prove that parallelogam ABCD is equilateral if the diagonal AC bisects angle A

Question

i need help with this proof and yall are smart so please help

Answer 1

Heather, let's draw this and label the angles on either side of the diagonal at A 1 and 2 and the angles at C 3 and 4, so that 2 and 3 are on the same side of the diagonal.

 

Then...

 

We know that Angle 1 is congruent to Angle 2, by the definition of bisection.

 

Angle 1 is congruent to Angle 3 because they are alternate interior angles.

 

Since Angle 1 is congruent to both 2 and 3, then Angle 2 is congruent to Angle 3.

 

Since Angles 2 and 3 are congruent and they are base angles of Triangle ACD, then Triangle ACD is isosceles.

 

AC and AD are congruent because of the definition of an isosceles triange.

 

AD and BC are congruent because they are opposite sides of a parallelogram.

 

CD and AB are congruent for the same reason.

 

AC, AD, CD, and AB are congruent, from the three steps before.

 

Therefore, ABCD is equilateral.