AnSWErS to tHE StudEnt'S Book ExErCiSES - Hodder Plus Home

4 a) (i) Angle QPX 5 angle XRS(Alternate angles.)Angle PQX 5 angle RSX(Alternate angles.)PQ 5 SR (Given.)So triangle PQX is congruent to triangleRSX (ASA).(ii) Angle QRX 5 angle SPX(Alternate angles.)Angle RQX 5 angle PSX(Alternate angles.)QR 5 SP (Given.)So triangle RQX is congruent to triangleSPX (ASA).b) SX 5 XQ -(Since triangles PQX and RSX are congruent.)PX 5 XR(Since triangles RQX and SPX are congruent.)So X is the midpoint of SQ and PR.SR 5 RQ (PQRS is a rhombus.)Angle RSX 5 angle RQX(Triangle QSR is isosceles.)XR is common to both triangle SXR andtriangle QXR.So triangle SXR is congruent to triangleQXR (SAS).So angle SXR 5 angle QXR.Angle SXR 5 angle QXR 5 90°(Since SQ is a straight line.)5 A and B6 AP 5 PB 5 BQ 5 QC 5 CR 5 RD 5 DS 5 AS(Because P, Q, R and S are the midpoints of thesides of the square ABCD)So all the triangles are isosceles and congruentand all the marked angles are 45°.SAPBQD R CSo the interior angles of the quadrilateral PQRSare 90°(Angles on a straight line.)All the sides, PS, SR, RQ and QP, are equal.So PQRS is a square.7 Angle DCB 5 angle DCA(CD bisects angle ACB.)Angle DCA 5 angle CAE(Alternate angles.)Angle DCB 5 angle AEC(Corresponding angles.)So angle AEC 5 angle CAESo triangle ACE is isosceles and AC 5 CE.8 Angle ACB 5 angle ABC 5 70°(Base angles of an isosceles triangle are equal.)Angle ACD 5 40°(Alternate angles.)So angle BCD 5 110°So angle DBC 5 angle BDC 5 35°(Base angles of an isosceles triangle are equal.)9 Angle BCD 5 122°(Alternate angles.)Angle DCE 5 58°(Angles on a straight line add up to 180°.)Angle BCY 5 58°(Vertically opposite angles are equal.)Angle ABC 5 58°(Angles on a straight line add up to 180°.)Angle BAD 5 89°(Angle sum of triangle BAE.)10 Angle BDC 5 80°(Base angles of an isosceles triangle are equal.)So angle DBC 5 20°(Angles in a triangle add up to 180°.)So angle ABC 5 40°(BD bisects angle ABC.)So angle BAC 5 60°(Angle sum of triangle BAC.)11 Angle BRQ 5 125°(Angles in a quadrilateral add up to 360°.)Angle BRP 5 55°(Angles on straight line add up to 180°.)So a 5 145°(Exterior angle of triangle is equal to the sum ofthe opposite, interior angles.)Mixed exercise 37 (page 505)1 x 5 22° (Alternate angles.)y 5 26° (Base angles of an isosceles triangle(PRS) are equal and Angles in atriangle add up to 180°.)z 5 48° (Alternate angles.)2 x 5 70° (Angles in a triangle add up to 180°.)y 5 60° (Alternate angles.)z 5 130° (Vertically opposite angles.)3 Angle DCB 5 80°(Angles in a quadrilateral add up to 360°.)So angle ACB 5 angle ACD 5 40°(Given that CA bisects angle DCB.)Angle DAC 5 10°(Angles in a triangle add up to 180°.)So angle CAB 5 70°(80° – 10°.)Since CAB 5 ABC, triangle CAB is isosceles.4 a 5 87° (Allied angles.)b 5 74° (Alternate angles.)5 x 5 100°(Angles in a quadrilateral add up to 360°.)234Answers to the Student’s Book exercisesHigher WJEC GCSE Mathematics © 2010, Hodder Education