4-gons (Quadrilaterals)

The One Thing

Quadrilateral problems are solved by combining precise definitions with parallel-line angle relationships, congruent triangles, the midpoint theorem and central symmetry. In particular, the defining properties of a parallelogram generate criteria that can be used both to prove its properties and to establish that a quadrilateral is a parallelogram.

Definitions and Results

• Quadrilateral: A polygon with four sides, four vertices and four interior angles. The sum of its interior angles is $360^{\circ }$.
• Parallelogram: A quadrilateral in which both pairs of opposite sides are parallel.
• Opposite sides: Sides of a quadrilateral that do not share a common vertex.
• Opposite angles: Angles whose vertices are opposite each other in a quadrilateral.
• Adjacent angles: Two angles next to each other that share a side and a vertex.
• Diagonal: A line segment joining two non-adjacent vertices of a polygon. A quadrilateral has two diagonals.
• Transversal: A line that intersects two or more other lines at distinct points and establishes relationships between the angles formed.
• Midpoint: The point that divides a line segment into two equal parts.
• Midpoint theorem: If $D$ and $E$ are the midpoints of $AB$ and $AC$ in triangle $ABC$, then

$$DE\parallel BC\text{and}DE=\frac{1}{2}BC.$$

• Converse of the midpoint theorem: A line drawn through the midpoint of one side of a triangle and parallel to another side bisects the third side.
• Central symmetry: A figure has central symmetry if it matches itself after a rotation of $180^{\circ }$ about a fixed point called the centre of symmetry.
• Centre of symmetry of a parallelogram: The point where the diagonals of a parallelogram intersect. Each diagonal bisects the other at this point.
• Congruent triangles: Triangles having the same shape and size. Congruence can establish equal sides or equal angles in quadrilateral proofs.
• Cyclic order of vertices: The order in which vertices are connected around a quadrilateral, such as $A$-$B$-$C$-$D$.

• Properties of a parallelogram:

- Opposite sides are equal: $$AB=CD,BC=AD.$$ - Opposite angles are equal: $$\angle A=\angle C,\angle B=\angle D.$$ - Adjacent angles are supplementary: $$\angle A+\angle B=180^{\circ }.$$ - The diagonals bisect each other. If diagonals $AC$ and $BD$ meet at $O$, then $$AO=OC,BO=OD.$$ - Each diagonal divides the parallelogram into two congruent triangles.

• Criteria for proving that a quadrilateral is a parallelogram: A quadrilateral is a parallelogram if any one of the following is established:

- both pairs of opposite sides are parallel; - both pairs of opposite sides are equal; - both pairs of opposite angles are equal; - one pair of opposite sides is both equal and parallel; - the diagonals bisect each other.

• Parallel-line angle facts: If a pair of lines is parallel, alternate interior angles are equal and co-interior angles are supplementary.

• Midpoint theorem applications:

- If $DE=6\text{ cm}$, then $BC=12\text{ cm}$. - If $BC=15\text{ cm}$, then $DE=7.5\text{ cm}$. - The converse midpoint theorem can be used to prove that a point is the midpoint of a side.

• Central symmetry in a parallelogram: The diagonals intersect at the centre of symmetry. A $180^{\circ }$ rotation about this point maps each vertex to the opposite vertex.

• Special quadrilaterals:

- A rectangle, rhombus and square are special types of parallelograms and therefore possess all basic parallelogram properties. - A rectangle has four right angles. - A rhombus has four equal sides. - A square has four equal sides and four right angles. - A kite generally has one pair of equal opposite angles and diagonals with special properties, but it is not necessarily a parallelogram. - A trapezium is a quadrilateral with at least one pair of parallel sides; it is not necessarily a parallelogram.

Worked Methods

Proving properties of a parallelogram using a diagonal

• Consider a parallelogram and draw one diagonal.
• Use the parallel opposite sides to identify equal alternate interior angles.
• Note that the diagonal is common to the two resulting triangles.
• Apply triangle congruence.
• Use the congruent triangles to establish equal opposite sides, equal opposite angles or other required properties.
• Since each diagonal divides the parallelogram into two congruent triangles, the diagonal proves several parallelogram results simultaneously.

