CAT Angles & Lines : Basic Concepts and Notes PDF
| Table of contents |
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| Angles |
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| Properties of Lines and Intersecting Lines |
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| Short Worked Examples |
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| Key Formulae and Facts (Quick Reference) |
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| Summary |
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Angles
Angles are formed by two rays (or line-segments) having a common end point called the vertex. Angles are classified by their measure and by their geometric relations to other angles and lines. Understanding angles and the properties of lines is fundamental for geometry and has direct applications in civil engineering, computer science (computational geometry), and electrical engineering (layout and orthogonality).
Classification by Measure
- Acute angle - an angle whose measure is less than 90°.
- Right angle - an angle whose measure is exactly 90°. It is also called a perpendicular relationship when formed by two lines.
- Obtuse angle - an angle whose measure is greater than 90° but less than 180°.
- Straight angle - an angle whose measure is exactly 180°. It is formed by two opposite rays and represents a straight line.
- Reflex angle - an angle whose measure is greater than 180° but less than 360°.
- Full rotation - an angle of 360°, representing one full turn.
Special Pairs of Angles
- Complementary angles - Two angles are complementary if the sum of their measures is 90°. If ∠A + ∠B = 90°, then ∠A is the complement of ∠B and vice versa.
- Supplementary angles - Two angles are supplementary if the sum of their measures is 180°. If ∠A + ∠B = 180°, then ∠A is the supplement of ∠B and vice versa.
- Adjacent angles - Two angles are adjacent if they share a common vertex and a common arm, and their other arms lie on either side of the common arm.
- Linear pair - A pair of adjacent angles whose non-common arms are opposite rays. The angles of a linear pair are supplementary, so their measures add up to 180°.
- Vertically opposite (vertical) angles - When two lines intersect, they form two pairs of opposite angles. Each pair of vertically opposite angles are equal in measure.
Linear Pair and Adjacent Angles - Illustration and Explanation
A linear pair is formed when two angles share a common vertex and one common arm, and the other two arms are in opposite directions forming a straight line. The sum of angles in a linear pair is 180°. For example, if rays OA and OB form angles ∠AOP and ∠POB such that OA and OB are opposite rays about point O, then ∠AOP and ∠POB form a linear pair and are supplementary.
Two angles are adjacent if they share a common arm and vertex and do not overlap. For instance, ∠ABC and ∠BCD are adjacent because they share the arm BC.
Properties of Lines and Intersecting Lines
A line extends indefinitely in both directions and consists of infinitely many points. Two distinct lines in a plane can either be parallel (no intersection) or intersecting (meet at a point).
Intersecting Lines: Vertical Angles and Linear Pairs
When two lines intersect at a point, they form four angles. Opposite (vertical) angles are equal in measure. Also, each pair of adjacent angles formed at the intersection is a linear pair and therefore supplementary.
For example, if two lines intersect at R forming angles labeled ∠PRQ, ∠QRS, ∠SRT, and ∠TRP, then ∠PRQ = ∠SRT and ∠QRS = ∠TRP because these are vertically opposite angles.
In the figure above, if ∠x and ∠y are adjacent forming a straight line, then ∠x + ∠y = 180°.
Perpendicular Lines
When two lines meet to form a right angle, each of the four angles at the intersection is 90°. Such lines are called perpendicular and are denoted by L1 ⟂ L2.
Perpendicularity is used widely: in civil engineering for orthogonal components of structures and in electrical engineering for orthogonal routing of PCB tracks and to maintain right-angle connections in layouts and drawings.
Parallel Lines
Parallel lines are coplanar lines that do not meet, no matter how far extended. The symbol for parallel is ∥. For example, L1 ∥ L2.
Parallel Lines and a Transversal
If a transversal (a line that intersects two lines) cuts two parallel lines, several pairs of related angles are formed with consistent relationships. These relationships are essential in proofs and problem solving.
- Corresponding angles - angles that occupy the same relative position at each intersection. Corresponding angles are equal when the lines are parallel (for example, ∠1 and ∠5).
- Alternate interior angles - interior angles on opposite sides of the transversal. Alternate interior angles are equal when the lines are parallel (for example, ∠2 and ∠5 in many standard labellings).
- Alternate exterior angles - exterior angles on opposite sides of the transversal. Alternate exterior angles are equal when the lines are parallel (for example, ∠3 and ∠6).
- Consecutive interior (co-interior or same-side interior) angles - interior angles on the same side of the transversal. These are supplementary when the lines are parallel (their measures sum to 180°).
- Vertically opposite angles - formed at each intersection; vertical pairs are equal.
- For parallel lines cut by a transversal: corresponding angles are equal; alternate interior angles are equal; alternate exterior angles are equal; and co-interior angles are supplementary.
- Typical equalities obtained are ∠1 = ∠5, ∠2 = ∠6 (depending on labelling), and ∠3 = ∠4 for vertical pairs in standard diagrams.
Short Worked Examples
Example 1. Two angles are complementary. One angle is 35°. Find the other angle.
Sol.
The sum of complementary angles is 90°.
Other angle = 90° − 35°.
Other angle = 55°.
Example 2. Two lines intersect. One angle is 120°. Find the measures of the other three angles.
Sol.
Vertically opposite angle = 120°.
Each adjacent angle to 120° forms a linear pair with it, so each adjacent angle = 180° − 120° = 60°.
The remaining angle (opposite this 60°) is vertically opposite to it and hence is also 60°.
Key Formulae and Facts (Quick Reference)
- Complementary: ∠A + ∠B = 90°.
- Supplementary: ∠A + ∠B = 180°.
- Linear pair: adjacent angles whose non-common arms form a straight line; they are supplementary.
- Vertical angles: equal in measure.
- Parallel lines with transversal: corresponding, alternate interior and alternate exterior angles are equal; co-interior angles are supplementary.
- Perpendicular lines: meet at 90° and are denoted by the symbol ⟂.
Summary
Understanding the types of angles and the relationships formed by intersecting, parallel and perpendicular lines is essential for geometric reasoning and for practical applications in engineering and computing. Remember the basic equalities and supplementary conditions: they provide the toolkit for solving many geometry problems and for applying geometry in technical fields.
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FAQs on CAT Angles & Lines : Basic Concepts and Notes PDF
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