Geometric Twins

Chapter Overview

Congruent figures are geometric twins: they match exactly in shape and size. Triangle congruence criteria allow us to establish an exact match without measuring every part.

This chapter belongs to Ganita Prakash, Grade 7. Focus on explaining each step, checking whether an answer is reasonable, and comparing more than one solution method.

Learning Objectives

After studying this chapter, you should be able to:

• explain and apply congruence;
• explain and apply triangle criteria;
• explain and apply corresponding parts;
• explain and apply isosceles and equilateral triangles;
• communicate the reasoning behind a solution clearly;
• check answers using estimation, substitution, or a second method.

Key Concepts

| Concept | Meaning | |---|---| | Congruence | Two figures are congruent if one can be placed exactly over the other using translation, rotation or reflection. | | Triangle criteria | SSS, SAS, ASA and RHS provide sufficient information to prove two triangles congruent. | | Corresponding parts | The order of letters in a congruence statement identifies matching vertices, sides and angles. | | Isosceles and equilateral triangles | Angles opposite equal sides are equal; an equilateral triangle has all sides and all angles equal. |

Detailed Explanation

Congruence

Two figures are congruent if one can be placed exactly over the other using translation, rotation or reflection.

Triangle criteria

SSS, SAS, ASA and RHS provide sufficient information to prove two triangles congruent.

Corresponding parts

The order of letters in a congruence statement identifies matching vertices, sides and angles.

Isosceles and equilateral triangles

Angles opposite equal sides are equal; an equilateral triangle has all sides and all angles equal.

Important Rules and Formulae

• AAA establishes similarity, not congruence.
• SSA is generally insufficient because different triangles may satisfy it.
• RHS applies only to right triangles and uses the hypotenuse plus one corresponding side.

Worked Examples

Example 1

Problem: Triangles ABC and PQR have AB = PQ, BC = QR and AC = PR. What proves congruence?

Solution: SSS congruence, with △ABC ≅ △PQR.

Example 2

Problem: In isosceles triangle ABC, AB = AC and angle B is 52°. Find angle C and angle A.

Solution: Angle C = 52°. Angle A = 180° − 104° = 76°.

Common Mistakes

• Writing corresponding vertices in the wrong order.
• Using SSA or AAA as a congruence criterion.
• Assuming equal areas guarantee congruence.

Quick Revision

• Congruence: Two figures are congruent if one can be placed exactly over the other using translation, rotation or reflection.
• Triangle criteria: SSS, SAS, ASA and RHS provide sufficient information to prove two triangles congruent.
• Corresponding parts: The order of letters in a congruence statement identifies matching vertices, sides and angles.
• Isosceles and equilateral triangles: Angles opposite equal sides are equal; an equilateral triangle has all sides and all angles equal.

Practice Questions

1. What does CPCT mean?
2. Which criterion uses two sides and the included angle?
3. Are two squares with side 5 cm congruent?
4. In congruent triangles, are corresponding angles equal?

Answers and Explanations

1. Corresponding Parts of Congruent Triangles.
2. SAS.
3. Yes.
4. Yes.

Self-Check

Explain one rule from this chapter in your own words, create a fresh example, solve it, and verify the answer. If the explanation and verification agree, the concept is understood rather than merely memorised.