Quadrilaterals · Lesson 12 of 13
Which Quadrilateral Is It?
“Classify quadrilaterals using overlapping properties, set relationships and counterexamples.”
• Classify figures from sufficient combinations of properties. • Use counterexamples to test claims. • Connect overlapping quadrilateral families. • Solve mixed construction, angle and congruence problems.
You now know several names for four-sided figures, and many of their properties overlap. A square has equal diagonals, but so does a non-square rectangle. A rhombus has perpendicular diagonals, but some kites do too. The challenge is to decide what the given information actually guarantees. Start by listing the facts, then ask whether a different kind of figure could satisfy them as well.
This lesson brings the earlier ideas together. You will use definitions, triangle congruence, parallel-line reasoning and counterexamples, rather than choosing a name only from the way a diagram looks.
Figure it Out
A property can be true for every member of a family without identifying that family uniquely. Equal diagonals are a rectangle property, but the full diagonal test also requires mutual bisection. A sufficient condition includes enough information to force the conclusion.
To disprove a statement about every figure, one valid counterexample is enough. To prove it, explain why the given conditions force the result in every case. A few successful measurements do not complete that proof.
| Given information | Strongest guaranteed family | Why |
|---|---|---|
| Four right angles | Rectangle | This is the angle definition; sides need not all be equal. |
| Four equal sides | Rhombus | This is the side definition; angles need not be right angles. |
| Diagonals bisect each other | Parallelogram | Opposite small triangles are congruent by SAS. |
| Equal diagonals that bisect each other | Rectangle | The isosceles-triangle argument gives four right angles. |
| Perpendicular diagonals that bisect each other | Rhombus | Four congruent small right triangles give equal sides. |
| Equal, perpendicular, mutually bisecting diagonals | Square | Both rectangle and rhombus conditions hold. |
In a Venn diagram, put rectangles and rhombuses inside parallelograms, with their overlap labelled squares. Under the kite definition used here, rhombuses lie inside kites too. A kite that is also a parallelogram must have all four sides equal: adjacent equalities from the kite and opposite equalities from the parallelogram connect all the sides.
A kite can also be a rectangle: its adjacent-side equalities, together with the rectangle’s opposite-side equalities, force four equal sides. With four right angles as well, the figure is a square.
The supplied answer key says no quadrilateral can be both a kite and a rectangle. That conflicts with the stated kite definition AB = BC and CD = DA: a square satisfies it and is also a rectangle. We follow the stated definition, so the answer is yes—a square. This source inconsistency is flagged rather than silently copied.
Problem
Must equal, bisecting diagonals give a square?
Problem
A quadrilateral has AB = CD and BC = DA. What must it be?
Problem
Both pairs of opposite angles are equal. Must the quadrilateral be a parallelogram?
Problem
PAIR and RODS are rectangles. O lies on AI and ∠ORI = 30°. Find ∠IOD.
Problem
In square CASE, U, X, W and V are the midpoints of CA, AS, SE and EC. What is the boundary U–X–W–V?
Problem
Does an inward-pointing quadrilateral still have angle sum 360°?
| Claim | Verdict | Reason |
|---|---|---|
| Equal, bisecting diagonals force a square. | False | A non-square rectangle is a counterexample. |
| Three right angles force a rectangle. | True | The fourth angle is 360° − 270° = 90°. |
| Mutually bisecting diagonals force a parallelogram. | True | SAS gives equal alternate angles and parallel opposite sides. |
| Perpendicular diagonals force a rhombus. | False | A kite can have one diagonal bisected but not the other. |
| Both pairs of opposite angles equal force a parallelogram. | True | The angle sum makes adjacent pairs supplementary. |
| All four angles equal force a rectangle. | True | Each angle is 360° ÷ 4 = 90°. |
| Every isosceles trapezium is a parallelogram. | False | Equal non-parallel legs do not provide a second parallel pair. |
Quiz
Which quadrilateral is both a rectangle and a rhombus?
Equal, bisecting diagonals guarantee a:
Perpendicular diagonals alone guarantee a rhombus.
If both pairs of opposite angles are equal, the quadrilateral is a:
Four equal sides plus one 90° angle force a:
Practice Problems
- A quadrilateral has equal diagonals that bisect each other. Give the strongest guaranteed name.
- Its diagonals bisect each other at 90° and have unequal lengths. Classify it.
- A quadrilateral has four equal sides and equal diagonals. Classify it.
- Give a counterexample to “perpendicular diagonals imply rhombus.”
- Which figures belong to both the kite and rectangle families under our definitions? Explain.
1. Equality and mutual bisection are the rectangle diagonal test. 2. The figure is a rectangle. It could be a square, but a square is not guaranteed.
1. Perpendicular mutual bisection gives four equal sides, so it is a rhombus. 2. Unequal diagonals exclude a square, so it is a non-square rhombus.
1. Four equal sides make it a rhombus, so its diagonals bisect each other at 90°. 2. The added equal-diagonal condition now supplies all three square conditions. 3. It is a square.
1. Construct AC with AO = OC = 3 cm and perpendicular BD with BO = 3 cm, DO = 5 cm. 2. The boundary is a kite and the diagonals are perpendicular. 3. But BO ≠ DO, contradicting the mutual bisection required in a rhombus. Thus this kite is a counterexample.
1. In a kite, AB = BC and CD = DA. In a rectangle, AB = CD and BC = DA. 2. Together these imply all four sides are equal; the rectangle also supplies four right angles. 3. The common figures are squares, despite the conflicting answer-key statement.
Key Takeaways
• Use all the given conditions before choosing the strongest guaranteed name. • A property of a family is not always a sufficient test for that family. • One valid counterexample disproves an “always” claim. • Squares are simultaneously rectangles, rhombuses, parallelograms and kites. • Congruence and parallel-line converses justify classification tests. • Concave quadrilaterals retain the 360° total when the reflex interior angle is counted.