Write a two column proof, a paragraph proof?

Question

Write a two column proof, a paragraph proof?

Given: BD⊥AC , D is the midpoint of AC.

Prove: BC ≅ BA

Answer 1

two column version of the proof:

 

 

            Statement                                          Reason

 

1) D is the midpoint of AC                                Given

 

2) AD = DC                                                  Definition of midpoint

 

3) BD = BD                                                   Reflexive

 

4) BD is perpendicular to AC                          Given

 

5) DBA and BDC are right angles                   definition of perpendicular

 

6) DBA = BDC                                               right angles are congruent

 

7)  Triangles ADB and CDB are                       SAS, LL, lA

     congruent right triangles

 

8) BC = AC                                                   CPCTC

 

 

 

Paragraph version of the same proof:

 

Since BD is perpendicular to AC, angles BDA are DBC are

congruent right angles.

 

Since D is the midpoint of AC, then AD = DC by definition

of midpoint.

 

DB is equal to itself by reflexive property, but is also shared

by right triangles by BDA and BDC.

 

Therefore, right triangles BDA and BDC are congruent

right triangles as two legs have been shown to be congruent.

(SAS applies where the angle in between is a right triangle,

as do LL and LA for right triangles)

 

By CPCTC , BC and BA are congruent.