Geometry

The One Thing

Geometry establishes conclusions about lines, angles, triangles, and quadrilaterals through definitions, stated properties, theorems, and logically justified steps. Diagrams assist visualisation, but measurements from a diagram cannot replace mathematical reasoning.

Definitions and Results

• Point: An exact position in space with no length, breadth, or thickness.
• Line: A straight path extending endlessly in both directions.
• Line Segment: A part of a line with two fixed endpoints.
• Ray: A part of a line with one endpoint extending endlessly in one direction.
• Intersecting Lines: Two lines that meet at a point.
• Parallel Lines: Lines in the same plane that never meet, however far they are extended.
• Transversal: A line that intersects two or more lines at different points.
• Angle: The figure formed by two rays having a common endpoint called the vertex.
• Acute Angle: An angle greater than $0^{\circ }$ and less than $90^{\circ }$.
• Right Angle: An angle measuring exactly $90^{\circ }$.
• Obtuse Angle: An angle greater than $90^{\circ }$ and less than $180^{\circ }$.
• Straight Angle: An angle measuring $180^{\circ }$.
• Reflex Angle: An angle greater than $180^{\circ }$ and less than $360^{\circ }$.
• Complementary Angles: Two angles whose sum is $90^{\circ }$.
• Supplementary Angles: Two angles whose sum is $180^{\circ }$.
• Vertically Opposite Angles: Opposite angles formed when two lines intersect; they are equal.
• Linear Pair: A pair of adjacent angles whose non-common arms form a straight line; their sum is $180^{\circ }$.
• Triangle: A closed figure formed by three line segments.
• Congruent Figures: Figures having the same shape and the same size.
• Congruence Criteria: Conditions used to prove that two triangles are congruent, such as SSS, SAS, ASA, and RHS.
• SSS Congruence: Two triangles are congruent if their three corresponding sides are equal.
• SAS Congruence: Two triangles are congruent if two corresponding sides and the included angle are equal.
• ASA Congruence: Two triangles are congruent if two corresponding angles and the included side are equal.
• RHS Congruence: Two right-angled triangles are congruent if their hypotenuse and one corresponding side are equal.
• Isosceles Triangle: A triangle with two equal sides; its base angles are equal.
• Equilateral Triangle: A triangle with three equal sides and three angles of $60^{\circ }$ each.
• Right-Angled Triangle: A triangle containing one angle of $90^{\circ }$.
• Angle Sum Property of a Triangle: The three interior angles of a triangle add up to $180^{\circ }$.
• Exterior Angle of a Triangle: An angle formed by one side of a triangle and the extension of an adjacent side.
• Exterior Angle Theorem: An exterior angle of a triangle equals the sum of its two interior opposite angles.
• Quadrilateral: A closed figure formed by four line segments.
• Parallelogram: A quadrilateral in which both pairs of opposite sides are parallel.
• Rectangle: A parallelogram with four right angles.
• Square: A quadrilateral with four equal sides and four right angles.
• Rhombus: A parallelogram with all four sides equal.
• Trapezium: A quadrilateral with one pair of opposite sides parallel.
• Kite: A quadrilateral with two pairs of adjacent equal sides.
• Diagonal: A line segment joining two non-adjacent vertices of a polygon.
• Midpoint Theorem: The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
• Proof: A logical sequence of statements that establishes the truth of a geometric result.
• Converse: A statement formed by reversing the hypothesis and conclusion of a theorem.

Important results include:

• Angles on a straight line add up to $180^{\circ }$.
• Angles around a point add up to $360^{\circ }$.
• Vertically opposite angles are equal.
• When two parallel lines are cut by a transversal:

- corresponding angles are equal; - alternate interior angles are equal; - interior angles on the same side of the transversal are supplementary.

• Conversely:

- equal corresponding angles imply that two lines are parallel; - equal alternate interior angles imply that two lines are parallel; - supplementary interior angles on the same side of a transversal imply that two lines are parallel.

