Download MS2013 Review - Main Theorems These are the main theorems we

MS2013 Review - Main Theorems
These are the main theorems we studied so far. Aim to know all the definitions of the
notions featured in these theorems, as well as the statements of the theorems, by heart. Be
prepared to reproduce a proof or part of a proof of any of these theorems.
1. Euclid’s 5 postulates :
Postulate (1) A unique straight line segment can be drawn joining any two distinct points.
Postulate (2) Any straight line segment is contained in a unique straight line.
Postulate (3) Given any straight line segment, a unique circle can be drawn having the segment as
radius and one endpoint as center.
Postulate (4) All right angles are congruent.
Postulate (5) Given a point not on a given line, there exists a unique line through that point parallel
to the given line.
Theorem 1.1. Parallel lines Two distinct lines crossed by another line at the points P and
Q are parallel ⇐⇒ the following equivalent properties hold
(A) Two alternate angles at P and Q are equal;
(B) Two corresponding angles at P and Q are equal.
(C) Two interior consecutive angles add up to 180◦
Theorem 1.2. Angles of a Triangle The sum of all the interior angles of a triangle is 180◦ .
Theorem 1.3. Cases Sufficient for Congruent Triangles
(A) S.A.S. (Two pairs of sides and the angles between them are equal, respectively).
(B) A.A.S. (Two pairs of angles and a pair of sides are equal, respectively).
(C) S.S.S. (Each of the three pairs of sides are equal).
Theorem 1.4. Isosceles Triangle Consider a triangle ABC. The following two statements
are equivalent:
(A) △ABC is isosceles with |AB| = |AC|.
(B) B̂ = Ĉ.
Theorem 1.5. Parallelogram Let ABCD be a quadrilateral. The following statements are
equivalent:
(A) ABCD is a parallelogram.
(B) One pair of opposites sides are parallel and equal.
(C) The diagonals AC and BD intersect at their midpoint.
(D) Opposite sides are equal.
(E) Opposite angles are equal.
Theorem 1.6. Rectangle Let ABCD be a quadrilateral. The following statements are equivalent:
(A) ABCD is a rectangle (i.e. parallelogram with a right angle).
(B) The diagonals AC and BD intersect at their midpoint and |AC| = |BD|.
1
2
Theorem 1.7. Rhombus Let ABCD be a quadrilateral. Show that the following statements
are equivalent:
(A) ABCD is a rhombus (i.e. parallelogram with equal sides).
(B) The diagonals AC and BD intersect at their midpoint and AC ⊥ BD.
⌢
Lemma 1.8. (Angle on a circle.) Let A, B, C be three points on a circle. Let BC denote the
arc which does not contain A. Then
⌢
\
BOC
BC
[ =
BAC
=:
.
2
2
Theorem 1.9. Cyclic quadrilateral Let ABCD be a quadrilateral. The following are equivalent:
(A) The quadrilateral ABCD is cyclic.
(B) The angle formed by a diagonal with a side is equal with that formed by the other diagonal
with the opposite side.
(C) The sum of two opposite angles is 180◦ .
Theorem 1.10. The circumcircle of a triangle. The perpendicular bisectors of the sides
in a triangle ABC intersect at a point O, which is the centre of the unique circle containing the
vertices of the triangle. The point O is called the circumcentre of the triangle, and the circles
is called the circumcircle of the triangle.
Example 1.11. The circumcentre of a right angled triangle is always the midpoint of the hypothenuse.
Proposition 1.12. Midline. A midline in a triangle is parallel to, and equal to half of, the
side opposite to it.
MS2013 – Sample Test 1
Please include all steps of the proofs. Please justify all statements.
1.(a) [ 10 marks] All the properties of a parallelogram.
(b) [ 20 marks] Prove that a quadrilateral whose diagonals bisect each other is a parallelogram.
(c) [ 20 marks] If all the vertices of a parallelogram are on a circle, what can you say about
that parallelogram? Justify your answer.
2. In △ABC, consider the line AD ⊥ BC where D is on the line BC. Let M, N and P be the
midpoints of the sides AB, AC and BC respectively.
a) [ 10 marks] Prove that |MD| = 12 |AB|.
b) [ 20 marks] Prove that MNP D is an isosceles trapezium.
c) [ 20 marks] Prove that there is a circle passing through the vertices of MNP D.