Given: ABCD is an isosceles trapezoid, GP and EQ are perpendicular to AB and DC, P and Q are midpoints of AB and DC

Prove: Triangle AGP is congruent to triangle DEQ

Perpendicular lines form right angles
Substitution
Algebra
Definition
The base angles of an isosceles trapezoid are equal
ASA (Angle-Side-Angle)
Segment addition theorem
Substitution
Substitution
Definition
BA = CD
AP = BP and DQ = CQ
BA = BP + AP and CD = CQ + DQ
BA = 2(PA) and CD = 2(QD)
PA = (1/2)BA and QD = (1/2)CD
PA = (1/2)BA and QD = (1/2)BA
PA = QD
Angle APG = Angle EQD = 90
Angle A = Angle D
Triangle AGP is congruent to triangle DEQ