4.In the diagram, ABCDis a trapezium with AB = CDProve that (i) BADis congruent to CDA(ii) . . , , s) ) BC) ) .
24 5.In the diagram, BCDEis a rectangle. AEDis isosceles with AD = AE. AFBand AGCare straight lines. Show that (i) ABEis congruent to ACD, (ii) AEFis congruent to ADG. (Pg256Q13)Proof(i)AED = ADE (base s of isos. AED) BEF = CDG = 90(s of rectangle) AED+ BEF = ADE + CDGHence, AEB = ADC.AE = AD (given) BE = CD (opp sides of rectangle) Hence ABEis congruent to ACD. (SAS) (ii) EAF = DAG(corr s of congruent s) AE = AD (given) AEF = ADG (base s of isos ) Hence AEFis congruent to ADG. (ASA) 6.In the diagram, ABCDand, CDEFare parallelograms. The point Xis the mid-point of ACand of BD, and the point Y is the mid-point of CEand of DF. Prove that (i) AEis parallel to XY, (ii) AEis parallel to BF, (iii) BAEFis a parallelogram. (Pg256Q14)Proof(i) AE// XY(Mid-point Theorem) (ii) XY// BF(Mid-point Theorem) Hence, AE// XY// BF. (iii) AB// CDand CD// EFHence, AB// EF. BAEFis a parallelogram. sgexamguru.com
25 7.In the diagram below, ABCDis a quadrilateral. AE = BEand BF = FC. The diagonal ACintersects DEat X and DFat Y, so that AX = XY = YC. Prove that (i) BXDYis a parallelogram, (ii) AXBis similar to CYD. (iii) ABCDis a parallelogram.(Pg270Q5)Proof(i) EX// BY(Mid-point Theorem) FY// BX(Mid-point Theorem) Hence, BXDYis a parallelogram. (ii)AXB = XYF(corr. s, BX// FY) = CYD(vert. opp. s) AX = CY(given) BX = DY(opp. sides of parallelogram)AXBis congruent to CYD. (SAS)(iii)BAX = DCY. (corr s of congruent s) AB// CD(alt s are equal) AB = CD(corr sides of congruent s) ABCDis a parallelogram. sgexamguru.com
26 Singapore Chinese Girls’ SchoolSecondary 4 Additional Mathematics Plane Geometry Name: ( ) Class: Sec 4_____ Date: ________________ Worksheet 7: Angle Properties of Circles Some Angle Properties of Circles PropertyAbbreviation 1.An angle in a semicircle is a right angle. in a semicircle 2.An angle at the centre is twice any angle at the circumference.at centre = 2at ce3.Angles in the same segment are equal.s in the same segment 4.The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.Ext of a cyclic quad 5.Angles in opposite segments are supplementary.x+ y= 180s in opp segments (or opp s of a cyclic quad) sgexamguru.com
27 PropertyAbbreviation 6.A tangent of a circle is perpendicular to the radius drawn to the point of tendency. OAB = 90tan rad 7.If PQand PRare two tangents to a circle centred at O, then PQ= PR OPQ = OPRtan from an ext pt The Alternate Segment Theorem (Tangent-Chord Theorem) An angle between a tangent, ATBand a chord, TPthrough the point of contact, T, is equal to the angle in the alternate segment.