Practice Maths

L27 — Applications of Deductive Geometry

Key Terms

Parallelogram
A quadrilateral with both pairs of opposite sides parallel and equal; diagonals bisect each other.
Rectangle
A parallelogram with all angles 90° and equal diagonals.
Rhombus
A parallelogram with all sides equal; diagonals bisect each other at right angles.
Square
Both a rectangle and a rhombus — all sides equal, all angles 90°, diagonals equal and perpendicular.
Proving a quadrilateral type
Showing the minimum sufficient conditions (e.g. diagonals bisect each other ⇒ parallelogram).
Coordinate geometry
Using coordinates, distance, midpoint, and slope formulas to prove geometric properties algebraically.

Quadrilateral properties

ShapeKey properties
ParallelogramOpposite sides parallel & equal; opposite angles equal; diagonals bisect each other
RectangleParallelogram + all angles 90°; diagonals equal
RhombusParallelogram + all sides equal; diagonals bisect at 90°
SquareRectangle + rhombus; diagonals equal, perpendicular, bisect angles
TrapeziumOne pair of parallel sides
KiteTwo pairs of adjacent equal sides; one diagonal bisects the other at 90°

Proving quadrilateral types

To prove it is a…Show…
ParallelogramBoth pairs opposite sides parallel, OR both pairs opposite sides equal, OR diagonals bisect each other, OR one pair opposite sides both parallel and equal
RectangleParallelogram with one right angle (or equal diagonals)
RhombusParallelogram with adjacent sides equal (or diagonals bisect at 90°)
SquareRectangle with adjacent sides equal
Quadrilateral Parallelogram Rectangle Rhombus Square
Quadrilateral hierarchy — each shape inherits properties of the one above
Hot Tip: To prove a quadrilateral is a specific type, show only the minimum sufficient conditions — e.g. for a parallelogram, showing the diagonals bisect each other is enough. Don't prove more than necessary.

Worked Example 1 — Proving a quadrilateral is a parallelogram

ABCD has AB = CD = 8 cm and AB ∥ CD. Prove ABCD is a parallelogram.

Step 1: AB ∥ CD (given).

Step 2: AB = CD (given).

Step 3: One pair of opposite sides is both parallel and equal. ∴ ABCD is a parallelogram. □

Worked Example 2 — Using coordinates to verify geometry

A(0,0), B(4,0), C(6,3), D(2,3). Show ABCD is a parallelogram.

Method 1 (Midpoints of diagonals): Diagonal AC: midpoint = (3, 1.5). Diagonal BD: midpoint = (3, 1.5). Diagonals share a midpoint ⇒ they bisect each other ⇒ ABCD is a parallelogram. □

Method 2 (Slope): Slope AB = 0/4 = 0. Slope DC = 0/4 = 0. AB ∥ DC. Slope AD = 3/2. Slope BC = 3/2. AD ∥ BC. Both pairs of opposite sides parallel ⇒ parallelogram. □

Worked Example 3 — Proving a rhombus

PQRS is a parallelogram with PQ = QR. Prove PQRS is a rhombus.

Step 1: PQRS is a parallelogram (given), so PQ = SR and QR = PS (opposite sides equal).

Step 2: PQ = QR (given). ∴ PQ = QR = RS = SP. All four sides equal.

Step 3: ∴ PQRS is a rhombus. □

Worked Example 4 — Applying similarity to find unknown lengths

In ▵ABC, D on AB and E on BC such that DE ∥ AC. BD = 5, DA = 3, BE = 4. Find EC.

Step 1: DE ∥ AC ⇒ ▵BDE ~ ▵BAC (AA: common angle at B, corresponding angles).

Step 2: Scale factor k = BD/BA = 5/8.

Step 3: BE/BC = 5/8 ⇒ BC = BE × 8/5 = 4 × 8/5 = 6.4. EC = BC − BE = 6.4 − 4 = 2.4.

Worked Example 5 — Intercept theorem (ratio form)

Two transversals cut three parallel lines. The intercepts on the first transversal are 4 cm and 6 cm. The intercepts on the second transversal are 5 cm and x cm. Find x.

Step 1: By the intercept theorem (equal ratios on parallel lines): 4/6 = 5/x.

Step 2: x = 6 × 5/4 = 30/4 = 7.5 cm.

See Answers ➔

  1. Quadrilateral properties. Fluency

    • (a) Name the quadrilateral where all sides are equal and all angles are 90°.
    • (b) A parallelogram has one angle of 70°. Find all four angles.
    • (c) A rhombus has diagonals of length 10 cm and 24 cm. Find the side length.
    • (d) A rectangle has diagonals of length 13 cm and sides 5 cm. Find the other side.

