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Geometry 1 Name: _______________________________ Homework #1 on Proofs The following Definitions, Postulates and Theorems can be used in this assignment: Definition of Midpoint: M is the midpoint of AB if and only if M is between A and B and AM BM . Definition of Bisects: A line bisects a segment if it intersects the segment at its midpoint, and a ray bisects an angle if it is interior to the angle, its vertex coincides with the vertex of the angle, and it forms two congruent angles. o Definition of Complementary: Two angles are complementary if their measures add to 90 . Definition of Supplementary: Two angles are supplementary if their measures add to 180o. Definition of Adjacent Angles: Two angles are adjacent if they have a common side. Definition of Linear Pair: Two angles form a linear pair if they are adjacent and their noncommon sides are opposite rays (form a line). Definition of Parallelogram: A quadrilateral is a parallelogram if opposite sides are parallel. Transitive: If two segments or angles are congruent to a third segment or angle then they are congruent to each other. Vertical Angle Theorem: Vertical angles are congruent. Linear Pair Theorem: The angles in a linear pair are supplementary.. Parallel Postulate: Given a point and a line not containing that point, there is one and only one line through the given point that is parallel to the given line. Parallel Lines: Two lines cut by a transversal are parallel if and only if corresponding angles are congruent. Two lines cut by a transversal are parallel if and only if alternate interior angles are congruent. Two lines cut by a transversal are parallel if and only if alternate exterior angles are congruent. Two lines cut by a transversal are parallel if and only if same-side interior angles are supplementary. Two lines cut by a transversal are parallel if and only if same-side exterior angles are supplementary. Page 1 of 8 Problems: 1. Mark the diagrams according to the given information, and fill in the missing statements or reasons for each of the following proofs. Q Given: P and Q are supplementary. R Prove: R and S are supplementary. Proof: 2. P S Statement: Reason: 1. P and Q are supplementary. Given 2. QR | | PS 3. R and S are supplementary. B Given: BAP QBC and PAC QBC Prove: AP bisects BAC P A Proof: Statement: Reason: 1. BAP QBC Given 2. PAC QBC 3. BAP PAC 4. AP bisects BAC Page 2 of 8 Q C 3 3. Given: 1 2 and 1 || 2 Prove: 1 3 4 2 1 1 2 3 Proof: Statement: Reason: 1. 1 || 2 Given 2. 2 3 3. 1 2 4. 1 3 Page 3 of 8 4. 3 Given: 1 || 2 , 1 2 , and 3 4 Prove: 1 4 1 4 1 1 Proof: Statement: 1. 1 || 2 2. 2 3 3. 1 3 4. 3 4 5. 1 4 2 3 2 Reason: Page 4 of 8 5 4 5. C Given: ACE CED and EAD ADC D Prove: ACDE is a parallelogram. A Proof: Statement: 1. E Reason: ACE CED 2. Two lines cut by a transversal are parallel if and only if alternate interior angles are congruent. 3. Given 4. CD | | AE 5. ACDE is a parallelogram. Page 5 of 8 6. Given: A ACB and D DCE Prove: A Statement: 1. A ACB 2. ACB DCE 3. A DCE 4. D DCE 5. A D D C AB | | DE Proof: 6. B E Reason: Two lines cut by a transversal are parallel if and only if alternate interior angles are congruent. Page 6 of 8 A 7. Given: BD bisects EBC and CBD AEB Prove: AE | | BD E Proof: Statement: 1. B Reason: BD bisects EBC Definition of “bisects.” 2. 3. CBD AEB 4. EBD AEB 5. AE | | BD Transitive Property Page 7 of 8 D C 8. Prove that the angles of a triangle add to 180o: D Given: ABC with a mBAC , b mABC , c mACB d Prove: a + b + c = 180o A Proof: Statement: E B b e a c C Reason: 1. Let DE be the line containing point B that is parallel to AC . 2. Let d mABD and e mCBD 3. d=a Definition of d and e. These are alternate interior angles for parallel lines DE and AC with transversal BC . 4. 5. DBA and ABC are adjacent angles. Definition of adjacent angles 6. mDBC d b Angle Addition Postulate 7. DBC and CBE are a linear pair. Definition of linear pair 8. DBC and CBE are supplementary. 9. mDBC e 180 10. d + b + e= 180o Substitution using steps 6 and 9. 11. a + b + c= 180o Substitution using steps 3, 4 and 10. Page 8 of 8