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Chapter 4 Proof Practice This is to give students more practice with proofs. 1) Print out the proofs on heavy paper or cardstock and put in a paper protector 2) Print out the justifications and statements on cardstock. (I have done statements on one color, justifications on another, or make it a little more challenging and have them all the same color – students should be able to figure out which is which). 3) Have students work on the proofs in groups or pairs. (I have several of each proof printed so there is enough for the whole class to work in pairs). 4) Either have students write down the proof once they are done or just check it off in some way so you know which ones they have completed. 5) Occasionally I had students waiting over another students shoulder for a particular number towards the end. But generally students were engaged and on task. 6) I ended up taking out the triangle congruence postulates as I expected students to know which one it was, you can do it either way. Also, you could give them just the statements or just the justifications to make it more challenging. Chapter 4 Proof #1 Given: IE GH ; EF HF ; F is the midpoint of GI Prove: EFI HFG Proof 1 Statements Proof 1 Justifications IE GH Given EF HF Given F is the midpoint of GI Given FG FI EFI HFG Proof 2 statements Defintion of a midpoint A point is the midpoint if and only if it divides the segment into 2 equal parts. SSS Postulate If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent. proof 2 justifications WZ ZS Given SD DW Given ZD ZD Reflexive Property of Congruence WZD SDZ SSS Postulate If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent. Chapter 4 Proof #2 Given: WZ ZS SD DW Prove: WZD SDZ Chapter 4 Proof #3 Given: AB CD , BAC DCA Prove: ABC CDA Proof 3 Statements Proof 3 Justifications AB CD Given BAC DCA Given AC AC ABC CDA Proof 4 Statements X is the midpoint of AG and NR AX XG NX XR Reflexive Property of Congruence SAS Postulate If the two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent. Proof 4 justifications Given Defintion of a midpoint A point is the midpoint if and only if it divides the segment into 2 equal parts. Defintion of a midpoint A point is the midpoint if and only if it divides the segment into 2 equal parts. AXN GXR Vertical Angles Theorem Vertical Angles are congruent AXN GXR If the two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent. SAS Postulate Chapter 4 Proof #4 Given: X is the midpoint of AG and NR Prove: AXN GXR Chapter 4 Proof #5 Given: GK bisects JGM , GJ GM Prove: GJK GMK Proof #5 Statements Proof #5 Justifications GJ GM Given GK bisects JGM Given GK GK GJK GMK JGK MGK Proof #6 Statements Reflexive Property of Congruence SAS Postulate If the two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent. Definition of an Angle Bisector A segment is the angle bisector if and only if it divides an angle into 2 congruent angles. Proof #6 Justifications AC CE Definition of an Segment Bisector A segment is the segment bisector if and only if it divides a segment into 2 segments. AE and BD bisect each other Given CD CB Definition of an Segment Bisector A segment is the segment bisector if and only if it divides a segment into 2 segments. SAS Postulate ACB ECD If the two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent. ACB ECD Vertical Angles Theorem Vertical Angles are congruent Chapter 4 Proof #6 Given: AE and BD bisect each other Prove: ACB ECD Chapter 4 Proof #7 Given: FG || KL ; FG KL Prove: FGK KLF Proof #7 Statements Proof #7 Justifications FG || KL Given FG KL Given FK FK FGK KLF GFK LKF Proof #8 Statements Reflexive Property of Congruence SAS Postulate If the two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent. Alternate Interior Angles Theorem If two parallel lines are intersected by a transversal then alternate interior angles are congruent. Proof #8 Justifications BAC DAC Given AC BD Given AC AC ABC ADC Reflexive Property of Congruence ASA Postulate If the two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent. BCA DCA All Right Angles are Congruent BCA and DCA are right angles Definition of Perpendicular Lines Two segments are perpendicular if and only if they intersect to form 4 right angles. Chapter 4 Proof #8 Given: BAC DAC AC BD Prove: ABC ADC Chapter 4 Proof #9 Given: V Y WZ bisects VWY Prove: VWZ YWZ Proof #9 Statements Proof #9 Justifications V Y Given WZ bisects VWY Given WZ WZ VWZ YWZ VWZ YWZ Proof #10 Statements Reflexive Property of Congruence AAS Theorem If the two angles and a non-included side of one triangle are congruent to the two angles and a non-included side of another triangle, then the triangles are congruent. Definition of an Angle Bisector A segment is the angle bisector if and only if it divides an angle into 2 congruent angles. Proof #10 Justifications PQ QS Given RS QS Given T is the midpoint of PR PQT RST PQT RST PQT is a right angle RST is a right angle PT TR QTP STR Given AAS Theorem If the two angles and a non-included side of one triangle are congruent to the two angles and a non-included side of another triangle, then the triangles are congruent. All Right Angles are Congruent Definition of Perpendicular Lines Two segments are perpendicular if and only if they intersect to form 4 right angles. Definition of Perpendicular Lines Two segments are perpendicular if and only if they intersect to form 4 right angles. Defintion of a midpoint A point is the midpoint if and only if it divides the segment into 2 equal parts. Vertical Angles Theorem Vertical angles are congruent Chapter 4 Proof #10 Given: PQ QS; RS QS, T is the midpoint of PR Prove: PQT RST Chapter 4 Proof #11 Given: N P ; MO QO Prove: MON QOP Proof #11 N P Given MO QO Given MON QOP MON QOP Vertical Angles are congruent AAS Theorem If the two angles and a non-included side of one triangle are congruent to the two angles and a non-included side of another triangle, then the triangles are congruent Proof #12 E D Given AE BD Given AE || BD Given AEB BDC If the two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent. EAB DBC ASA Postulate Corresponding Angles Postulate If two parallel lines are intersected by a transversal then corresponding angles are congruent. Proof #13 DH bisects BDF Given 1 2 Given DH DH Reflexive Property of Congruence BDH FDH If the two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent. BDH FDH ASA Postulate Definition of an Angle Bisector A segment is the angle bisector if and only if it divides the angle into two congruent angles. Proof #12 Given: E D AE BD AE || BD Prove: AEB BDC Proof #13 Given: DH bisects BDF 1 2 Prove: BDH FDH