Chapter 7: Triangles

Summary

• Two figures are congruent if they are of the same shape and size.
• Two circles with the same radius are congruent.
• Two squares with the same side lengths are congruent.
• Congruence of triangles can be expressed as: A ABC ≡ A PQR when corresponding vertices are equal.
• SAS Congruence Rule: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, they are congruent.
• ASA Congruence Rule: If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, they are congruent.
• AAS Congruence Rule: If two angles and one side of one triangle are equal to two angles and the corresponding side of another triangle, they are congruent.
• Angles opposite equal sides of a triangle are equal.
• Sides opposite equal angles of a triangle are equal.
• Each angle of an equilateral triangle is 60°.
• SSS Congruence Rule: If three sides of one triangle are equal to three sides of another triangle, they are congruent.
• RHS Congruence Rule: In two right triangles, if the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, they are congruent.

Chapter 7: Triangles

7.1 Introduction

• A triangle is a closed figure formed by three intersecting lines.
• It has three sides, three angles, and three vertices.
• Example: In triangle ABC, sides are AB, BC, CA; angles are ∠A, ∠B, ∠C.

7.2 Congruence of Triangles

• Definition: Two figures are congruent if they are of the same shape and size.
• Congruent Figures: Examples include two circles of the same radius and two squares of the same side length.
• Symbolic Representation: If triangles ABC and PQR are congruent, it is expressed as △ABC ≡ △PQR.

Criteria for Congruence

1. SAS (Side-Angle-Side) Rule: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.
2. ASA (Angle-Side-Angle) Rule: If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.
3. AAS (Angle-Angle-Side) Rule: If two angles and one side of one triangle are equal to two angles and the corresponding side of another triangle, then the triangles are congruent.
4. SSS (Side-Side-Side) Rule: If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.
5. RHS (Right angle-Hypotenuse-Side) Rule: In two right triangles, if the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the triangles are congruent.

7.4 Some Properties of a Triangle

• Angles opposite to equal sides of a triangle are equal.
• Sides opposite to equal angles of a triangle are equal.
• Each angle of an equilateral triangle is 60°.

7.6 Summary

• Two figures are congruent if they are of the same shape and size.
• Congruence can be established using various rules (SAS, ASA, AAS, SSS, RHS).
• Properties of triangles include relationships between angles and sides.

Common Mistakes and Exam Tips for Triangles

Common Pitfalls

• Incorrect Correspondence: When stating congruence, ensure the correspondence of vertices is correct. For example, writing A DEF ≡ A ABC is incorrect if the vertices do not correspond properly.
• Assuming Congruence from Angles Alone: Remember that equality of three angles is not sufficient for congruence of triangles. At least one side must be equal.
• Misapplying Congruence Rules: Ensure you apply the correct congruence rule (SAS, ASA, AAS, SSS, RHS) based on the given information.

Tips for Success

• Draw Diagrams: Visualize the problem by drawing accurate diagrams. This can help in understanding the relationships between sides and angles.
• Check Conditions for Congruence: Always verify that the conditions for the specific congruence rule you are using are met (e.g., for SAS, ensure the included angle is between the two sides).
• Use CPCTC: Remember that in congruent triangles, corresponding parts are equal. Use CPCTC (Corresponding Parts of Congruent Triangles are Congruent) to justify your answers.
• Practice with Examples: Work through various examples to familiarize yourself with identifying congruent triangles and applying the rules correctly.