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Geometry: triangles (congruency, similarity, Pythagoras theorem) — KCSE Mathematics

KCSE Mathematics · 104 practice questions · 4 syllabus objectives

38 easy34 medium32 hard

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

State the conditions for congruent triangles (SSS, SAS, ASA, AAS, RHS) and use them to prove triangles are congruent

State the conditions for similar triangles; calculate unknown sides and areas using ratios of similarity

State and apply Pythagoras' theorem (c² = a² + b²); identify right-angled triangles and calculate unknown sides

Geometry: triangles (congruency, similarity, Pythagoras theorem)

Sample Questions

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1
easySHORT ANSWER2 marks

State the length of the hypotenuse in a right-angled triangle where one leg measures 6 cm and the other leg measures 8 cm. (2 marks)

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Part (a) — 2 marks
Apply Pythagoras' theorem: c² = a² + b² = 6² + 8² = 36 + 64 = 100 (1 mk)
Therefore, c = √100 = 10 cm (1 mk)
2
easySHORT ANSWER4 marks

State the conditions under which two triangles are similar and provide an example for each condition. (4 marks)

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Part (a) — 2 marks
AA (Angle-Angle): Two angles of one triangle are equal to two angles of another triangle. (1 mk)
SSS (Side-Side-Side): The sides of one triangle are in proportion to the sides of another triangle. (1 mk)
Part (b) — 2 marks
Example for AA: Triangle ABC and Triangle DEF, where angle A = angle D and angle B = angle E. (1 mk)
Example for SSS: Triangle XYZ has sides of lengths 3 cm, 4 cm, and 5 cm, and Triangle PQR has sides of lengths 6 cm, 8 cm, and 10 cm. (1 mk)
3
easySHORT ANSWER4 marks

List the conditions under which two triangles are congruent. (4 marks)

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Part (a) — 4 marks
Side-Side-Side (SSS): All three sides of one triangle are equal to the corresponding three sides of another triangle. (1 mk)
Side-Angle-Side (SAS): Two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of another triangle. (1 mk)
Angle-Side-Angle (ASA): Two angles and the included side of one triangle are equal to the corresponding two angles and included side of another triangle. (1 mk)
Angle-Angle-Side (AAS): Two angles and a non-included side of one triangle are equal to the corresponding two angles and non-included side of another triangle. (1 mk)
4

In triangle ABC, angle A measures 30° and angle B measures 60°. Identify the measure of angle C. (2 marks)

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Master Geometry: triangles (congruency, similarity, Pythagoras theorem) for KCSE

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