Question

for each figure, choose whether it is a parallelogram or not necessarily a parallelogram.

first figure (top right: quadrilateral abcd with diagonal bc, angles at b and c marked):

• parallelogram
• not necessarily a parallelogram

second figure (top left: quadrilateral ghij with two sides marked equal and two sides with arrows indicating parallel):

• parallelogram
• not necessarily a parallelogram

third figure (bottom right: quadrilateral wxyz with different side markings: two sides with double ticks, two sides with single ticks):

• parallelogram
• not necessarily a parallelogram

fourth figure (bottom left: quadrilateral klmn with a diagonal, angles at m and l marked, and two sides with arrows indicating parallel):

• parallelogram
• not necessarily a parallelogram

Explanation:

Step1: Analyze figure ABCD

In quadrilateral ABCD, $\angle ACB = \angle CBD$ (alternate interior angles), so $AB \parallel CD$. Also, $\angle ABC = \angle BCD$ implies $AB \parallel CD$ and $\triangle ABC \cong \triangle DCB$ by ASA, so $AB=CD$. A quadrilateral with one pair of sides parallel and equal is a parallelogram.

Step2: Analyze figure GHIJ

In quadrilateral GHIJ, $GH \parallel IJ$ (arrow marks) and $GH=IJ$ (tick marks). A quadrilateral with one pair of sides parallel and equal is a parallelogram.

Step3: Analyze figure WXYZ

In quadrilateral WXYZ, $WX=YZ$ and $VW=XY$ but the equal sides are not confirmed to be parallel. This only shows two pairs of equal sides, which could be a kite or other quadrilateral, not necessarily a parallelogram.

Step4: Analyze figure KLMN

In quadrilateral KLMN, $MN \parallel KL$ (arrow marks) but we only know $\angle NML = \angle MLK$ which only confirms one pair of parallel sides, not that the sides are equal or the other pair is parallel. This could be a trapezoid, not necessarily a parallelogram.

Answer:

1. Figure ABCD: Parallelogram
2. Figure GHIJ: Parallelogram
3. Figure WXYZ: Not necessarily a parallelogram
4. Figure KLMN: Not necessarily a parallelogram