a) Definition of bisect
b) Vertcial angles are congruent
c) SAS congruence
d) Converse of Alternate interior angles theorem
e) CPCTC
f) Definition of parallelogram
Explanation:
AC and BD are diagonals that intersect each other. Dividing each diagoanl into equal halves
BP = PD and AP = PC
Reason: Definition of bisect
∠APB and ∠CPD (vertical angles)
∠APD and ∠CPB (vertical angles)
∠APB = ∠CPD, ∠APD = ∠CPB
Reason: Vertcial angles are congruent
ΔAPB = ΔCPD, ΔBPC = ΔDPA
In ΔAPB, we have two sides and one included angle congruent to two sides and one included angle in ΔCPD
Two sides and one included angle of ΔBPC congruent to two sides and one included angle in ΔDPA
Reason: SAS (side-angle-side) congruence
∠ABD = ∠CDP, ∠ADP = ∠CBP
∠ABD and ∠CDP are alternate angles
∠ADP and ∠CBP are alternate angles
Reason: Converse of Alternate interior angles theorem
AB || CD, AD || BC
opposite sides of a parallelogram are parallel
correponding parts of congruent triangles are congruent
Reason: CPCTC
Therefore ABCD is a parallologram
Reason: Definition of parallelogram