• In a triangle, the sum of the interior angles is $180^{\circ }$.
• An exterior angle of a triangle equals the sum of the two interior opposite angles.
• In a triangle, the greater side lies opposite the greater angle, and the greater angle lies opposite the greater side.
• The sum of any two sides of a triangle is greater than the third side.
• In an isosceles triangle, equal sides have equal opposite angles, and equal angles have equal opposite sides.
• The diagonals of a parallelogram bisect each other.
• Opposite sides of a parallelogram are equal and parallel.
• Opposite angles of a parallelogram are equal, while adjacent angles are supplementary.
• A quadrilateral can be divided into two triangles; therefore, the sum of its interior angles is $360^{\circ }$.
• The diagonals of a rectangle are equal and bisect each other.
• The diagonals of a rhombus are perpendicular bisectors of each other.
• The diagonals of a square are equal, perpendicular, and bisect each other.
• A quadrilateral is a parallelogram if:

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

• For congruent triangles, corresponding sides and corresponding angles are equal by CPCTC: corresponding parts of congruent triangles are congruent.
• The midpoint theorem has a converse: a line through the midpoint of one side of a triangle and parallel to another side bisects the third side.

Worked Methods

1. Determining unknown angles from intersecting lines

• Identify whether the relevant angles lie on a straight line, around a point, or at the intersection of two lines.
• Use the appropriate result:

- angles on a straight line sum to $180^{\circ }$; - angles around a point sum to $360^{\circ }$; - vertically opposite angles are equal.

• Form an equation for the unknown angle.
• Solve the equation and state the result with its justification.

2. Using a transversal with parallel lines

• Establish that the two lines are parallel, or use the given parallel condition.
• Identify the pair of angles involved.
• Apply the correct relationship:

- corresponding angles are equal; - alternate interior angles are equal; - interior angles on the same side of the transversal are supplementary.

• Calculate the unknown angle.
• If parallelism must be proved, use the converse of the relevant result: equal corresponding angles, equal alternate interior angles, or supplementary interior angles on the same side.

3. Applying the angle sum property of a triangle

• Identify the three interior angles of the triangle.
• Use

$$\angle A+\angle B+\angle C=180^{\circ }.$$

• Substitute the known angles.
• Rearrange to find the unknown angle.
• Justify the result by the angle sum property of a triangle.

4. Applying the exterior angle theorem

• Identify the exterior angle formed by one side of the triangle and the extension of an adjacent side.
• Identify the two interior opposite angles.
• Use

$$\text{exterior angle}={\text{interior opposite angle}}_1+{\text{interior opposite angle}}_2.$$

• Substitute the known values and calculate the unknown angle.
• Distinguish the interior opposite angles from the interior angle adjacent to the exterior angle.

5. Comparing sides and angles in a triangle

• Compare the relevant sides and angles.
• Use the result that the greater side lies opposite the greater angle.
• Conversely, the greater angle lies opposite the greater side.
• State the corresponding ordering of sides or angles.

For a possible triangle inequality check:

• Identify the three side lengths.
• Add each pair of sides.
• Verify that the sum of any two sides is greater than the third side.
• Conclude whether the stated side lengths satisfy the triangle condition.

6. Proving triangles congruent

• Identify the corresponding vertices and sides.
• Record the equal information given in the problem.
• Select the applicable congruence criterion:

- SSS: three corresponding sides are equal; - SAS: two corresponding sides and the included angle are equal; - ASA: two corresponding angles and the included side are equal; - RHS: the triangles are right-angled and have equal hypotenuses and one corresponding equal side.

• State that the triangles are congruent using the selected criterion.
• Use CPCTC to conclude that corresponding sides and corresponding angles are equal.

7. Using properties of an isosceles or equilateral triangle

For an isosceles triangle:

• Identify the two equal sides.
• Use the fact that their opposite angles are equal.
• Alternatively, if two angles are equal, conclude that their opposite sides are equal.
• Apply the angle sum property if a numerical angle is required.

For an equilateral triangle:

• Use the fact that all three sides are equal.
• Use the fact that all three angles are $60^{\circ }$.
• Apply these equalities to determine the required side or angle relationship.

