1.2.7 · D5Basic Geometry
Question bank — Quadrilaterals — square, rectangle, parallelogram, rhombus, trapezium, kite
Throughout, remember three anchor pictures the parent built:
- A parallelogram is a four-post fence where both pairs of opposite posts run parallel.
- Its diagonals are the two lines joining opposite corners; in a parallelogram they cut each other exactly in half.
- Perpendicular height is the straight-up distance between a base and the side opposite it — not the slanted side length.
True or false — justify
Every square is a rhombus
True — a square has all four sides equal, which is exactly the defining condition of a rhombus; it simply also has right angles, making it a special rhombus.
Every rhombus is a square
False — a rhombus only needs equal sides, so it can lean over with angles like and ; only the upright case with angles is a square.
Every rectangle is a parallelogram
True — a rectangle has both pairs of opposite sides parallel (that is forced by having four right angles), so it inherits every parallelogram property.
A trapezium can also be a parallelogram
False under the "exactly one pair parallel" definition — a parallelogram has two pairs of parallel sides, so it violates the "exactly one" requirement of a trapezium.
The diagonals of every parallelogram are equal in length
False — they always bisect each other, but they are only equal when the parallelogram is a rectangle; a slanted parallelogram has one long and one short diagonal.
The diagonals of a rhombus always meet at right angles
True — a rhombus is a parallelogram (so diagonals bisect) with equal sides, and that extra symmetry forces the crossing to be perpendicular, splitting it into four congruent right triangles.
The diagonals of a kite always meet at right angles
True — the axis of symmetry of a kite runs along one diagonal and reflects the other onto itself, which can only happen if they cross at .
A kite's diagonals always bisect each other
False — only the symmetry diagonal is bisected by the other; the symmetry diagonal itself is cut into two unequal pieces in a general kite.
Opposite angles of a kite are always equal
False — only the one pair of angles between the unequal sides is equal (the pair pierced by the axis of symmetry); the other pair is generally unequal.
If a quadrilateral has all four sides equal it must be a square
False — equal sides give a rhombus; you also need right angles to promote it to a square.
The interior angles of any quadrilateral sum to
True — one diagonal splits it into two triangles ( each), and this works for any four-sided shape, however irregular.
A parallelogram with one right angle is automatically a rectangle
True — consecutive angles are supplementary, so a single forces its neighbour to , and opposite-angle equality then makes all four .
Spot the error
"This parallelogram has base and slant side , so its area is ."
Wrong — area is base times perpendicular height, not slant side. The slant side is longer than the height, so overstates the true area with .
"Both bases of this trapezium are parallel, so it must be a rectangle."
Parallel bases only fix one pair of parallel sides; the legs are slanted and the angles are not , so it stays a trapezium.
"The rhombus has diagonals and , so its side is ."
Wrong tool — you must use Basic Geometry - Pythagorean Theorem on the half-diagonals and , giving side , not an average.
"Kite area needs base times height like a triangle."
No — a kite's diagonals are perpendicular, so the correct shortcut is ; base-times-height would require a height that isn't naturally defined here.
"All parallelograms have equal diagonals because opposite sides are equal."
Equal opposite sides do not force equal diagonals; equal diagonals is the extra condition that specifically identifies a rectangle.
"To prove opposite sides of a parallelogram are equal, just say they're parallel."
Parallelism alone is not enough — two parallel lines can carry segments of any length; you need the ASA congruence of the two triangles (via alternate angles) to force equality.
Why questions
Why does the perpendicular height, not the slant side, decide a parallelogram's area?
Because cutting the slanted triangle off one end and sliding it to the other rebuilds the shape into a rectangle of base and height ; only that vertical distance measures how "tall" the covered region is.
Why is the trapezium area and not just the height times one base?
Joining a second copy upside-down makes a parallelogram of base and height ; halving its area gives one trapezium, so the two bases must be averaged.
Why does splitting a quadrilateral by a diagonal prove the angle sum?
The diagonal builds two triangles whose angles together reassemble every corner angle of the quadrilateral, so their totals add to with none left over or double-counted.
Why do a rhombus's diagonals let you find its side with Pythagoras?
The perpendicular bisecting diagonals cut it into four right triangles whose legs are the two half-diagonals, so each side is the hypotenuse .
Why is every square simultaneously a rectangle AND a rhombus?
It satisfies both extra conditions at once — all angles (rectangle) and all sides equal (rhombus) — so it sits in the overlap of both families.
Why can't a kite (in general) tile-pair into a parallelogram the way a trapezium does?
A trapezium's pairing relies on its one parallel pair; a general kite has no parallel sides to align, so the diagonal-based route is used instead.
Edge cases
Is a shape with four equal sides and one angle a square?
Yes — equal sides make it a rhombus, and a single right angle in a parallelogram forces all four to , upgrading it to a square.
What happens to a parallelogram's area if its slant is pushed until it lies flat (height )?
The area shrinks to ; the shape degenerates into a doubled line segment with no enclosed region.
Can a "kite" have both pairs of adjacent sides equal to the same length?
Yes, and then all four sides are equal — it becomes a rhombus, the special symmetric kite where both diagonals now bisect each other.
If a trapezium's two parallel bases become equal (), what shape emerges?
The legs become parallel too, so it turns into a parallelogram; consistently, the area matches the parallelogram formula .
Does the diagonal-splitting angle proof still work for a "dented" (non-convex) quadrilateral?
You must pick the diagonal that stays inside the shape, but such a diagonal exists and still yields two triangles, so the sum holds.
Is a single straight line ever a valid quadrilateral?
No — collapsing the four posts onto one line leaves zero area and no genuine corners, so it fails the "four-sided polygon" requirement; it is a degenerate limit, not a quadrilateral.
Recall Quick self-test
Name the extra condition that turns a parallelogram into (a) a rectangle, (b) a rhombus, (c) a square. ::: (a) all angles ; (b) all sides equal; (c) both at once. Which two quadrilaterals share the area formula , and why? ::: Rhombus and kite — both have perpendicular diagonals.