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4-Ext

Proving Constructions Valid. 4-Ext. Lesson Presentation. Holt Geometry. Objective. Use congruent triangles to prove constructions valid. When performing a compass and straight edge construction, the compass setting remains the same width until you change it.

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4-Ext

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  1. Proving Constructions Valid 4-Ext Lesson Presentation Holt Geometry

  2. Objective Use congruent triangles to prove constructions valid.

  3. 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.

  4. 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. You have learned that there exists exactly one midpoint on any line segment.

  5. Remember! To construct a midpoint, see the construction of a perpendicular bisector on p. 172.

  6. Given: Diagram showing the steps in the construction Prove:CD AB Example 1: Proving the Construction of a Midpoint

  7. 1. Draw AC, BC. 2. AC BC 3. AD BD 4. CD CD 8. CD AB Example 1 Continued Statements Reasons 1. Through any two points there is exactly one line. 2. Same compass setting used 3. Same compass setting used 4. Reflex. Prop. of  5. ∆ADC ∆BDC 5. SSS Steps 2, 3, 4 6. ADC BDC 6. CPCTC 7. ADCand BDC are rt. s 7.  s that form a lin. pair are rt. s. 8. Def. of 

  8. Given: Prove: CD is the perpendicular bisector of AB. Check It Out! Example 1

  9. 1. Draw AC, BC, AD, and BD. 2. AC BC  AD  BD 3. CD CD 6. CM CM Check It Out! Example 1 Continued Statements Reasons 1. Through any two points there is exactly one line. 2. Same compass setting used 3. Reflex. Prop. of  4. ∆ADC ∆BDC 4. SSS Steps2, 3 5. ADC BDC 5. CPCTC 6. Reflex. Prop. of  7. ∆ACMand ∆BCM 7. SAS Steps 2, 5, 6 8. AMC BMC 8. CPCTC

  10. 10. AC BC 11. AM BM 12. CDbisects AB Check It Out! Example 1 Continued Statements Reasons 9. AMC and BMC are rt. s 9.  s supp.  rt. s 10. Def. of  11. CPCTC 12. Def. of bisector

  11. Example 2: Proving the Construction of an Angle Given: diagram showing the steps in the construction Prove: DA

  12. Since there is a straight line through any two points, you can draw BC and EF. The same compass setting was used to construct AC, AB, DF, and DE, so ACABDFDE. The same compass setting was used to construct BC and EF, so BCEF. Therefore ABCDEF by SSS, and DA by CPCTC. Example 2 Continued

  13. Draw BD and CD (through any two points. there is exactly one line). Since the same compass setting was used, ABAC and BDCD. ADAD by the Reflexive Property of Congruence. So ABDACD by SSS, and BADCAD by CPCTC. Therefore AD bisects BAC by the definition of an angle bisector. Check It Out! Example 2 Prove the construction for bisecting an angle.

  14. Remember! To review the construction of an angle congruent to another angle, see p. 22.

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