STUDYFORGE Grade 10 Mathematics
CAPS EXAM-FOCUSED STUDY NOTES Euclidean Geometry
Quadrilaterals & Triangles — Proof Reasons
CAPS SCOPE
In Grade 10 Euclidean Geometry riders, you are often given a quadrilateral (or a triangle) with some
sides, angles or diagonals marked, and asked to prove what type of shape it is. The reason you
write is just as important as the conclusion — a correct conclusion with the wrong or missing reason
does not get full marks. This summary focuses only on those reasons: what you must show to
justify each classification.
Parallelogram Rectangle Rhombus Kite Trapezium
REASONS FOR EACH QUADRILATERAL
Any one of the reasons listed for a shape is enough on its own to prove that classification — you do
not need to show all of them.
Quadrilateral Reason(s) that prove it
Parallelogram • Both pairs of opposite sides are parallel.
• Both pairs of opposite sides are equal.
• One pair of opposite sides is both equal and parallel.
• The diagonals bisect each other.
• Both pairs of opposite angles are equal.
Rectangle • A parallelogram with one angle equal to 90°.
• A parallelogram with equal diagonals.
• All four angles are equal to 90°.
Rhombus • A parallelogram with a pair of adjacent sides equal (so all four sides are equal).
• The diagonals bisect each other at 90°.
• The diagonals bisect the interior angles.
Square • A rectangle with a pair of adjacent sides equal.
• A rhombus with one angle equal to 90°.
• The diagonals are equal, bisect each other, and are perpendicular.
Kite • Two pairs of adjacent sides are equal (and it is not a rhombus).
• One diagonal bisects the other at 90°.
• One diagonal bisects the two angles it passes through.
Trapezium • Exactly one pair of opposite sides is parallel.
StudyForge — Quadrilaterals & Triangles: Proof Reasons (Grade 10) Page 1
CAPS EXAM-FOCUSED STUDY NOTES Euclidean Geometry
Quadrilaterals & Triangles — Proof Reasons
CAPS SCOPE
In Grade 10 Euclidean Geometry riders, you are often given a quadrilateral (or a triangle) with some
sides, angles or diagonals marked, and asked to prove what type of shape it is. The reason you
write is just as important as the conclusion — a correct conclusion with the wrong or missing reason
does not get full marks. This summary focuses only on those reasons: what you must show to
justify each classification.
Parallelogram Rectangle Rhombus Kite Trapezium
REASONS FOR EACH QUADRILATERAL
Any one of the reasons listed for a shape is enough on its own to prove that classification — you do
not need to show all of them.
Quadrilateral Reason(s) that prove it
Parallelogram • Both pairs of opposite sides are parallel.
• Both pairs of opposite sides are equal.
• One pair of opposite sides is both equal and parallel.
• The diagonals bisect each other.
• Both pairs of opposite angles are equal.
Rectangle • A parallelogram with one angle equal to 90°.
• A parallelogram with equal diagonals.
• All four angles are equal to 90°.
Rhombus • A parallelogram with a pair of adjacent sides equal (so all four sides are equal).
• The diagonals bisect each other at 90°.
• The diagonals bisect the interior angles.
Square • A rectangle with a pair of adjacent sides equal.
• A rhombus with one angle equal to 90°.
• The diagonals are equal, bisect each other, and are perpendicular.
Kite • Two pairs of adjacent sides are equal (and it is not a rhombus).
• One diagonal bisects the other at 90°.
• One diagonal bisects the two angles it passes through.
Trapezium • Exactly one pair of opposite sides is parallel.
StudyForge — Quadrilaterals & Triangles: Proof Reasons (Grade 10) Page 1
