Quadrilaterals: questions with answers

A quadrilateral must have four straight sides. Side UV is curved, so it prevents XUVW from being a quadrilateral. An inward corner is allowed.

A quadrilateral is a simple closed figure with four straight sides. A corner pointing inward only makes the figure a dented one, so it stays a quadrilateral. So the 2 outlines with four straight sides are all quadrilaterals. The other 2 outlines are not, because one side of each is curved instead of straight. The count is 2.

The diagonals of a rhombus bisect each other, so ES = 2 × 2 = 4 cm. The diagonals need not be equal. Knowing ES alone does not determine the length of AY.

In every rhombus the four sides are equal. That makes the triangles on either side of a diagonal congruent, so the two angles at the crossing point are equal. Those two angles lie on a straight line, so each one is 90 degrees. This reasoning never uses a corner angle, so EASY and GAME both give 90 degrees. Half of 100 and half of 140 are parts of corner angles, which the diagonals do bisect, not the crossing angle.

A parallelogram with a 60-degree angle has each pair of adjacent angles adding up to 180 degrees. Its 60-degree angle means it is not a rectangle, since every rectangle has four right angles. A square is also a rectangle.

Equal, parallel opposite sides make PEAR a parallelogram. A rectangle needs all four angles to be 90 degrees, but the angle at P is 30 degrees. A rhombus needs all four sides equal, and that is not given. Only the name parallelogram must hold.

The strips are the diagonals of the outline. They remain equal and bisect each other as they turn. These properties guarantee a rectangle for every crossing angle greater than 0 degrees and less than 180 degrees. At 40 degrees, isosceles triangles give corner parts of 20 and 70 degrees, adding to 90 degrees. At 0 or 180 degrees, the endpoints lie on one straight line.

A rectangle has equal diagonals that bisect each other. Each diagonal has two equal halves, so all four half-diagonals are equal. OE is half of ES. Since ES equals AY, OE is half as long as AY. The crossing angle is not needed.

A square is also a rectangle, so its diagonals are equal. Therefore, PR = SQ = 6 cm. A square is also a rhombus; a rhombus can have equal diagonals.

Three right angles total 270 degrees, so the fourth would have to be 90 degrees. This contradicts the given angle of 120 degrees. Two right angles are possible: the remaining angle is 360 - 180 - 120 = 60 degrees. Therefore, the greatest number of right angles is 2.

Diagonals that bisect each other make a parallelogram. Equal diagonals make it a rectangle; perpendicular diagonals make it a rhombus. Thus it is a square. A square is also a rectangle, rhombus and parallelogram. With the inclusive definitions in the question, it is also a kite and a trapezium. All six names therefore apply.

If SQ bisects angle S, parallel sides RS and PQ give equal angles PSQ and SQP. These equal angles in triangle SPQ give SP = PQ. Opposite sides of a parallelogram are equal, so PQ = RS. Therefore, RS must be 5 cm. This length makes all four sides equal, forming a rhombus whose diagonals bisect its corner angles.

In a parallelogram, opposite angles are equal and adjacent angles total 180 degrees. The first two angles listed, both 145 degrees, are at adjacent corners, so they should total 180 degrees, not 290. Swapping the second and third angles gives 145, 35, 145 and 35 degrees. Now each pair of adjacent angles totals 180 degrees, and the opposite angles are equal. Swapping the first and second angles changes nothing, because both of them are 145 degrees.

A trapezium needs at least one pair of parallel opposite sides. PEAR has two such pairs, so it is a trapezium. ABCD has no parallel opposite sides, so it is not a trapezium. The number of trapeziums is 1.

Four equal sides make a rhombus. A rhombus is a square only when its angles are right angles. Here the angle at S is 120 degrees, not 90 degrees, so SPQR is a rhombus and not a square. It is not a rectangle either, since a rectangle needs four right angles.