Chapter 5Introduction to Deductive Geometry 5.1

Chapter 5Introduction to Deductive

Geometry

Warm-up Exercise

1.In each of the following figures, AOBis a straight line. Find the unknowns.

(a)(b)(c)

2.Find the unknowns in each of the following figures.

(a)(b)(c)

3.In each of the following figures, AOB, COD and EOFare straight lines. Find the unknowns.

(a)(b)(c)

4.In each of the following, write down the congruent triangles and state the reason.

(a)

(b)

Build-up Exercise

[ This part provides two extra sets of questions for each exercise in the textbook, namely Elementary Setand Advanced Set. You may choose to complete any ONE set according to your need. ]

Exercise 5A

Elementary Set

Level 1

1.To complete each of the following proofs, fill in the blanks with suitable theorem.

(a)Prove that AOC is a straight line.

Proof:

AOBBOC9090(given)

180

AOCis a straight line.()

(b)In the figure, AOD is a straight line. Prove that x60.

Proof:

xxx180( )

3x180

x60

2.In the figure, express AOCin terms of y.

3.In the figure, AOB is a straight line. Express y in terms of x.

4.In the figure, express y in terms of x.

5.In the figure, AOB and CODare straight lines. Express y in terms of x.

6.In the figure, prove that AOB is a straight line.

7.In the figure, if xy90, prove that AOB is a straight line.

8.In the figure, COD is a straight line.Prove that AOB is a straight line.

9.In the figure, AOB is a straight line. Prove that COD is a right angle.

10.In the figure,AOBCOD. Ifxy180,prove that AOB is a right angle.

Level 2

11.In the figure, prove that AOB is a straight line.

12.In the figure, AOB is a straight line. Prove that OC is an angle bisector of BOD.

13.In the figure, AOBis a straight line. If BOC is 3 times the size of AOD,

(a)find a;

(b)prove that DOC is a right angle.

Solution:

(a)2a 2a45

45

a

(b)a

AOD

BOC

DOC180()

DOC90

DOC is a right angle.

14.In the figure, prove that AOB is a right angle.

Proof:

AOCBOCreflexangleAOB360()

 360

a

2aa

90

AOB is a right angle.

Advanced Set

Level 1

1.In the figure, AOB and COD are straight lines. Express BOD in terms of x and y.

2.In the figure, AOB is a straight line. Express y in terms of x.

3.In the figure, expressBOC in terms of x.

4.In the figure, AOF, EOD and BOCare right angles. Prove that a30.

5.In the figure, AOBand COD are straight lines. If xyz180, prove that EOFis a straight line.

6.In the figure, if ab90, prove thatAOB is a straight line.

7.In the figure, if xy90, prove that AOB is a straight line.

Level 2

8.In the figure, prove that AODis a right angle.

9.In the figure, DOF is a straight line. If AOC is a right angle, prove that BOCis 3 times the size of EOF.

10.In the figure, prove that AOCis a right angle.

11.(a)Find x as shown in the figure.

(b)Prove that AOC and BODare straight lines.

(c)Prove that ACBD.

Exercise 5B

Elementary Set

Level 1

1.In the figure, BADDCB. ABDand CDB are right angles.

(a)Prove that ABDCDB.

(b)Prove that ADBC.

2.In the figure, ACBD and BCCD.

(a)Write down a pair of congruent triangles with a proof.

(b)Prove that ABAD.

3.In the figure, ABAD and BCDC. Prove that BACDAC.

4.In the figure, PQTS. QPTand STPare right angles. Prove that PSTQ.

5.In the figure, APOSB and CROQD are straight lines.OPOS and OQOR. Prove that PQRS.

6.In the figure, BCDE is a straight line. ABAE, BCED and BACEAD. Prove that ACDADC.

7.In the figure, BACCDBandACBDBC.Prove that ACDB.

8.In the figure, AC and BD intersect at E.ABADDC and ABDDCA. Prove that EAED.

Proof:

Let ABDx,

then DCAx.

