1. The document provides an overview of grade 11 mathematics concepts related to Euclidean geometry and circle geometry. 2. It outlines several circle geometry theorems that will be proven, including theorems about angles subtended by chords and arcs, relationships between angles of cyclic quadrilaterals, and properties of tangents. 3. Steps for solving geometry problems are provided, such as using diagrams, marking equal angles and segments, and working backwards from what is required to prove.

1. Mathematics
Grade 11
EUCLIDEAN GEOMETRY

2. Presented
By
Avhafarei Thavhanyedza
Saint Georges Conference Centre
03 March 2017

3. TOPIC OVERVIEW
1: Revise Grade 10 work & earlier grades
2. Concepts learnt from earlier grades
(and tan-chord theorem) must be used as
axioms
3. Use theorems & their converses to
solve riders

4. Proof the theorems for Circle
Geometry
The line drawn from the centre of a circle
perpendicular to a chord, bisects the chord (┴
from centre)
The Perpendicular bisector of a chord passes
through the centre of a circle
The angle subtended by an arc at the centre of
the circle is double the size of the angle
subtended by the same arc at the
circumference (angle at centre = 2angle at
circum)

5. Proofs of theorems (Cont ---)
Angles subtended by a chord of a circle on
the same side of the chord are equal
The opposite angles of a cyclic quad are
supplementary (opp angle cyclic quad)
Two tangents from the same point are
equal in length (2 tan to circle)
The angle between tangent & chord = the
angle in the alternate segment (tan-chord
theorem)

6. PARTS OF THE CIRCLE

7. THEOREMS (Sketches)

8. THEOREM 1 (b)

9. ANGLE AT CENTRE THEOREM

10. Point of clarity (1)

11. Point of clarity (2)

12. Cont ---

13. Cont ---

14. Cont ---

15. POINT OF CLARITY

16. A Quadrilateral-Any four-sided figure
A Cyclic Quadrilateral is a four-sided figure
with all its vertices lying on the circumference
of a circle

17. Opposite angles of a cyclic
quad are supplementary

18. Exterior angles of a cyclic quad
(ext angle of cyclic quad)

19. TANGENT

20. Radius is perpendicular to tangent

21. TAN, SAME POINT

22. TAN-CHORD THEOREM

23. Hints in solving problems
Use x and y on the diagram
Mark all equal angles with x’s and y’s and
try to solve the problem
Use colours on the diagram
Mark equal angles with red open dots, red
closed dots, blue stars, etc

24. STEPS IN SOLVING RIDERS

25. General advice for solving
riders
Highlight all parallel lines in the same
colour, to find alternate and corresponding
angles equal.
Use suitable markers to mark angles equal
to each other in the same colour.
Look at what you are required to prove, and
select a strategy to solve the problem.

26. Cont ---
Make a preliminary test to see if your
strategy has a chance of success.
Always re-check the given to make sure
that you have used ALL the given
information.
Look at previous parts of questions for
clues. You may need to further extend the
solutions or conclusions to these
questions.

27. Centre of the circle:
Mark off all radii- and consequently, all base
angles of isosceles triangles formed by radii,
equal.
If given that a chord is bisected- mark off the
90° angle formed by the perpendicular line
from the centre of circle.
For all diameters, mark off angles subtended
at the circumference equal to 90°.
Check each angle at the centre (mark it as
2x) and mark off the angle at the
circumference as x (using x may be
cumbersome).
Mark the 90° angle formed between radius
and tangent, if you are given a tangent.

28. TANGENT
Mark off all radii perpendicular to tangent.
Check if any two tangents come from the
same point, and mark them equal, (also
mark off the base angles of isosceles
triangle formed as a result).
Mark the angle between the tangent and
chord and the angles that they will be
equal to in the alternate segment.

29. CYCLIC QUADRILATERAL
Remember you may not always be told that
quadrilaterals in a given circle are cyclic. You
would have to check the circle in your diagram
to see if there are four or more points on the
circle forming cyclic quadrilaterals.
Mark off all angles in the same segment
equal.
Mark off all exterior angles = to interior
opposite angles.
Remember that opposite angles of cyclic
quad. are supplementary.
Work backwards from the 'required to
prove'

30. If asked to prove two line
segments equal:
If they are in the same triangles try proving
the base angles of that triangle equal (i.e.
the angles opposite the sides you want to
prove equal.)
If they are in two different triangles - check
to see if triangles are congruent.
Check if the angles subtended by the
chords are equal, or if you can prove them
equal.

31. If asked to prove two angles equal
If they are in the same triangle, check to
see if these angles are the base angles of
an isosceles triangle - are the sides equal.
If you are trying to prove ÐA = ÐB, then
find all angles equal to , and all angles
equal to ÐA, and all angles equal to ÐB;
now check if there is an angle equal to
both ÐA & ÐB.

32. USING AL ALGEBRAIC APPROACH
Sometimes you may need to prove an
equation that has some numerical value in
it: e.g. ÐA = 90° – 2ÐB
This may require you to start with an equation
that already has a numerical value: e.g.
Sum of angles of triangle equal to 180°.
Opposite angles of cyclic quad are
supplementary.
Co-interior angles are supplementary.
Manipulate the relevant numerical equation
using substitution of equal quantities to arrive at
the required equation.

33. RIDER 1

34. RIDER 2

35. SOLUTION 1

36. SOLUTION 2

37. Hints in examinations
Proofs of theorems can be asked in
examinations, but their converses cannot
be asked.