Angles and Constructions | Year 8 Mathematics IGCSE (Cambridge) PDF Download

Types of Angles

Angles are formed when two lines or rays meet at a common point called the vertex.
Here are the main types of angles:

• Acute Angles: An acute angle is an angle that is less than 90 degrees.
  • Example: A 45-degree angle is an acute angle.
• Right Angles: A right angle is exactly 90 degrees. It looks like the corner of a square or rectangle.
  • Example: The angles in a piece of paper are right angles.
• Obtuse Angles: An obtuse angle is more than 90 degrees but less than 180 degrees.
  • Example: A 120-degree angle is an obtuse angle.
• Straight Angles: A straight angle is exactly 180 degrees, forming a straight line.
  • Example: The angle formed by a straight line is a straight angle.
• Reflex Angles: A reflex angle is more than 180 degrees but less than 360 degrees.
  • Example: A 270-degree angle is a reflex angle.
• Full Rotation: A full rotation is 360 degrees.
  • Example: A complete circle represents a full rotation.

Angle Properties

Understanding the properties of angles helps in solving various geometric problems.
Here are some important properties:

• Complementary Angles: Two angles are complementary if their sum is 90 degrees.
  • Example: If one angle is 30 degrees, the complementary angle is 60 degrees (30° + 60° = 90°).
• Supplementary Angles: Two angles are supplementary if their sum is 180 degrees.
  • Example: If one angle is 110 degrees, the supplementary angle is 70 degrees (110° + 70° = 180°).
• Adjacent Angles: Adjacent angles share a common side and vertex but do not overlap.
  • Example: In a 'T' shape, the two angles on either side of the vertical line are adjacent.
• Vertically Opposite Angles: When two lines intersect, they form two pairs of vertically opposite angles. These angles are equal.
  • Example: If two intersecting lines form angles of 50 degrees and 130 degrees, the vertically opposite angles are also 50 degrees and 130 degrees.
• Angles on a Straight Line: The sum of angles on a straight line is 180 degrees.
  • Example: If one angle on a straight line is 70 degrees, the other angle is 110 degrees (70° + 110° = 180°).
• Angles Around a Point: The sum of angles around a point is 360 degrees.
  • Example: If three angles around a point are 90 degrees, 120 degrees, and 60 degrees, the fourth angle is 90 degrees (90° + 120° + 60° + 90° = 360°).

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Constructing Geometric Figures

Constructing geometric figures involves using tools like a ruler, compass, and protractor.
Here are some basic constructions:

Constructing a Perpendicular Bisector

• Draw a line segment AB.
• Place the compass point on A and draw an arc above and below the line.
• Without changing the compass width, place the compass point on B and draw another set of arcs intersecting the first set.
• Draw a line through the intersection points. This line is the perpendicular bisector of AB.

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Constructing an Angle Bisector

• Draw an angle ∠ABC.
• Place the compass point on B and draw an arc that intersects both sides of the angle.
• Place the compass point on one intersection and draw an arc inside the angle.
• Without changing the compass width, place the compass point on the other intersection and draw another arc intersecting the first arc.
• Draw a line from B through the intersection of the arcs. This line bisects ∠ABC.

Constructing a Triangle Given Three Sides (SSS)

• Draw the base of the triangle using a ruler.
• Place the compass point on one end of the base and draw an arc with a radius equal to one of the other sides.
• Without changing the compass width, place the compass point on the other end of the base and draw another arc intersecting the first arc.
• Draw lines connecting the intersection point of the arcs to the ends of the base.

Constructing a Triangle Given Two Angles and a Side (ASA)

• Draw the given side of the triangle using a ruler.
• Use a protractor to draw one of the given angles at one end of the base.
• Use the protractor to draw the second angle at the other end of the base.
• Extend the sides of the angles until they intersect. This forms the triangle.

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Examples

Example 1: Constructing a Perpendicular Bisector

• Draw a line segment AB of 8 cm.
• Follow the steps for constructing a perpendicular bisector.
• The perpendicular bisector will divide AB into two equal parts of 4 cm each.

Example 2: Constructing an Angle Bisector

• Draw an angle ∠PQR of 60 degrees.
• Follow the steps for constructing an angle bisector.
• The angle bisector will divide ∠PQR into two equal angles of 30 degrees each.

Example 3: Constructing a Triangle Given Three Sides

• Given sides of 5 cm, 6 cm, and 7 cm.
• Follow the steps for constructing a triangle with these sides.
• Verify that the triangle has the correct side lengths.