Question

State true or false. A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.

Answer 1

Hint: Use the property of the triangles to solve for the problem involving figures with straight sides. Also some of the properties of polygons like parallelogram , square , rectangle, rhombus etc.are the same.
Quadrilateral is a flat shape having 4 sides.

Parallelogram definition : parallelogram is flat shape having four straight sides , with the opposite sides are parallel and equal to each other.

Complete step-by-step answer:
Let ABCD be a quadrilateral in which

AB = DC & \[AB\parallel DC\]
DA = CB & \[DA\parallel CB\]
Then \[AB \cong DC\]& \[DA \cong CB\]
Construct a segment DB , constructing a diagonal \[DB \cong DB\] (Reflexive property)
\[\Delta ABD \cong \Delta DBC\] by s-s-s (side-side-side) postulate
s-s-s postulate – if three sides of one triangle are congruent to the three sides of another triangle , then the two triangles are congruent.
\[{\angle ADB \cong \angle DBC}\], by CPCTC (corresponding parts of a congruent triangle are congruent )
Also , \[{\angle ABD \cong \angle BDC}\]
Means \[AB\parallel DC\]& \[DA\parallel CB\] ( converse of the Alternate Interior Angles )

The given statement is true.

Note: Few properties
Properties of parallelogram:-
(i) opposite sides are congruent
(ii) opposite angles are congruent
(iii) if one angle is right, then all the angles are right.
(iv) The diagonals of a parallelogram bisect each other.
(v) Each diagonal of a parallelogram separates it into two congruent triangles .
Properties of square:-
(i). All sides of the square are equal.
(ii). All four angles of the square are equal i.g. Right angle
(iii). Opposite sides are equal and parallel in length.
(iv). Diagonal bisects its angles.
Properties of rhombus :
(i).All sides are equal.
(ii). The opposite sides are parallel.
(iii). Opposite angles are equal.