Proving Constructions

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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
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