Quadrilaterals · Lesson 13 of 13
Chapter Summary and Practice
“A complete revision of quadrilateral definitions, angle rules, diagonal properties, classifications and mixed problem solving.”
• Recover the definitions and methods from all twelve teaching lessons. • Choose the correct side, angle or diagonal rule. • Review constructions and triangle-joining investigations. • Solve mixed problems and justify family relationships.
If you return to quadrilaterals after a few weeks, the names and diagonal rules may feel mixed together. Rebuild them in the order you learned them: start with a four-sided boundary, add right angles or equal sides, and then examine the diagonals. Each extra condition narrows the possible family. This page brings the main definitions, proofs, constructions and distinctions back into one connected revision path.
Read each short review, then pause before the worked examples and try to choose a useful fact yourself. If a rule seems uncertain, look for the triangle or pair of parallel lines that explains it.
4.1 Rectangles and Squares
From Lesson 1: a quadrilateral is a simple closed polygon with four straight sides. Its adjacent sides share a vertex, its opposite sides do not, and its diagonals connect opposite vertices. A rectangle has four right angles; its opposite sides are equal and parallel. Turning a figure does not change these properties.
From Lesson 2: rectangle diagonals are equal and bisect each other. Equal means AC = BD; mutual bisection means AO = OC and BO = OD. The full diagonal test needs both facts. Equal, midpoint-connected strips form a rectangle at any crossing angle strictly between 0° and 180°.
The carpenter’s argument uses isosceles triangles: for one central angle x, a corner is split into 90° − x/2 and x/2, totalling 90°. Congruent opposite small triangles give equal opposite sides. Measurements suggest this pattern; the proof establishes it for every allowed x.
From Lesson 3: a square has four equal sides and four right angles. It is both a rectangle and, later, a rhombus. Its diagonals are equal, bisect each other and meet at 90°. They also bisect vertex angles into 45° halves. Two perpendicular diameters of one circle give this same square construction.
Problem
Equal diagonals of 8 cm bisect each other at 40°. What is formed, and how are the corners split?
4.2 Angles in a Quadrilateral
From Lesson 4: a suitable internal diagonal divides a quadrilateral into two triangles. Their angle totals add to 360°. A missing angle is 360° minus the sum of the other three. Three right angles force the fourth to be a right angle.
All four equal angles must each be 360° ÷ 4 = 90°, which identifies a rectangle but does not force equal sides. In a concave figure, include the reflex interior angle, not the smaller angle outside the boundary.
4.3 More Quadrilaterals with Parallel Opposite Sides
From Lesson 5: a parallelogram has both pairs of opposite sides parallel. Opposite sides are equal, opposite angles are equal and each adjacent angle pair totals 180°. A starting angle x gives the sequence x, 180° − x, x, 180° − x. A diagonal and congruence justify the side equalities.
From Lesson 6: its diagonals bisect each other, but need not be equal or perpendicular. Mutual bisection also identifies a parallelogram in reverse: opposite small triangles are congruent by SAS, giving equal alternate angles and parallel sides.
To construct from adjacent sides and their included angle, draw those two sides first and complete the other two with parallel lines. To construct from diagonals, make their midpoints coincide at the specified crossing angle and join consecutive endpoints.
Problem
A parallelogram has one angle 42° and diagonals 14 cm and 10 cm. Find the other angles and half-diagonals.
4.4 Quadrilaterals with Equal Sidelengths
From Lesson 7: a rhombus has four equal sides. SSS congruence across a diagonal gives alternate-angle equalities, proving that opposite sides are parallel. It is therefore a parallelogram and inherits the opposite-angle and adjacent-angle rules.
A square lies in the overlap of rectangles and rhombuses: it has the right angles of the first and equal sides of the second. A rhombus with one right angle must be a square because its angle relationships force three more right angles.
From Lesson 8: rhombus diagonals bisect each other, bisect the vertex angles and meet at 90°. Equal sides and SSS prove perpendicularity; equal adjacent angles along a straight line must each be 90°. Its diagonals need not be equal. Adding equality makes the rhombus a square.
Problem
A rhombus diagonal makes 20° with a side at A. Find all whole vertex angles.
4.5 Playing with Quadrilaterals
From Lesson 9: equal perpendicular diagonals with a common midpoint give a square. Extending one diagonal equally at both ends preserves perpendicularity and mutual bisection, producing a rhombus with unequal diagonals. Extending only one endpoint loses the midpoint condition.
When joining triangles, the joined side is internal. Two equilateral triangles make a rhombus with 60° and 120° angles. Two 8–8–6 triangles joined along their 6 cm bases make a rhombus; a half-turn join along an 8 cm side makes a parallelogram.
With two 6–9–12 triangles, reflection across their joined 12 cm edge gives adjacent equal side pairs and a kite. A half-turn join gives opposite equal side pairs and a parallelogram. Specify the matching orientation as well as the shared edge.
4.6 Kite and Trapezium
Kite
From Lesson 10: a kite can be labelled ABCD with AB = BC and CD = DA. The diagonal BD joining the equal-pair meeting points bisects angles B and D and perpendicularly bisects AC. The reverse bisection is not required.
To construct a convex kite, draw one diagonal and its perpendicular bisector. Place the other diagonal’s endpoints on opposite sides of the first, with the chosen two distances adding to its given length. An unequal split creates a non-rhombus kite.
With this inclusive definition, a rhombus is a kite and a square is both a kite and a rectangle. The supplied answer key’s contrary statement is inconsistent with its definition; use the definitions to justify the square example.