8. Proving properties of quadrilaterals by division into triangles

• Draw a diagonal joining two non-adjacent vertices of the quadrilateral.
• Divide the quadrilateral into two triangles.
• Apply the triangle angle sum property to both triangles.
• Add the two angle sums:

$$180^{\circ }+180^{\circ }=360^{\circ }.$$

• Conclude that the sum of the interior angles of the quadrilateral is $360^{\circ }$.

9. Establishing parallelogram properties

To use the properties of a parallelogram:

• Identify that both pairs of opposite sides are parallel.
• Apply the relevant result:

- opposite sides are equal and parallel; - opposite angles are equal; - adjacent angles are supplementary; - diagonals bisect each other.

To prove that a quadrilateral is a parallelogram, use one of the valid criteria:

• Show that both pairs of opposite sides are equal; or
• Show that one pair of opposite sides is both equal and parallel; or
• Show that the diagonals bisect each other.
• Conclude that the quadrilateral is a parallelogram.

10. Using special quadrilateral diagonals

• For a rectangle, use that its diagonals are equal and bisect each other.
• For a rhombus, use that its diagonals are perpendicular bisectors of each other.
• For a square, use that its diagonals are equal, perpendicular, and bisect each other.
• For a general parallelogram, use that its diagonals bisect each other.

11. Applying the midpoint theorem

• Identify the two midpoints of sides of a triangle.
• Join those midpoints.
• Apply the midpoint theorem:

- the joining segment is parallel to the third side; - its length is half the length of the third side.

• State both conclusions where required.

For the converse:

• Identify a line through the midpoint of one side.
• Confirm that it is parallel to another side.
• Conclude that it bisects the third side.

12. Writing a geometric proof

• State the given information.
• State the result to be proved.
• Set out each logical step in order.
• Justify each step using a definition, axiom, theorem, or property.
• Avoid assuming a conclusion merely because it appears true in the diagram.
• State the final conclusion clearly.
• Use a converse only when that converse is known to be true.

Where It Goes Wrong

• Treating measurements taken from a diagram as proof, rather than justifying the result through definitions, theorems, and properties.
• Confusing corresponding angles, alternate interior angles, and interior angles on the same side of a transversal.
• Forgetting the conditions required for parallel-line results: the lines must be parallel, or the relevant converse condition must be established.
• Applying a triangle congruence criterion without checking its exact requirements, especially the included angle in SAS and the right-angle condition in RHS.
• Failing to use CPCTC after proving triangles congruent, or matching corresponding sides and angles incorrectly.
• Using the converse of a theorem without verifying that the converse is known to be true.

What Gets Asked

This material supports questions requiring students to:

• Define points, lines, line segments, rays, angles, transversals, triangles, quadrilaterals, diagonals, proofs, and converses.
• Classify angles as acute, right, obtuse, straight, or reflex.
• Use complementary and supplementary angle relationships.
• Calculate unknown angles formed by intersecting lines, a linear pair, or angles around a point.
• Apply vertically opposite angle properties.
• Use corresponding, alternate interior, and same-side interior angle relationships with parallel lines and a transversal.
• Prove that two lines are parallel using the corresponding-angle, alternate-interior-angle, or same-side-interior-angle converse.
• Calculate unknown angles using the angle sum property of a triangle and the exterior angle theorem.
• Compare sides and angles in a triangle and test the triangle inequality.
• Use properties of isosceles, equilateral, and right-angled triangles.
• Prove triangles congruent using SSS, SAS, ASA, or RHS.
• Deduce further equalities using CPCTC.
• Establish properties of parallelograms, rectangles, rhombi, squares, trapeziums, and kites.
• Prove that a quadrilateral is a parallelogram using equal opposite sides, one pair of equal and parallel opposite sides, or diagonals that bisect each other.
• Use diagonals to distinguish properties of parallelograms, rectangles, rhombi, and squares.
• Calculate the interior-angle sum of a quadrilateral by dividing it into two triangles.
• Apply the midpoint theorem and its converse.
• Write formal proofs by identifying the given information, the statement to be proved, the theorem or property used, and the conclusion.