CIE IGCSE Mathematics (0580) Angles Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Angles Study Notes
LEARNING OBJECTIVE
- Study the properties of Angles
Key Concepts:
- Geometrical Angle Properties
Geometrical Angle Properties
Geometrical Angle Properties
The following fundamental angle rules are used frequently in geometry to calculate unknown angles. All statements must be supported by correct geometrical reasoning and terminology using three-letter angle notation (e.g., ∠ABC).
1. Sum of Angles at a Point = 360°
All angles that meet at a single point on a flat surface make a full rotation.
- Rule: \( \angle AOB + \angle BOC + \angle COD + \ldots = 360^\circ \)
Proof:
- Consider a full circle around point \( O \).
- The total measure around a point is one full revolution = \( 360^\circ \).
2. Sum of Angles on a Straight Line = 180°
Angles on a straight line form a half-turn, or straight angle.
- Rule: \( \angle ABC + \angle CBD = 180^\circ \), when A-B-C-D are collinear.
Proof:
- A straight angle equals half of a full circle = \( 180^\circ \).
- The angles lying along the line must sum to this.
3. Vertically Opposite Angles are Equal
When two straight lines intersect, the opposite (facing) angles are equal.
- Rule: If two lines intersect at \( O \), then \( \angle AOC = \angle BOD \)
Proof:
- Let two lines intersect forming four angles around point \( O \).
- From angle-on-a-line rule: \( \angle AOC + \angle COB = 180^\circ \)
- Also, \( \angle BOD + \angle COB = 180^\circ \)
- Therefore, \( \angle AOC = \angle BOD \) (by subtraction)
4. Sum of Angles in a Triangle = 180°
Any triangle’s interior angles always add up to 180°.
- Rule: \( \angle ABC + \angle BCA + \angle CAB = 180^\circ \)
Proof:
- Draw triangle \( ABC \).
- Draw a line parallel to \( BC \) through point \( A \).
- Alternate angles formed are equal to angles at \( B \) and \( C \).
- Hence the three interior angles lie on a straight line: sum = \( 180^\circ \)
5. Sum of Angles in a Quadrilateral = 360°
Any quadrilateral’s four interior angles add up to 360°.
- Rule: \( \angle ABC + \angle BCD + \angle CDA + \angle DAB = 360^\circ \)
Proof:
- Draw diagonal \( AC \), dividing the quadrilateral into triangles \( ABC \) and \( CDA \).
- Each triangle has angle sum = \( 180^\circ \), so total = \( 180^\circ + 180^\circ = 360^\circ \)
Example:
In the figure, a straight line passes through point B. If \( \angle ABC = 65^\circ \), find \( \angle CBD \).
▶️ Answer/Explanation
Angles on a straight line add to \( 180^\circ \):
\( \angle ABC + \angle CBD = 180^\circ \)
\( 65^\circ + \angle CBD = 180^\circ \Rightarrow \angle CBD = 180^\circ – 65^\circ = 115^\circ \)
Answer: \( 115^\circ \)
Example:
Two lines intersect at a point. One of the angles formed is \( 48^\circ \). Find its vertically opposite angle.
▶️ Answer/Explanation
Vertically opposite angles are equal:
Answer: \( 48^\circ \)
Example:
Triangle \( ABC \) has angles \( \angle ABC = 70^\circ \), \( \angle BCA = 50^\circ \). Find \( \angle CAB \).
▶️ Answer/Explanation
Sum of interior angles in a triangle is \( 180^\circ \):
\( \angle CAB = 180^\circ – (70^\circ + 50^\circ) = 180^\circ – 120^\circ = 60^\circ \)
Answer: \( 60^\circ \)
Angles Formed by Parallel Lines
Angles Formed by Parallel Lines
When a pair of parallel lines is intersected by a transversal (a line crossing both), several important angle relationships are formed. These relationships help in calculating unknown angles and giving justifications.
