementary DEC and BEC are supplementary Reason i. j. Definition of Supplementary Angles k. l. m BEC = m BEC m. m AEB = m DEC n. o. Definition of Congruent Angles AEB DEC Transitive Property of Equality m AEB + m BEC = 180 m DEC + m BEC = 180 m AEB + m BEC = m DEC + m BEC
5 Congruent Complements Theorem (Paragraph Proof) Below is a paragraph proof proving the Congruent Complements Theorem. The Congruent Complements Theorem states that if two angles are complementary to the same or congruent angles, then the two angles are congruent. : MQN and NQO are complementary OQP and NQO are complementary Prove: MQN OQP N M O Q P It is that MQN and NQO are complementary and OQP and NQO are complementary. By, m MQN + m NQO = 90 and. By the Transitive Property of Equality,. m NQO = m NQO by the. m MQN m OQP by the. By definition of congruent angles,. Definition of Complementary Angles m MQN + m NQO = m OQP + m NQO given m OQP + m NQO = 90 MQN OQP
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