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Chapter Analysis
Intermediate13 pages • EnglishQuick Summary
Chapter 6, titled 'Lines and Angles', discusses the properties and relationships between lines and angles. It introduces basic terms such as collinear points, various types of angles, and adjacent angles. Key concepts covered include linear pair axioms, theorems about vertically opposite angles, and behavior of angles when lines intersect or run parallel. The chapter concludes with examples and exercises to reinforce these concepts.
Key Topics
- •Linear pair of angles
- •Vertically opposite angles
- •Parallel and transversal lines
- •Corresponding angles
- •Theorems on parallel lines
- •Intersecting lines
- •Basic angle concepts
- •Math in real-life applications
Learning Objectives
- ✓Understand and apply the concept of linear pair axiom
- ✓Prove theorems related to vertically opposite angles
- ✓Use corresponding angles to prove lines are parallel
- ✓Identify and calculate angles formed by intersecting lines
- ✓Recognize and utilize angle properties in practical scenarios
Questions in Chapter
In Fig. 6.23, if AB || CD, CD || EF and y : z = 3 : 7, find x.
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In Fig. 6.24, if AB || CD, EF ⊥ CD and ∠ GED = 126°, find ∠ AGE, ∠ GEF and ∠ FGE.
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In Fig. 6.25, if PQ || ST, ∠ PQR = 110° and ∠ RST = 130°, find ∠ QRS.
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In Fig. 6.26, if AB || CD, ∠ APQ = 50° and ∠ PRD = 127°, find x and y.
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In Fig. 6.27, PQ and RS are two mirrors placed parallel to each other. An incident ray AB strikes the mirror PQ at B, the reflected ray moves along the path BC and strikes the mirror RS at C and again reflects back along CD. Prove that AB || CD.
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Additional Practice Questions
Explain how the concept of vertically opposite angles can be applied in real-life architectural designs.
mediumAnswer: In architectural design, understanding vertically opposite angles helps in ensuring that structures are aligned correctly. For example, when designing the framework for roofs or for complex structures like domes, knowing that intersecting beams create equal opposing angles ensures that the weight distribution is even, thus maintaining balance and symmetry.
Describe the significance of corresponding angles when constructing parallel lines.
easyAnswer: Corresponding angles are crucial when constructing parallel lines because they help verify parallelism. If a transversal intersects two lines creating equal corresponding angles, it guarantees that the lines are parallel. This principle is used in various engineering fields to design parallel beams and streets.
Draw your own diagram of two intersecting lines and use it to find the measure of unknown angles using the linear pair axiom.
mediumAnswer: To solve, you need to identify linear pairs in your diagram. Suppose line AB and line CD intersect at point O. If ∠AOC is 60°, then ∠BOD = 180° - 60° = 120° using the linear pair axiom, which states that adjacent angles on a straight line sum up to 180°.
Consider two parallel lines cut by a transversal. If one interior angle is 70°, what are the measures of all other angles formed?
hardAnswer: If two parallel lines are cut by a transversal and one interior angle is 70°, then the angle adjacent to it on the same line of the transversal is 110° (supplementary angle). The corresponding angle on the opposite side of the transversal will also be 70°, and its adjacent angle will be 110°. The vertically opposite angles are equal, confirming these values.
Define and explain the concept of a straight angle using a real-world example.
easyAnswer: A straight angle is an angle of 180°, forming a straight line. It can be seen when a door is fully open, forming a straight line with the frame – representing half the complete circle (360°) and demonstrating the balance and symmetry present in design and nature.