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

KCSE Mathematics · 104 practice questions · 4 syllabus objectives · 4 revision lessons

38 easy34 medium32 hard

Last updated · Aligned to the KNEC KCSE syllabus

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)

Revision Notes

Concise lesson notes for Geometry: triangles (congruency, similarity, Pythagoras theorem), written to the KCSE Mathematics marking standard. Read the first lesson free below.

Conditions for Congruent Triangles

In geometry, two triangles are said to be congruent if they have the same size and shape. The conditions for triangle congruence are:

  • SSS (Side-Side-Side): If all three sides of one triangle are equal to the three sides of another triangle, the triangles are congruent.
  • SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
  • ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.
  • AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are equal to two angles and a corresponding side of another triangle, the triangles are congruent.
  • RHS (Right angle-Hypotenuse-Side): If the hypotenuse and one side of a right triangle are equal to the hypotenuse and one side of another right triangle, the triangles are congruent.

To prove that triangles are congruent, identify the corresponding sides and angles that meet any of the above conditions.

Key points to remember

  • Congruent triangles have equal corresponding sides and angles.
  • SSS, SAS, ASA, AAS, and RHS are congruence conditions.
  • Use congruence conditions to establish triangle similarity.

Worked example

Question: Prove that triangles ABC and DEF are congruent given that AB = DE, AC = DF, and angle A = angle D.

  • By SAS condition, since AB = DE, AC = DF, and angle A = angle D, triangles ABC and DEF are congruent.

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Lesson 2: Conditions for Similar Triangles

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

Similar triangles have the same shape but not necessarily the same size. To establish that two triangles are similar, the following conditions must be met:

  • Angle-Angle (AA) Criterion: If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
  • Side-Side-Side (SSS) Criterion: If the corresponding sides of two triangles are in proportion, the triangles are similar.
  • Side-Angle-Side (SAS) Criterion: If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in proportion, the triangles are similar.

When calculating unknown sides and areas, use the ratio of similarity. If two triangles are similar with a ratio of k:1, then:

  • The ratio of their corresponding sides is k:1.
  • The ratio of their areas is k²:1.

For example, if triangles ABC and DEF are similar with a ratio of 2:3, and side AB = 4 cm, then side DE can be calculated as follows:

  • DE = (3/2) × AB = (3/2) × 4 cm = 6 cm.
  • Similar triangles have equal corresponding angles.
  • Corresponding sides of similar triangles are proportional.
  • The area ratio of similar triangles is the square of the side ratio.
  • AA, SSS, and SAS are criteria for similarity.

Given triangles PQR and STU are similar. If PQ = 5 cm, and ST = 10 cm, find the ratio of their areas.

  • Area ratio = (10/5)² = 2² = 4:1.
Lesson 3: Understanding Pythagoras' Theorem

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

Pythagoras' theorem is a fundamental principle in geometry that applies to right-angled triangles. It states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This can be expressed as: c² = a² + b².

To apply this theorem, first, identify the right-angled triangle. The right angle is typically marked with a small square. Once identified, label the sides:

  • c: the hypotenuse (the longest side opposite the right angle)
  • a and b: the other two sides.

To find an unknown side, rearrange the theorem as needed. For example, if you know the lengths of sides a and c, you can find b using b = √(c² - a²).

This theorem is not only useful in solving problems but also in real-life applications such as construction and navigation.

  • Pythagoras' theorem applies only to right-angled triangles.
  • Formula: c² = a² + b², where c is the hypotenuse.
  • Identify the right angle to label sides correctly.
  • Rearrange the formula to find unknown sides.
  • Useful in practical applications like construction.

Question: In a right-angled triangle, if a = 3 cm and b = 4 cm, find c.

  • Using Pythagoras' theorem: c² = a² + b²
  • c² = 3² + 4² = 9 + 16 = 25
  • c = √25 = 5 cm.
Lesson 4: Understanding Triangle Properties: Congruency and Similarity

Objective: Geometry: triangles (congruency, similarity, Pythagoras theorem)

In geometry, triangles can be classified based on their properties of congruency and similarity. Congruent triangles are triangles that are identical in shape and size, meaning all corresponding sides and angles are equal. To prove two triangles are congruent, you can use criteria such as:

  • SSS (Side-Side-Side): All three sides are equal.
  • SAS (Side-Angle-Side): Two sides and the included angle are equal.
  • ASA (Angle-Side-Angle): Two angles and the included side are equal.

On the other hand, similar triangles have the same shape but not necessarily the same size. Their corresponding angles are equal, and their corresponding sides are in proportion. To prove triangles are similar, you can use the following criteria:

  • AA (Angle-Angle): Two angles of one triangle are equal to two angles of another.
  • SAS (Side-Angle-Side): Two sides are in proportion and the included angle is equal.

Understanding these properties allows for solving various geometric problems effectively.

  • Congruent triangles have equal corresponding sides and angles.
  • Similarity in triangles means equal angles and proportional sides.
  • SSS, SAS, and ASA are congruency criteria.
  • AA and SAS are used to prove similarity.

Question: Prove that triangles ABC and DEF are congruent if AB = DE, AC = DF, and angle A = angle D.

  • By SAS criterion, since two sides and the included angle are equal, triangles ABC and DEF are congruent.

Sample Questions

Read 3 questions and answers free. Sign up to access all 104 questions with full KNEC-style marking schemes and a personalised study plan.

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)

Answer & marking scheme

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)

Answer & marking scheme

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)

Answer & marking scheme

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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Frequently asked questions

What does the KCSE Mathematics topic "Geometry: triangles (congruency, similarity, Pythagoras theorem)" cover?

Geometry: triangles (congruency, similarity, Pythagoras theorem) covers 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, and more, all aligned to the official KNEC KCSE Mathematics syllabus.

How many practice questions are available for Geometry: triangles (congruency, similarity, Pythagoras theorem)?

HighMarks has 104 Geometry: triangles (congruency, similarity, Pythagoras theorem) practice questions for KCSE Mathematics, each with a full marking scheme. The first 3 are free; sign up to access the rest, plus all KCSE mock exams and past papers.

Are these aligned with the KNEC KCSE syllabus?

Yes. Every objective on this page is taken directly from the official KNEC KCSE Mathematics syllabus. Practice questions match the KCSE exam format and are graded against the standard KNEC marking scheme.

How should I revise Geometry: triangles (congruency, similarity, Pythagoras theorem) for the KCSE exam?

Start with the revision notes on this page to refresh the core concepts, then work through the practice questions in increasing difficulty. Sign up for HighMarks to get a personalised study plan that adapts to the topics you keep getting wrong, plus mock exams, subject-wide practice, and detailed performance tracking. See pricing.

Why Practise Geometry: triangles (congruency, similarity, Pythagoras theorem)?

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