Geometry and Trigonometry

Edwin Ogie Library

Geometry and Trigonometry E‑Book

Topics Covered:
• Properties of lines and angles.
• Polygons (triangles, quadrilaterals, general polygons).
• Circle theorems (central and inscribed angles, cyclic quadrilaterals, intersecting chords).
• Geometric constructions (including special angles such as 30º, 45º, 60º, 75º, 90º).

Page 1: Introduction

Geometry is the study of shapes, sizes, and the properties of space. Trigonometry focuses on the relationships between angles and sides in triangles. This e‑book introduces the fundamentals of Euclidean geometry and trigonometry and explains key concepts with clear examples and illustrations.

Page 2: Properties of Lines and Angles

Lines may be parallel, perpendicular, or intersecting. Angles are classified as acute (< 90º), right (90º), obtuse (> 90º), straight (180º), or reflex (> 180º). Vertical angles, formed when two lines intersect, are equal.

Page 3: Polygons

A polygon is a closed figure with straight sides. Triangles have 3 sides (with an angle sum of 180º), quadrilaterals have 4 sides (360º), and an n-gon has an interior angle sum of (n-2)×180º. Polygons can be regular (all sides and angles equal) or irregular.

Example Diagram:

Page 4: Circles and Angle Properties

A circle is the set of points equidistant from a fixed center. Important concepts include:

• Central Angles: Angles with their vertex at the center.
• Inscribed Angles: Angles with their vertex on the circle, measuring half the intercepted arc.
• Intersecting Chords: The products of the segments of one chord equal those of the other chord.

Page 5: Cyclic Quadrilaterals and Intersecting Chords

A cyclic quadrilateral is a four-sided figure with all vertices on a circle. Its key property is that opposite angles add to 180º.

Page 6: Geometric Constructions Overview

Geometric constructions use only a compass and straightedge. They allow you to accurately create figures, bisect angles, and construct special angles such as 30º, 45º, 60º, 75º, and 90º.

Such constructions form the basis of many geometric proofs and practical applications.

Page 7: Construction Procedures for Special Angles

Common construction techniques include:

• 30º: Construct an equilateral triangle and bisect one angle.
• 45º: Construct a 90º angle and bisect it.
• 60º: Use an equilateral triangle directly.
• 75º: Combine a 30º and a 45º angle.
• 90º: Construct perpendicular lines.

Page 8: Worked Example 1 – Identifying Lines and Angles

Problem: In a diagram with intersecting lines, identify the parallel lines and the angles formed by a transversal.

Solution:

Identify the lines that never meet as parallel. When a transversal crosses parallel lines, corresponding angles are equal. (See Page 2’s illustration of parallel lines.)

Page 9: Worked Example 2 – Sum of Interior Angles

Problem: Calculate the sum of the interior angles of a heptagon (7-sided polygon).

Solution:

Sum = (n - 2) × 180º = (7 - 2) × 180º = 5 × 180º = 900º.

Page 10: Worked Example 3 – Inscribed Angle Theorem

Problem: In a circle, if an inscribed angle intercepts an arc of 80º, what is the measure of the inscribed angle?

Solution:

By the inscribed angle theorem, the inscribed angle = 80º/2 = 40º.

Page 11: Worked Example 4 – Cyclic Quadrilateral

Problem: In a cyclic quadrilateral, if one angle is 80º, determine its opposite angle.

Solution:

Opposite angles in a cyclic quadrilateral sum to 180º; hence, the opposite angle = 180º - 80º = 100º.

Page 12: Worked Example 5 – Constructing a 45º Angle

Problem: Explain how to construct a 45º angle from a 90º angle.

Solution:

Construct a 90º angle (using perpendicular lines) and then bisect it with a compass and straightedge. The bisector divides the 90º angle into two 45º angles.

Page 13: Worked Example 6 – Constructing a Perpendicular Bisector

Problem: Outline the steps to construct the perpendicular bisector of a given line segment.

Solution:

1. With the compass, choose a radius greater than half the segment’s length.
2. Draw arcs from both endpoints.
3. The intersection points of the arcs, when connected, form the perpendicular bisector.

Page 14: Worked Examples 7-10 – Additional Constructions and Calculations

Example 7: Constructing a 30º Angle

Problem: Describe a method to construct a 30º angle using an equilateral triangle.

Solution:

Construct an equilateral triangle (all angles 60º) and then bisect one of its angles to get 30º.

Example 8: Constructing a 75º Angle

Problem: Explain how to construct a 75º angle by combining known angles.

Solution:

Construct a 30º angle and a 45º angle separately, then place them adjacent so that their sum is 75º.

Example 9: Angle Sum in a Quadrilateral

Problem: Calculate the sum of the interior angles of a quadrilateral.

Solution:

Sum = (4-2) × 180º = 360º.

Example 10: Finding an Exterior Angle of a Regular Polygon

Problem: If a regular octagon has interior angles of 135º, what is each exterior angle?

Solution:

Exterior angle = 180º - 135º = 45º.

Page 15: Worked Examples 11-15 and Summary

Example 11: Using the Inscribed Angle Theorem

Problem: In a circle, if an inscribed angle intercepts an arc of 100º, find the inscribed angle.

Solution:

Inscribed angle = 100º / 2 = 50º.

Example 12: Sum of Interior Angles of a Decagon

Problem: Calculate the sum of the interior angles of a decagon (10-sided polygon).

Solution:

Sum = (10 - 2) × 180º = 8 × 180º = 1440º.

Example 13: Vertical Angles

Problem: Prove that vertical angles formed by two intersecting lines are equal.

Solution:

When two lines intersect, the opposite (vertical) angles are congruent due to the linear pair postulate and the fact that the sum of angles on a straight line is 180º.

Example 14: Adjacent Angles

Problem: Define adjacent angles and provide an example.

Solution:

Adjacent angles share a common vertex and a common side. For example, if two angles are formed by the rays AB, AC, and AD with AC common, then ∠BAC and ∠CAD are adjacent.

Example 15: Construction – Perpendicular Bisector of a Line

Problem: Construct the perpendicular bisector of a given line segment and explain the steps.

Solution:

1. With a compass, choose a radius more than half the segment's length.
2. Draw arcs from both endpoints.
3. Mark the intersection points of these arcs.
4. Draw a straight line through these intersection points. This line is the perpendicular bisector.

Summary

The above examples illustrate key concepts in geometry and trigonometry including properties of lines and angles, polygon angle sums, circle theorems, and basic construction techniques.

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