Properties of Quadrilaterals

Establish properties of quadrilaterals using congruent triangles and angle properties, and solve related problems explaining reasoning.

Every property of a quadrilateral can be proved by splitting it into triangles. The diagonal is almost always the first move.

Builds onYr 8 · Congruence and Similarity

Method

Draw a diagonal first

It turns one quadrilateral into two triangles you can work with.

Angle sum is 360

(4-2) \times 180 = 360

Two triangles, so twice 180.

Give a reason for every line

\text{statement} + \text{reason}

The reason is the marked half.

Names overlap

A square is also a rectangle, a rhombus and a parallelogram.

e.g.

\text{Prove that opposite sides of a parallelogram are equal.}

Draw a diagonal to split it into two triangles. $\text{creates } \triangle ABC \text{ and } \triangle CDA$
The diagonal is shared by both triangles. $AC = CA \quad [\text{common side}]$
Use the parallel sides to find equal angles. $\angle BAC = \angle DCA \quad [\text{alternate angles}]$
Find a second pair the same way. $\angle BCA = \angle DAC \quad [\text{alternate angles}]$
Name the congruence test. $\triangle ABC \equiv \triangle CDA \quad [ASA]$
Conclude, since congruent triangles have equal sides. $\therefore AB = CD \text{ and } BC = DA$

State the reason beside every line.

\text{Three angles of a quadrilateral are } 80, 95, 110. \text{ Find the fourth.}

Answer

75^\circ \ [\text{angle sum} = 360]

\text{Is every square a rhombus?}

Answer

\text{Yes — all four sides are equal.}

\text{Is every rhombus a square?}

Answer

\text{No — the angles need not be } 90^\circ.

\text{Why do the diagonals of a rectangle have equal length?}

Answer

\text{They form congruent triangles } [SAS].

\text{What is the first move in most quadrilateral proofs?}

Answer

\text{Draw a diagonal.}

\text{Why is the angle sum } 360?

Answer

\text{Two triangles, each } 180.