Proving that a quadrilateral is a parallelogram

• Identify the information given about the quadrilateral in its cyclic order, such as $A$-$B$-$C$-$D$.
• Determine whether one valid parallelogram criterion can be established.
• Prove one of the following:

- both pairs of opposite sides are parallel; - both pairs of opposite sides are equal; - both pairs of opposite angles are equal; - one pair of opposite sides is equal and parallel; - the diagonals bisect each other.

• Conclude that the quadrilateral is a parallelogram.
• Justify each step using definitions, parallel-line angle rules, known theorems or congruence rather than relying only on the diagram.

Using parallel lines and a transversal

• Identify the two parallel lines and the transversal crossing them.
• Locate the relevant pair of angles.
• Use equal alternate interior angles where appropriate.
• Use supplementary co-interior angles where appropriate.
• Apply the resulting angle relationships to prove equal angles, supplementary adjacent angles or triangle congruence.

Applying the midpoint theorem

• Identify triangle $ABC$.
• Confirm that $D$ is the midpoint of $AB$ and $E$ is the midpoint of $AC$.
• Apply the theorem:

$$DE\parallel BC,DE=\frac{1}{2}BC.$$

• If $DE=6\text{ cm}$, calculate:

$$BC=2(DE)=2(6)=12\text{ cm}.$$

• If $BC=15\text{ cm}$, calculate:

$$DE=\frac{1}{2}(15)=7.5\text{ cm}.$$

Applying the converse midpoint theorem

• Identify a triangle and the midpoint of one of its sides.
• Draw or identify a line through that midpoint.
• Establish that the line is parallel to another side of the triangle.
• Apply the converse midpoint theorem.
• Conclude that the line bisects the third side, so the relevant point is its midpoint.

Using central symmetry in a parallelogram

• Identify the intersection $O$ of the diagonals.
• Use the diagonal-bisection property:

$$AO=OC,BO=OD.$$

• Observe that each pair of opposite vertices lies on one straight diagonal and is equally distant from $O$.
• Conclude that a $180^{\circ }$ rotation about $O$ maps each vertex to the opposite vertex.
• Therefore, $O$ is the centre of symmetry of the parallelogram.

Where It Goes Wrong

• Confusing opposite sides with adjacent sides: opposite sides do not share a common vertex, whereas adjacent angles share both a side and a vertex.
• Forgetting that the interior angles of any quadrilateral sum to $360^{\circ }$.
• Using a parallelogram property without checking its condition, such as assuming that a quadrilateral is a parallelogram merely because it has one pair of parallel sides, as in a trapezium.
• Omitting the parallel-line condition when applying alternate interior angles or co-interior angle rules.
• Applying the midpoint theorem without establishing that both points are midpoints of two sides of the same triangle.
• Treating a kite as necessarily being a parallelogram, despite its generally having only one pair of equal opposite angles and special diagonal properties.
• Relying only on the appearance of a diagram instead of giving justified statements based on definitions, theorems, angle rules and congruence.

What Gets Asked

• Define a quadrilateral, parallelogram, diagonal, midpoint, transversal, central symmetry or another named term.
• Calculate an unknown interior angle using the $360^{\circ }$ quadrilateral angle sum or supplementary adjacent angles.
• Prove that opposite sides or opposite angles of a parallelogram are equal.
• Prove that the diagonals of a parallelogram bisect each other.
• Use a diagonal and congruent triangles to establish parallelogram properties.
• Determine whether a quadrilateral is a parallelogram using one valid criterion.
• Apply alternate interior angles and co-interior angles formed by parallel lines and a transversal.
• Apply the midpoint theorem, including the cases $DE=6\text{ cm}\Rightarrow BC=12\text{ cm}$ and $BC=15\text{ cm}\Rightarrow DE=7.5\text{ cm}$.
• Use the converse midpoint theorem to prove that a point is a midpoint.
• Identify the centre of symmetry of a parallelogram and describe the effect of a $180^{\circ }$ rotation.
• Distinguish among a parallelogram, rectangle, rhombus, square, kite and trapezium according to their defining properties.