ADAB(given)

ADB ()

x

ADCD()

DAC ()

x

EADEDA

EAED()

9.Construct an isosceles right-angled triangle with a proof where its two equal sides are equal to the given line segment x.

10.In the figure, prove that the perimeter of quadrilateral ABCDis greater than that of ABD.

11.In the figure, ABCD is a straight line. EBEC. Prove that xy.

Proof:

EBCx180()

EBC

ECBy180()

ECB

EBEC(given)

EBCECB()

180x

xy

Level 2

12.In the figure, AED and BDC arestraight lines andAC=AB. Prove that ACEABE.

13.In the figure, AFC, AHD, BFGD and CGHE are straight lines. ABAE,BCED and ABCAED90.

(a)Prove that ABCAED.

(b)Prove that ACD is an isosceles triangle.

(c)Prove that BCDEDC.

14.In the figure, ABBCCDAD. AC and BD intersect at E.

(a)Prove that ABDCDB.

(b)Prove that ABCADC.

(c)Prove that ABECDE.

(d)Prove that BEDE and AECE.

15.In the figure, ABCDE is a regular pentagon.Given that all sides of a regular polygon are equal and all its interior angles are the same in size.

(a)Prove that ABCAED.

(b)Prove that ACD is an isosceles triangle.

16.In the figure, E is a point in convex quadrilateral ABCD. Prove thatEAEBECED(ABBCCDAD).

17.In the figure, BCD is a straight line. If ACBACD, prove that AB is the longest side of ABC.

Advanced Set

Level 1

1.In the figure, ACBD. ABC and DCB are right angles.

(a)Prove that ABCDCB.

(b)Hence, prove that ABDC.

2.In the figure, ABCDCB andBACCDB.Prove that ACDB.

3.In the figure, CDEB is a straight line.ADEAED and CDBE. Prove that ACAB.

4.In the figure, BDC is a straight line.BADCAD. Prove that ABC is an isosceles triangle.

5.In the figure, PQPR. OP is an angle bisector of QPR. Prove that OQRORQ.

6.In the figure, AOBand COD are straight lines. Given that OCAC and BDOD, prove that OACOBD.

7.In the figure, prove that the perimeter of quadrilateral ABCD is greater than 2BD.

8.Construct an isosceles triangle with base a and base angle x with a proof.

Level 2

9.In the figure, ABCDE is a pentagon where ABBCCDDEEA.ACDADC.ProvethatABCAED.

10.In the figure, ABCD is a quadrilateral. AC and BD intersect at E. ADBC and DACCBD.

(a)Prove that DECE.

(b)Prove that BDCACD.

11.In the figure, AHB, DGE, FBEC, FHID and AIGC are straight lines. AB and DE are perpendicular to FC. FBEC and FDAC.

(a)Prove that ABCDEF.

(b)Prove that FBHCEG.

(c)Prove that HAGD.

12.In the figure, ABAD and BCDC. AC and BD intersect at E.

(a)Prove that ABCADC.

(b)Prove that ABEADE.

(c)Prove that AC bisects BD.

13.In the figure, ABAD and CBCD. Prove that ABCADC.

14.In the figure, F and D are the points on AB and AC respectively. BED and CEF are straight lines. ABAC and ADAF. Prove that EDEF.

15.In the figure, ABC is a scalene triangle. D is a point on BC.Prove that 2ADABACBC.

16.In the figure, ABCD is a quadrilateral. Diagonals AC and BD intersect at E. Prove that.

17.ConsiderABC as shown in the figure. Construct DEF with a proof where , and .

18.Construct an isosceles triangle with height h and two equal sides of length a with a proof.

Chapter Test / (Time allowed: 1 hour)

SectionA(1) [3 marks each]

1.In the figure,AOB is a straight line. If c3a and b2a, prove that c90.

2.In the figure, AOB, COD and EOF are straight lines. ABEF. Prove that abc.

3.In the figure, AOB and COD are straight lines. Prove that ABCD.

4.In the figure, ABCD is a quadrilateral. AC and BD intersect at O. ADAB and DACBAC. Prove that DOOB.

