ABCD is a quadrilateral in which AB||DC and AD = BC. Prove that `angleA=angleB and angleC= angleD`.

Explanation

Draw CE \\\\ AD from point C with meets AB produced at E.
`{:(Now",",AB,"||",DC),(implies,AE,"||",DC):}`
In `squareAECD.`
`AE"||"DCand AD"||"EC`
`thereforesquareAECD"is a parallelogram."" "(because"pairof opposite sides are parallel")`
`implies" "AD=CE" "("opposite sides of a parallelogram")`
`and" "angleA+angleE=180^(@)" "("sumof co-interio angles")`
`But" "AD=BC" "(given)`
`therefore" "BC=CE`
`implies" "angle1=angle2" "("angles opposite to equal sides are equal")`
`Now," "angle1+angle3=180^(@)" "("sum of co-interior angles")`
`Also," "angle2+angle4=180^(@)" "(L.P.A.)`
`therefore" "angle1+angle3=angle2+angle4`
`implies" "angle3=angle4" "(because angle1=angle2)`