SOLUTION: Given: ∠BAD ≈ ∠CAD line AD perpendicular to line BC Prove Triangle ABC is isosceles

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Question 758773: Given: ∠BAD ≈ ∠CAD
line AD perpendicular to line BC
Prove Triangle ABC is isosceles
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
Since given that AD is perpendicular to BC => angle ADC=90degrees and angle ADB=90 degrees,

so angle ADC is congruent to angle ADB

The measure of angle BAD is equal to 180 minus 90 minus the measure of angle ADB,
but since the measure of ADB=+ADC,

the measure of angle BAD is equal to 180 minus 90 minus the measure of angle ADC

However, that is exactly the measure of angle CAD.

Hence angle BAD is congruent to angle CAD.

Since AD+=+AD, triangle ADC is congruent to triangle ADB by SAS.
Then AB+=+AC by CPCT, and finally ABC is isosceles by definition.

or:

Given: AD perpendicular to BC; angle BAD congruent to CAD
Prove: ABC is isosceles

Proof:
statement..........................................reason
1. angle BAD congruent to angle CAD ..........................given
2.AD is perpendicular to BC, => BDA is congruent to the angle CDA ..........all right angles are congruent
3. AD is congruent to AD.......................... reflexive property
4. triangle BAD congruent to triangle CAD ...........by SAS
5. AB is congruent to AC ...................corresponding parts of congruent triangles are congruent
6. triangle ABC is isosceles ....................it has two congruent sides