Edexcel Mathematics (4XMAH) -Unit 1 - 4.7 Geometrical Reasoning- Study Notes- New Syllabus
Edexcel Mathematics (4XMAH) -Unit 1 – 4.7 Geometrical Reasoning- Study Notes- New syllabus
Edexcel Mathematics (4XMAH) -Unit 1 – 4.7 Geometrical Reasoning- Study Notes -Edexcel iGCSE Mathematics – per latest Syllabus.
Key Concepts:
A provide reasons, using standard geometrical statements, to support numerical values for angles obtained in any geometrical context involving lines, polygons and circles
Edexcel iGCSE Mathematics -Concise Summary Notes- All Topics
Geometrical Reasoning (Giving Reasons for Angles)
In geometry you must not only find angles, but also justify each step using standard geometrical facts.
Common Angle Facts
Angles in Lines
- Angles on a straight line sum to \( 180^\circ \)
- Vertically opposite angles are equal
Parallel Lines
- Corresponding angles are equal
- Alternate angles are equal
- Co-interior angles sum to \( 180^\circ \)
Triangles
- Angles in a triangle sum to \( 180^\circ \)
- Isosceles triangle base angles are equal
- Exterior angle equals sum of opposite interior angles
Polygons
Sum of interior angles \( =(n-2)\times180^\circ \)
Circle Theorems
- Angle in a semicircle is \( 90^\circ \)
- Angles in the same segment are equal
- Angle at centre is twice angle at circumference
- Opposite angles in a cyclic quadrilateral sum to \( 180^\circ \)
- Radius is perpendicular to tangent
Example 1:
Two angles on a straight line are \( 3x \) and \( 60^\circ \). Find \( x \).
▶️ Answer/Explanation
Angles on a straight line sum to \( 180^\circ \).
\( 3x+60=180 \)
\( 3x=120 \)
\( x=40^\circ \)
Conclusion: \( x=40^\circ \).
Example 2:
The angles of a triangle are \( x \), \( x+20 \), and \( x+40 \). Find \( x \).
▶️ Answer/Explanation
Angles in a triangle sum to \( 180^\circ \).
\( x+(x+20)+(x+40)=180 \)
\( 3x+60=180 \)
\( 3x=120 \)
\( x=40^\circ \)
Conclusion: \( x=40^\circ \).
Example 3:
Opposite angles of a cyclic quadrilateral are \( 2x+10 \) and \( 3x-20 \). Find \( x \).
▶️ Answer/Explanation
Opposite angles in a cyclic quadrilateral sum to \( 180^\circ \).
\( (2x+10)+(3x-20)=180 \)
\( 5x-10=180 \)
\( 5x=190 \)
\( x=38^\circ \)
Conclusion: \( x=38^\circ \).