Key Angle Relationships:
- Corresponding Angles: Angles that are in the same position on each parallel line. They are equal.
\( \angle A = \angle B \) - Alternate Angles: Angles on opposite sides of the transversal, but between the two parallel lines. They are equal.
\( \angle A = \angle B \) - Co-interior Angles: Also called consecutive or allied interior angles. They are on the same side of the transversal and add up to \( 180^\circ \).
\( \angle A + \angle B = 180^\circ \)
These rules are fundamental in problems involving parallel lines and are often tested in IGCSE geometry questions.
Example:
Lines AB and CD are parallel. A transversal crosses them, forming an angle of \( 60^\circ \) at the top right of AB. Find the corresponding angle on CD.
▶️ Answer/Explanation
By the corresponding angle rule: \( \angle = 60^\circ \)
Answer: \( 60^\circ \)
Example:
A transversal cuts two parallel lines. One interior angle on the left of the transversal is \( 75^\circ \). Find the alternate interior angle on the right.
▶️ Answer/Explanation
By the alternate angle rule: \( \angle = 75^\circ \)
Answer: \( 75^\circ \)
Example:
Two parallel lines are cut by a transversal. The co-interior angle on one side is \( 112^\circ \). Find the angle on the same side below the other parallel line.
▶️ Answer/Explanation
Co-interior angles sum to \( 180^\circ \):
\( \angle = 180^\circ – 112^\circ = 68^\circ \)
Answer: \( 68^\circ \)
Example:
In the figure below, \( AB \) and \( CD \) are parallel lines, and a transversal intersects them forming angles.
The angle adjacent to angle \( x \) is \( 53^\circ \).
Use angle rules for parallel lines to find angles \( x \) and \( y \).
(a) Write down the size of angle \( x \). ………………………… °
(b) Give a reason for your answer.……………………………………
(c) Write down the size of angle \( y \). ………………………… °
(d) Give a reason for your answer.……………………………………
▶️Answer/Explanation
(a) The size of angle \( x \) is \( \boxed{53^\circ} \)
(b) Reason: \( x \) and the given \( 53^\circ \) angle are alternate interior angles, and alternate angles are equal when lines are parallel.
(c) The size of angle \( y \) is \( \boxed{53^\circ} \)
(d) Reason: \( y \) and the given \( 53^\circ \) angle are corresponding angles, and corresponding angles are equal when lines are parallel.
Angle Properties of Regular Polygons
Angle Properties of Regular Polygons
A regular polygon is a polygon with all sides equal in length and all angles equal in measure. Regular polygons have important angle properties that can be calculated using formulas.
Key Angle Properties:
- Sum of interior angles: \( (n – 2) \times 180^\circ \), where \( n \) is the number of sides
- Each interior angle (regular polygon): \( \frac{(n – 2) \times 180^\circ}{n} \)
- Each exterior angle (regular polygon): \( \frac{360^\circ}{n} \)
- Interior + Exterior angle: Always \( 180^\circ \)
Example :
Find the sum of interior angles of a regular octagon.
▶️ Answer/Explanation
\( n = 8 \)
Sum of interior angles = \( (8 – 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ \)
Answer: \( 1080^\circ \)
Example :
Find one interior angle of a regular decagon.
▶️ Answer/Explanation
\( n = 10 \)
Each interior angle = \( \frac{(10 – 2) \times 180^\circ}{10} = \frac{8 \times 180^\circ}{10} = \frac{1440^\circ}{10} = 144^\circ \)
Answer: \( 144^\circ \)
Example :
A regular polygon has each exterior angle equal to \( 30^\circ \). How many sides does it have?
▶️ Answer/Explanation
Each exterior angle = \( \frac{360^\circ}{n} \)
\( 30^\circ = \frac{360^\circ}{n} \Rightarrow n = \frac{360}{30} = 12 \)
Answer: 12 sides