Trapezium
From Lesson 11: a trapezium has at least one pair of parallel opposite sides. Thus parallelograms also qualify under this convention. For PQ ∥ SR, the same-leg pairs P + S and Q + R each total 180°.
An isosceles trapezium has equal non-parallel legs. The two angles along one base are equal, and each is supplementary to the neighbouring angle along a leg. Dropping perpendiculars creates a rectangle and two congruent right triangles, explaining the base-angle equality.
Problem
An isosceles trapezium has one lower base angle 76°. Find the other three angles.
Figure it Out
From Lesson 12: test a claim by asking whether all its conditions force the conclusion. Equal diagonals without mutual bisection are not the full rectangle test. Perpendicular diagonals without mutual bisection are not the full rhombus test. A counterexample disproves an “always” claim.
Equal opposite sides imply a parallelogram by SSS across a diagonal. Equal opposite angles imply one because the 360° total makes adjacent pairs supplementary. Three right angles, or four equal angles, force a rectangle. Four equal sides plus one right angle force a square.
For the square-inside-a-square task, equal corner triangles give four equal inner sides, and pairs of complementary angles give right inner corners. The same reasoning works with equal clockwise offsets along the outer sides. In overlapping rectangles, identify the right triangle first, then use the remaining right angle to finish the angle calculation.
| Family | Definition | Diagonal facts |
|---|---|---|
| Rectangle | Four right angles | Equal; bisect each other. |
| Square | Four equal sides and four right angles | Equal; mutually bisect at 90°; bisect vertex angles. |
| Parallelogram | Both opposite-side pairs parallel | Bisect each other; no required equality or right-angle crossing. |
| Rhombus | Four equal sides | Mutually bisect at 90°; bisect vertex angles; not necessarily equal. |
| Kite | Two pairs of equal adjacent sides | The special diagonal perpendicularly bisects the other and bisects two vertex angles. |
| Trapezium | At least one parallel opposite-side pair | No universal extra diagonal condition. |
| Isosceles trapezium | A trapezium with equal non-parallel legs | Use equal base angles and supplementary same-leg angles here. |
| Construction target | What to do | What to check |
|---|---|---|
| Rectangle from diagonals | Join equal diagonals at their midpoints. | Crossing is neither 0° nor 180°. |
| Square from one diagonal | Construct its perpendicular bisector; mark half the same length on each side. | Equal, perpendicular and mutually bisecting. |
| Parallelogram from two diagonals | Mark half of each length on opposite rays through O. | Both midpoints coincide; use the specified angle. |
| Rhombus from two diagonals | Use perpendicular lines through their common midpoint. | Half of each diagonal goes on each side. |
| Kite from two diagonals | Put one diagonal on the other’s perpendicular bisector. | For the convex case, its endpoints straddle the first diagonal. |
Problem
A quadrilateral has diagonals 10 cm and 6 cm that bisect each other at 90°. Name it and explain a construction.
Which Quad?
Gameplay
Quiz
Which quadrilateral has diagonals that are equal and bisect each other, but need not be perpendicular?
What is the strongest guaranteed family when a quadrilateral’s diagonals bisect each other?
A rhombus with one right angle is a:
The sum of a quadrilateral's interior angles is:
In an isosceles trapezium, one lower base angle is 70°. What is an upper base angle?
Practice Problems
- Three quadrilateral angles are 95°, 85° and 110°. Find the fourth and explain your method.
- Parallelogram ABCD has ∠A = 42°, AB = 8 cm and BC = 5 cm. Find its other angles and sides.
- Construct a non-rhombus kite with AC = 6 cm and BD = 8 cm, then state which diagonal is bisected.
- An isosceles trapezium has one lower base angle of 76°. Find the other three and explain why it need not be a parallelogram.
- A quadrilateral’s 10 cm and 6 cm diagonals bisect each other at 90°. Classify it, give its half-diagonal lengths, and state what change would make it a square.
1. Known total = 95° + 85° + 110° = 290°. 2. Every simple quadrilateral has angle total 360°. 3. The missing angle is 360° − 290° = 70°.
1. Adjacent angles B and D are 180° − 42° = 138°; C = A = 42°. 2. Opposite sides are equal: CD = 8 cm and DA = 5 cm.
1. Draw AC and its perpendicular bisector through midpoint O. 2. Place B and D on opposite sides with BO = 3 cm and DO = 5 cm, then join the endpoints. 3. BD bisects AC into 3 cm halves; AC does not bisect BD. This is a kite, not a rhombus.
1. The other lower angle is 76°, and both upper angles are 180° − 76° = 104°. 2. The equal non-parallel legs do not supply a second pair of parallel sides. 3. A trapezium with different base lengths satisfies these facts without being a parallelogram.
1. Perpendicular mutual bisection makes a rhombus. Its unequal diagonals show it is not a square. 2. Its half-diagonals are 5 cm, 5 cm, 3 cm and 3 cm. 3. Make both full diagonals equal while keeping the common midpoint and right-angle crossing. Then all square diagonal conditions hold.
Key Takeaways
• Begin with definitions: side equality, right angles and parallel pairs describe different conditions. • Every simple quadrilateral has interior-angle sum 360°; parallelogram and rhombus adjacent angles total 180°. • Rectangle diagonals are equal and bisecting; rhombus diagonals are perpendicular and bisecting; squares have all three conditions. • A kite has two equal adjacent-side pairs and generally only one-way diagonal bisection. • Use same-leg supplementary angles in trapeziums and equal same-base angles in the isosceles case. • Construction and joining activities are justified by the properties they preserve. • Use congruence, parallel-line facts and counterexamples to explain why a conclusion follows.
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Which Quadrilateral Is It?
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