Question

Which of the following statements are true?

• A. $△$ABD $\cong$ $△$BCD
• B. $△$BMC $\cong$ $△$CMD
• C. $△$AMB $\cong$ $△$CMD
• D. $△$AMB $\cong$ $△$BMC

Answer 1

Answer: (A), (C)

The correct options are
A $△$ABD $\cong$ $△$BCD
C $△$AMB $\cong$ $△$CMD

1. $△$ABD $\cong$ $△$BCD
AB = CD (Opposite sides of a parallelogram)
AD = BC (Opposite sides of a parallelogram)
BD = BD (Common side)
Triangles are congruent by SSS congruence criterion.

2. $△$BMC $\cong$ $△$CMD
The triangles are not congruent by SSS / SAS / ASA / RHS congruence criterion.

3. $△$AMB $\cong$ $△$CMD
AM = MC ( Diagonals of a parellogram bisect each other)
BM = MD ( Diagonals of a parellogram bisect each other)
$\angle$ AMD = $\angle$ CMD ( Vertically opposite angles)
By SAS congruence criterion, the triangles are congruent.

4. $△$AMB $\cong$ $△$BMC
The triangles are not congruent by SSS / SAS / ASA / RHS congruence criterion.

Hence, options $△$ABD $\cong$ $△$BCD and $△$AMB $\cong$ $△$CMD are correct.