Given: Quadrilateral ABCD; Angle A is conguent to Angle C; Angle B is congruent to Angle D
Prove: Quadrilateral ABCD is a Parallelogram.
D___________________ C
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A /# &/B
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Given: Quadrilateral ABCD
1. mÐA = mÐC
mÐB = mÐD They are measures of congruent angles.
mÐA + mÐB + mÐC + mÐD = 360° The sum of the measures of the interior
angles of a polygon with n sides has sum
of angles given by the expression
°. Quadrilaterals have 4
sides, which means that n=4. Therefore
the sum of the interior angles of a
quadrilateral is 
° which
equals 
° or 
°
mÐA + mÐB + mÐA + mÐB = 360° Equals may be substituted for equals. Here
we are given that mÐA = mÐC and that
mÐB = mÐD, and we made those substitutions
of equals for equals.
2mÐA + 2mÐB = 360° mÐA + mÐA = 2AÐ and mÐB + mÐB = 2ÐB and
substituting equals for equals.
mÐA + mÐB = 180° Dividing the equation through by 2.
(ÐA and ÐB are supplementary)
AD||BC When a transversal (AB) cuts two lines
(AD and BC), and the angles on the same
side of the transversal (ÐA and ÐB) are
suplementary, the lines are parallel.
m/C + m/B = 180° Substituting mÐC for mÐA in equation 2 steps back.
(ÐC and ÐB are supplementary)
AB||CD When a transversal (BC) cuts two lines
(AB and CD), and the angles on the same
side of the transversal (ÐC and ÐB) are
suplementary, the lines are parallel.
quadrilateral ABCD is a parallellogram.
Because both pairs of opposite sides are
parallel and that completes the definition
of a parallelogram.
Edwin
