ACE-Learning Lesson Brief
Simple Examples in Geometric Proof Introduction: The following concept maps summarise properties of triangles, circles and cyclic quadrilaterals. Make sure that you are very familiar with these properties before you do geometric proofs.
Equal corresponding angles
Similar s
have
Proportional correspond sides
Triangles
can be grouped into
Equal corresponding angles
Congruent s
have
Equal corresponding sides
Perpendicular bisector of a chord passes through centre
Symmetric properties Circles
have
consist of
Equal chords are equidistant from centre Tangents from an external point are equal in length The line joining the external point to the centre bisects the angles between the tangents
Angle properties
consist of
Angle at centre is twice angle on circumference
Angle in a semicircle is a right angle
Angles in the same segment are equal
Angles in opposite segments are supplementary
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Angle between the radius and the tangent is a right angle
1
ACE-Learning Lesson Brief Opposite angles are supplementary
Cyclic quadrilaterals
have the property that
The exterior angle is equal to the interior opposite angle
Congruent triangles Two triangles are congruent if all the corresponding sides are equal, all the corresponding angles are equal. In short, congruent triangles match identically in every aspect. Congruency Criteria: 1. SSS 2. SAS
3. ASA or AAS
4. RHS
Similar triangles Similarity Criteria: Two triangles are similar if all the corresponding angles are equal, all the ratios of the corresponding sides are equal.
1. All corresponding angles are equal.
A=P , B =Q , C =R
2. All corresponding sides are proportional.
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ACE-Learning Lesson Brief
AB BC CA = = PQ QR RP
Circles
Symmetric properties
Perpendicular bisector of chord passes through the centre. OC is the perpendicular bisector of a chord AB. OC AB AC = CB
Equal chords are equidistant from the centre. In the same circle, chords AB and CD of the same length are equidistant from the centre O. AB = CD OM = ON
Tangents from an external point to a circle are equal in length. The two tangents PA and PB are drawn to a circle from external point P where A and B are the points of tangency. PA = PB
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ACE-Learning Lesson Brief
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ACE-Learning Lesson Brief
The line joining the external point to the centre bisects the angle between the 2 tangents.
APB is the angle between the 2 tangents PA and PB. The line PO is the bisector of APB .
APO = BPO
Note that OP also bisects the angle between two the 2 radii OA and OB.
AOP = BOP
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ACE-Learning Lesson Brief
Circles
Angle properties
Angle at centre is twice the angle at the circumference. is an angle at the centre of a circle. subtended by the same arc.
POQ = 2 PAQ POQ PAQ
is an angle at the circumference
Angle in a semicircle is a right angle.
B
AC is a diameter of a semicircle. ABC =90 .
A C
Angles in the same segment are equal.
Both CAD and CBD are the same segment. CAD =CBD
D C
Angles in opposite segments are supplementary.
A and B are in opposite segments.
A + B = 180
Angle between the radius and the tangent is a right angle.
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ACE-Learning Lesson Brief
B is a point of tangency. Tangent AC is perpendicular to the radius OB.
OBC =90
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ACE-Learning Lesson Brief
Cyclic quadrilaterals Angles in opposite segments are supplementary. The angles in opposite segments are supplementary. a + c = 180o b + d = 180o
The exterior angle is equal to the interior opposite angle. i e is an exterior angle of a cyclic quadrilateral. i is the interior opposite angle relative to e. e=i
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ACE-Learning Lesson Brief
Practice Questions: 1. In the diagram, O is the centre of the circle, PT is parallel to QS and TQ = TS. Prove that
1 POQ, 2 PQ (b) T and RST are congruent, (a) RST =
(c) PQRT is an isosceles trapezium. 2. In quadrilateral ABCD, a line from B parallel to CD is drawn to meet a line from C parallel to AB at E. Prove that (a) ABE =ECD, (b) ABE is similar to ECD , (c) AB ED = AE EC . 3. In the figure below, PQRS is a trapezium where PS // QR. Straight lines drawn from P and Q intersect at T, the midpoint of SR. At Y, a line perpendicular to QR, passes through T to meet PS produced at X. Prove that (a) T is the midpoint of XY, (b) Area PTQ =
1 area of trapezium PQRS. 2
4. In the figure, quadrant ABC has a square JBKL inscribed in it. JK is the diagonal of the square. Prove that (a) JK2 = BC2 (b) Area of the square JBKL =
1 BC 2 2
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