  2. Coordinate geometry and quadrilaterals. Fluency

    • (a) A(0,0), B(5,0), C(5,4), D(0,4). What quadrilateral is ABCD?
    • (b) Find the midpoints of both diagonals of the quadrilateral in (a). What do you notice?
    • (c) E(1,1), F(4,1), G(5,4), H(2,4). Show EFGH is a parallelogram using slope.
    • (d) Show that EFGH in (c) is not a rectangle by computing one interior angle.

  3. Intercept theorem. Fluency

    • (a) Three parallel lines cut two transversals. Intercepts on first transversal: 6 and 9. Intercept on second transversal: 8 and y. Find y.
    • (b) In ▵ABC, D on AB and E on AC with DE ∥ BC. AD = 4, DB = 6, AE = 5. Find EC.
    • (c) In the same triangle, find DE if BC = 15.
    • (d) Two transversals are cut by three parallel lines giving intercepts 3, 5 on one and x, 10 on the other. Find x.

  4. Identify quadrilateral from properties. Fluency

    • (a) A quadrilateral has diagonals that are equal and bisect each other. What is it?
    • (b) A quadrilateral has diagonals that bisect each other at right angles. What could it be?
    • (c) A quadrilateral has exactly one pair of parallel sides and non-equal diagonals. What is it?
    • (d) A quadrilateral has two pairs of adjacent equal sides and one diagonal that bisects the other at right angles. What is it?

  5. Parallel intercept theorem in a diagram. Understanding

    Lines l1, l2, l3 are parallel. Two transversals cross all three. The intercepts on the left transversal are AB = 6 and BC = 9. The intercepts on the right transversal are PQ = x and QR = 12.

    l₁ l₂ l₃ A B C 6 9 P Q R x 12
    • (a) State the intercept theorem (basic proportionality theorem) as it applies here.
    • (b) Set up the proportion AB/BC = PQ/QR.
    • (c) Solve for x.
    • (d) If a third transversal has intercept ST = 10 between l1 and l2, find the intercept TU between l2 and l3.

  6. Proving a rectangle. Understanding

    ABCD is a parallelogram with AC = BD (diagonals equal). Prove ABCD is a rectangle.

    • (a) In ▵ABC and ▵DCB, list three equal parts and state the congruence.
    • (b) Hence show ∠ABC = ∠DCB.
    • (c) Since ABCD is a parallelogram, ∠ABC + ∠DCB = 180°. Find ∠ABC.
    • (d) Conclude that ABCD is a rectangle.

  7. Kite properties. Understanding

    ABCD is a kite with AB = AD and CB = CD.

    • (a) Prove ▵ABD ≅ ▵CBD.
    • (b) Hence prove ∠ABD = ∠CBD.
    • (c) Prove that the diagonal BD is the perpendicular bisector of AC.
    • (d) If ∠ABD = 34° and ∠ADB = 62°, find all four angles of the kite.

  8. Coordinate proof. Understanding

    P(0,0), Q(6,0), R(8,4), S(2,4).

    • (a) Find the length of all four sides.
    • (b) Find the slopes of all four sides and hence state the type of quadrilateral.
    • (c) Find the midpoints of the diagonals PR and QS. What does this confirm?
    • (d) Show PQRS is not a rectangle by checking the slope of PQ and PS.

  9. Tiling and geometry. Problem Solving

    A mosaic is made from congruent equilateral triangles of side 4 cm. They are arranged in a strip alternating point-up and point-down to form a parallelogram shape.

    • (a) What is the angle at each vertex of the equilateral triangle?
    • (b) Two triangles (one up, one down) form a rhombus. Find all angles of this rhombus.
    • (c) A row of 8 triangles (4 up, 4 down) forms a strip. Find the width (height) of the strip.
    • (d) The strip is 6 rows tall (6 strips stacked). Find the total area of the mosaic region.

  10. Bridge truss design. Problem Solving

    A bridge truss consists of triangular units. The main truss is a large isosceles triangle ABC with AB = AC = 25 m and BC = 30 m. A vertical support from A meets BC at M. Smaller triangles are formed by connecting midpoints of the sides.

    • (a) Find AM (the height of the main triangle) using Pythagoras’ theorem.
    • (b) The midpoints D, E, F of AB, AC, BC respectively form ▵DEF. Prove ▵DEF ~ ▵ABC and find the scale factor.
    • (c) Find the perimeter of ▵DEF.
    • (d) The total steel required equals the perimeter of ▵ABC plus the perimeter of ▵DEF plus AM. Calculate the total steel length.