Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Cardinal direction wikipedia , lookup
Rational trigonometry wikipedia , lookup
Lie sphere geometry wikipedia , lookup
Shape of the universe wikipedia , lookup
Analytic geometry wikipedia , lookup
Cartan connection wikipedia , lookup
Algebraic geometry wikipedia , lookup
History of the compass wikipedia , lookup
Geometrization conjecture wikipedia , lookup
Line (geometry) wikipedia , lookup
History of geometry wikipedia , lookup
4-Ext 4-ExtProving ProvingConstructions ConstructionsValid Valid Lesson Presentation Holt Geometry Holt Geometry 4-Ext Proving Constructions Valid Objective Use congruent triangles to prove constructions valid. Holt Geometry 4-Ext 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 4-Ext 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. You have learned that there exists exactly one midpoint on any line segment. Holt Geometry 4-Ext Proving Constructions Valid Remember! To construct a midpoint, see the construction of a perpendicular bisector on p. 172. Holt Geometry 4-Ext Proving Constructions Valid Example 1: Proving the Construction of a Midpoint Given: Diagram showing the steps in the construction Prove: CD AB Holt Geometry 4-Ext Proving Constructions Valid Example 1 Continued Statements Reasons 1. Draw AC, BC. 1. Through any two points there is exactly one line. 2. AC BC 2. Same compass setting used 3. AD BD 3. Same compass setting used 4. CD CD 4. Reflex. Prop. of 5. ∆ADC ∆BDC 5. SSS Steps 2, 3, 4 6. ADC BDC 6. CPCTC 7. ADC and BDC are rt. s 7. s that form a lin. pair are rt. s. 8. CD AB 8. Def. of Holt Geometry 4-Ext Proving Constructions Valid Check It Out! Example 1 Given: Prove: CD is the perpendicular bisector of AB. Holt Geometry 4-Ext Proving Constructions Valid Check It Out! Example 1 Continued Statements Reasons 1. Draw AC, BC, AD, and BD. 1. Through any two points there is exactly one line. 2. AC BC AD BD 2. Same compass setting used 3. CD CD 3. Reflex. Prop. of 4. ∆ADC ∆BDC 4. SSS Steps 2, 3 5. ADC BDC 5. CPCTC 6. CM CM 6. Reflex. Prop. of 7. ∆ACM and ∆BCM 7. SAS Steps 2, 5, 6 8. AMC BMC 8. CPCTC Holt Geometry 4-Ext Proving Constructions Valid Check It Out! Example 1 Continued Statements Reasons 9. AMC and BMC are rt. s 9. s supp. rt. s 10. AC BC 10. Def. of 11. AM BM 11. CPCTC 12. CD bisects AB 12. Def. of bisector Holt Geometry 4-Ext Proving Constructions Valid Example 2: Proving the Construction of an Angle Given: diagram showing the steps in the construction Prove: D A Holt Geometry 4-Ext Proving Constructions Valid Example 2 Continued 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. Holt Geometry 4-Ext Proving Constructions Valid Check It Out! Example 2 Prove the construction for bisecting an angle. 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. Holt Geometry 4-Ext Proving Constructions Valid Remember! To review the construction of an angle congruent to another angle, see p. 22. Holt Geometry