5.In the figure, PQRS is a quadrilateral. PTST,PQSRandQPTRST90. Prove that TQRTRQ.

6.In the figure, ABCis an acute-angled triangle.D is a point on AC such that BABD. Prove that BDCBDA.

SectionA(2) [6 marks each]

7.In the figure, VZYW is a straight line. UVWXWV and UYXZ. XZWand UYVare right angles.

(a)Prove that VZWY.

(b)Hence, prove that VZUWYX.

8.In the figure, ABCDCE. BCE is a straightline. BCCEAD. Let BACa,ABCb and ACBc.

(a)Prove that ABCCDA.

(b)Hence, prove thatDABABC180.

9.In the figure, ADB, AFC, BEF and CED are straight lines. AE is an angle bisector of BAC. ABFACD.

(a)Prove that ABEACE.

(b)Prove that ADEAFE.

10.In ABC as shown in the figure, BD bisects ABC and CD bisects ACB. Prove that BDCBAC.

Section B

11.In the figure, BCDE is a straight line. CAD is an isosceles triangle where ACAD. BACEAD.

(a)Prove that BACEAD.(4 marks)

(b)Prove that ABAE.(2 marks)

(c)Prove that.(7 marks)

Multiple Choice Questions [3 marks each]

Chapter 5Introduction to Deductive Geometry 5.1

12.In the figure, ab180. Which of the following must be true?

A.ab

B.ba

C.a is an obtuse angle, bis an acute angle.

D.AOBis a straight line.□

13.In the figure, AOB is a straight line. Which of the following must be true?

A.ab180

B.ab90

C.b3090

D.a6090□

14.In the figure, AOB and CODarestraight lines. Which of the following must be true?

A.acbd

B.abcd360

C.ac270

D.bd180□

15.Express CODin terms of a and b as shown in the figure.

A.90ab

B.180ab

C.270ab

D.360ab□

16.In the figure, AOC and BOD are straight lines. ADCD and ACBD. Which of the following is not necessarily true?

A.OAOB

B.DACDCA

C.OAOC

D.ABCB□

17.In the figure, PST, TUQ and SUR are straight lines.TUQ bisects PQR. RUQ65,TSU50,RURQ.Whichof the following must be true?

A.TUSTQP

B.TSUQRU

C.TURRQP

D.SPQPQR180□

18.Whichof the following is/are the minimum information required to prove that ACBDCB?

I.ABDB

II.ABCDBC

III.ACCD

A.I only

B.II only

C.I and II only

D.I, II and III□

19.In order toprove that OBCOCB, which of the following must be true?

I.CAOBAO

II.CBAC

III.ACAB

A.I only

B.II only

C.I and II only

D.I and III only□

20.In the figure, which of the following must be true?

A.ABACBC

B.ABAC

C.ABACBC

D.AB=ACBC□

21.In the figure, the perimeter of PQRshould be

A.greater than 2(PRQR).

B.less than 2PR.

C.less than 2QR.

D.less than 2(PQQR).□

22.In the figure, S is a point on QR. Which of the following must be true?

A.PSPQQRSR

B.PSPQQRSR

C.PQQRPSSR

D.PQQRPSSR□

23.In ABC as shown in the figure, ABAC. D and E are two points on AC and BC respectively. AEBC. AE and BD intersect at F. Which of the following is not necessarily true?

A.ACEABE

B.ABDDBC

C.AFBACE

D.BFADFA□

24.In the figure, D and Eare two points on BC.Which of the following must be true?

A.ACAEEC

B.2(ADAE)BCABAC

C.ABACBC2(ADAE)

D.ABACBCAEADDE□

25.Which set of the following lengths of line segments can form a triangle?

A.1, 2, 3

B.7, 8, 10

C.2, 4, 8

D.3, 7, 11□

26.In the figure, ACBABCBAC.Which of the following must be true?

I.BCAC

II.

III.ABis the longest side.

A.Ionly

B.III only

C.I and III only

D.II and III only□