Step-by-step proof:
Check the attached picture
Statement - Reason
Parallelogram ABCD - Given
Line AC - two points determine a line
AB║CD ; AD║BC - A parallelogram has 2 pairs of opposite sides parallel
∠1 ≅ ∠4 ; ∠2 ≅ ∠3 - line AC acts as a transversal for the parallel sides therefore, they are congruent through alternate interior angles
Line AC ≅ Line AC - Reflexive property
ΔABC ≅ ΔCDA - ASA (Angle - Side - Angle postulate)
∠D = ∠B - CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
m∠1 = m∠4 ; m∠2 = m∠3 - Congruent angles have congruent measures
m∠1 + m∠2 = m∠3 + m∠4 - Addition property of equality
m∠1 + m∠2 = m∠DAB m∠3 + m∠4 = m∠BCD - Angle Addition Postulate
m∠DAB = m∠BCD - Substitution
∠DAB = ∠BCD - Congruent angles have equal measures.
Hope this helps :)
Answer:
See below.
Step-by-step explanation:
The opposite sides of a parallelogram are parallel ( The definition of a parallelogram).
If the Parallelogram is ABCD then
m < A + m < B = 180 degrees ( 2 inside angles of the parallel lines AB and DC.
Similarly :
m < C + m < B = 180 ( The sides AD and BC are parallel).
Subtracting the 2 above equations:
m < A - m < C = 0
So we have proved that
< A is congruent to < C - which are opposite angles.
In the same way we can prove that the other 2 opposite angles B and D are also congruent.
