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Proving Constructions Valid Ch. 6 Proving Constructions Valid Lesson Presentation Holt Geometry Holt Geometry Proving Constructions Valid Objective Use congruent triangles to prove constructions valid. Holt Geometry Proving Constructions Valid When performing a compass and straight edge construction, the compass setting remains the same width until you change it. This fact allows you to construct a segment congruent to a given segment. You can assume that two distances constructed with the same compass setting are congruent. Holt Geometry Proving Constructions Valid The steps in the construction of a figure can be justified by combining: • the assumptions of compass and straightedge constructions, and • the postulates and theorems that are used for proving triangles congruent. Holt Geometry Proving Constructions Valid Your figure will be a post-construction drawing, including arcs. You will then have to draw line segments connecting points in your figure so that you can create triangles that appear to be congruent. Drawing line segments will be actual steps in your proof. The reason for introducing a new line segment is the theorem that states “through any two points there is exactly one line.” Holt Geometry Proving Constructions Valid Given: BAC, and AD by construction Prove: AD is the angle bisector of BAC. Holt Geometry Proving Constructions Valid Example 1 Continued Statements Reasons 1. BAC , and AD as constructed 1. Given 2. Draw BD and CD. 2. Through any two points there is exactly one line. 3. AC AB ; CD BD 3. Same compass setting used 4. AD AD 4. Reflex. Prop. of 5. ∆ADC ∆ADB 5. SSS Steps 3, 4 6. DAC DAB 6. CPCTE 7. AD is the angle bisector of BAC. 7. angles angle bisector Holt Geometry Proving Constructions Valid Check It Out! Example 1 Given: AB, and CD as constructed Prove: CD is the perpendicular bisector of AB. Holt Geometry Proving Constructions Valid Example 1 Continued Statements Reasons 1. AB, and CD as constructed 1. Given 2. Draw AC, BC, AD, and BD. 2. Through any two points there is exactly one line. 3. AC BC AD BD 3. Same compass setting used 4. CD CD 4. Reflex. Prop. of 5. ∆ADC ∆BDC 5. SSS Steps 3, 4 6. ACD BCD 6. CPCTE 7. CM CM 7. Reflex. Prop. of 8. ∆ACM ∆BCM 8. SAS Steps 2, 5, 6 Holt Geometry Proving Constructions Valid Example 1 Continued Statements 9. AMC BMC Reasons 9. CPCTC 10. AMC and BMC are lin. pr. 10. Def. of linear pair 11. 2 ’s in linear pr = ―> 11. AB DC sides 12. AM BM 12. CPCTE 13. CD bisects AB 13. Def. of bisector 14. CD is the perpendicular bisector of AB. 14. Def. of bisector Holt Geometry