Question 237403: Prove triangle ABC is isoceles
Given: angle BAD is congruent to angle CAD
line segment AD is perpendicular to line segment BC
The answers that I originally gave were:
statement: BD is congruent to CD
reason: definition of a bisector
statement: angle BAD is congruent angle CAD
reason: ray AD is the bisector of angle BAC (given)
statement: AD is equal to AD
reason: reflexive property of equality
statement: triangle ABD is congruent to triangle ACD
reason: SAS of congruency of triangles
statement: angle ABD is congruent to angle ACD
reason: CACT
grading comments: While the effort is appreciated, the proof contains several errors. revise. Found 2 solutions by Edwin McCravy, solver91311:Answer by Edwin McCravy(6936) (Show Source):
statement: ∠BAD ≅ ∠CAD
reason: given
statement: AD ≅ AD
reason: a line segment is congruent to itself
statement: m∠ADB = 90°
reason: given AD is perpendicular to BC
statement: m∠ADC = 90°
reason: given AD is perpendicular to BC
statement: ∠ADB ≅ ∠ADC
reason: both angles have measure 90°, i.e., they are right angles.
statement: ∆ABD ≅ ∆ACD
reason: ASA of congruency of triangles
statement: AB ≅ AC
reason: CSCT. or corresponding sides of congruent triangles.
statement: ∆ABC is isosceles
reason: ∆ABC has two sides congruent.
Edwin
Your first step is in error. Where did you get the information that anything bisected anything else? Just because in the picture AD looks like a bisector of BC, and the fact that AD will ultimately turn out to be a bisector of BC, unless you are given the fact that it is a bisector or can prove from the information that you were given that it is a bisector, you cannot assume that it is.
"Objection! Assumes facts not in evidence!"
"Sustained."
Your second step: Again with the bisector thing. No, BAD congruent to CAD because it is given that BAD congruent to CAD.
Your third step is properly constructed and justified.
Your fourth step is the crux of your problem. Instead of SAS, you should be using ASA.
Just follow the bouncing ball:
1. BAD congruent to CAD -- Given
2. AD congruent to AD -- Reflexive property of equality
3. BDA is a right angle -- Definition of perpendicular
4. CDA is a right angle -- Definition of perpendicular
5. BDA congruent to CDA -- Transitive property of equality
6. Triangle BDA congruent to Triangle CDA -- ASA
7. AB congruent to AC -- Corresponding sides of congruent triangles
8. Triangle ABC is isosceles -- Definition of isosceles
