7. Triangles Class IX Mathematics

Exercise 7.1

Q.No. 1. In a quadrilateral ABCD, AC = AD and AB bisect <A (see Fig. 7.16). Show that △ABC⩭△ABD. What can you say about BC and BD.

Solution:

Consider about the △ABC and △ABD.

AB = AB {Common}

AC = AD {Given}

<CAB = <DAB {AB bisect <A}

△ABC ⩭ △ABD {Under SAS}

BC = BD.

Q.No. 2. ABCD is a quadrilateral in which AD = BC and <DAB = <CBA (see Fig. 7.17). Prove that

(i) △ABD ⩭ △BAC

(ii) BD = AC

(iii) <ABD = <BAC.

Solution:

(i)

Consider in △ABD and △BAC.

AB = AB {Common}

AD = BC {Given}

<DAB = <CBA {Given}

△ABD ⩭ △BAC. {Under SAS}Proved.

(ii)

BD = AC {△ABD ⩭ △BAC} Proved.

(iii)

<ABD = <BAC {△ABD ⩭ △BAC}Proved.

Q.No. 3. AD and BC are equal perpendicular to a line segment AB(see Fig. 7.18). Show that CD bisect AB.

Solution:

Consider about the △BOC and △AOD.

BC = AD {Given}

<BOC = <AOD {Vertically opposite angles}

<OBC = <OAD { Each right angles}

△BOC ⩭ △AOD {Under ASA}

BO = AO

∴ CD bisect AB Proved.

Q.No. 4. l and m are two parallel lines intersected by another pair of parallel lines p and q (see Fig. 7.19). Show that △ ABC ⩭ △CDA.

Solution:
Consider about the △ABC and △CDA.
<BAC = <DCA {Alternate pair}
AC = AC {Common}
<ABC = <ADC {Opposite angles of a parallelogram}
∴△ABC⩭△CDA. Proved.

Q.No. 5. Line l is the bisector of an angle <A and B is any point on l. BP and BQ are perpendiculars from B to the arms of <A (see Fig. 7.20).
Show that:
(i) △APB⩭△AQB
(ii) BP = BQ or B is equidistant from the arms <A.

Solution:
(i)
Consider about the △APB and △AQB.
AB = AB {Common}
<APB = <AQB {Each right angles}
<PAB = <QAB {l is the bisector of <A}
∴△APB⩭△AQB {Under ASA} Proved.

(ii)
BP = BQ {△APB⩭△AQB} Proved.

Continuous ............................