Apply angle facts, triangle congruence, similarity and properties of…

The idea

This topic links angle facts, congruence, similarity and quadrilateral properties so you can conjecture and derive results about angles and sides. You use angle facts such as angles on a straight line summing to 180°, angles in a triangle summing to 180° and vertically opposite angles being equal. Congruence means identical shape and size; similarity means the same shape with equal corresponding angles and proportional sides. Quadrilateral properties include opposite sides of a parallelogram being equal and parallel, and diagonals of a rhombus bisecting at right angles. From these you derive Pythagoras’ theorem and prove base angles of an isosceles triangle are equal. You then use known results to build simple proofs, for example proving a triangle is isosceles by showing two angles are equal.

What earns the marks

• Apply angle facts correctly: angles on a straight line sum to 180°, angles around a point sum to 360°, vertically opposite angles are equal, and angles in a triangle sum to 180°.
• Use congruence criteria (SSS, SAS, ASA, RHS) to justify that two triangles are identical in shape and size, and use similarity to set up proportional sides when corresponding angles are equal.
• Apply properties of quadrilaterals, such as opposite sides of a parallelogram being equal and parallel, or diagonals of a rhombus bisecting at right angles, to find unknown angles and sides.
• Derive Pythagoras’ theorem from similarity or area arguments, and use it to find a missing side in a right-angled triangle, for example a² + b² = c² where c is the hypotenuse.
• Prove that the base angles of an isosceles triangle are equal, for example by drawing the line of symmetry from the apex to the midpoint of the base and using congruent triangles.
• Construct a simple proof by starting from known results, stating each step with a reason, and reaching the required conclusion without gaps.

Common mistakes and how to put them right

• Misidentifying the hypotenuse in Pythagoras’ theorem.

  Put it right by checking the right angle first.

• Assuming two triangles are congruent when only two sides and a non-included angle match.

  Put it right by using a valid congruence criterion such as SAS (included angle) or SSS.

• Treating similar shapes as congruent.

  Put it right by remembering similar shapes have equal angles but sides in proportion, so lengths may differ by a scale factor.

• Using a quadrilateral property that does not apply, such as assuming diagonals of every parallelogram bisect at right angles.

  Put it right by checking the specific quadrilateral type, since that property belongs to a rhombus.

Exam tips

• Write a reason for every angle fact you use, for example ‘angles in a triangle sum to 180°’, so the examiner can follow your proof.
• When using Pythagoras’ theorem, label the hypotenuse c and substitute carefully; check whether you need to add or subtract squares to find a shorter side.
• For proof questions, work backwards from the required result to plan steps, then write the proof forwards with clear logical order.
• Sketch the figure and mark equal angles or sides; this helps you spot congruent triangles and choose the correct criterion.