Given: a quadrilateral ABCD with AB = BC = 3.
To prove:
Quadrilateral ABCD is a parallelogram.
AC + BD = AD
Proof:
1)
Since AB = BC, angle ABC = angle BCA (Isosceles triangle property).
Therefore, angle ABC = angle BCD (vertically opposite angles).
Since angle ABC = angle BCD, quadrilateral ABCD has opposite sides that are parallel (consecutive interior angles theorem).
Therefore, quadrilateral ABCD is a parallelogram.
In parallelogram ABCD, opposite sides are equal in length.
So, AC = BD and AD = BC.
Therefore, AC + BD = BD + BD = 2BD.
Also, AD = BC = 3.
Therefore, AC + BD = 2BD = 3.
