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Chapter Analysis
Intermediate20 pages • EnglishQuick Summary
This chapter discusses the congruence of triangles, explaining various criteria such as the SAS, ASA, AAS, and RHS rules for determining congruence. It relates these rules to practical applications and provides numerous examples and exercises to deepen understanding. The chapter also explores the properties of isosceles and equilateral triangles, elaborating on the significance of congruent figures in real-world contexts, like manufacturing and construction.
Key Topics
- •Congruence of triangles
- •SAS congruence rule
- •ASA congruence rule
- •AAS congruence rule
- •RHS congruence rule
- •Properties of isosceles triangles
- •Properties of equilateral triangles
- •Real-world applications of congruence
Learning Objectives
- ✓Understand and apply criteria for triangle congruence.
- ✓Utilize congruence rules in problem-solving scenarios.
- ✓Recognize the significance of congruent figures in real-world contexts.
- ✓Identify and prove properties of isosceles and equilateral triangles.
- ✓Develop spatial reasoning skills through geometric constructions.
Questions in Chapter
1. In quadrilateral ACBD, if AC = AD and AB bisects ∠A, show that ∠ABC ≅ ∠ABD. What can you say about BC and BD?
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2. ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA (see Fig. 7.17). Prove that (i) ∆ABD ≅ ∆BAC (ii) BD = AC (iii) ∠ABD = ∠BAC.
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3. AD and BC are equal perpendiculars to a line segment AB (see Fig. 7.18). Show that CD bisects AB.
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4. l and m are two parallel lines intersected by another pair of parallel lines p and q (see Fig. 7.19). Show that ∆ABC ≅ ∆CDA.
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5. Line l is the bisector of an angle ∠A and B is any point on l. BP and BQ are perpendiculars from B to the arms of ∠A (see Fig. 7.20). Show that: (i) ∆APB ≅ ∆AQB (ii) BP = BQ or B is equidistant from the arms of ∠A.
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Additional Practice Questions
What is the importance of the congruence of triangles in engineering?
mediumAnswer: Triangular congruence is essential in engineering for ensuring stability and integrity in structures. Congruent triangles ensure uniformity in parts, which is crucial for components to fit together precisely without gaps, contributing to structural strength.
Describe a real-world scenario where the AAS congruence rule is applicable.
easyAnswer: In construction, when setting up roof trusses, knowing two angles and a corresponding non-included side can help ensure that each side of the truss is identical and fits precisely, providing stability and assisting in predictability of load bearing.
Explain the RHS congruence rule with a practical example.
mediumAnswer: In architecture, the RHS rule is used for creating identical right-angled components, like frames or brackets. Knowing the hypotenuse and one leg of a right triangle ensures identical supports are produced, simplifying assembly and ensuring equal load distribution.
How does knowing the congruence properties of an isosceles triangle assist in problem-solving?
mediumAnswer: Understanding that the angles opposite the equal sides in an isosceles triangle are also equal helps in determining unknown angles, which can be critical in geometric proofs and when designing objects that have reflective symmetry.
Why is congruence of shapes significant in the manufacturing of everyday products?
mediumAnswer: In manufacturing, congruence ensures that parts can be interchanged without the need for modification. This is crucial for mass production where different items, like car parts, must fit perfectly together, ensuring functionality and reducing production costs.