Chapter 7: Triangles

EXERCISE 7.1

1. In quadrilateral ABCD , AC=AD and AB bisects ∠A (see Fig. 7.16) .Show that ∆ABC≅∆ABD . What can you say about BC and BD ?

Solution : Given, ABCD is a quadrilateral , AC=AD and AB bisects ∠A . Then show we that

 ∆ABC≅∆ABD 

Proof: Since ,  AB bisects ∠A .

So, ∠BAC=∠BAD

In ∆ABC and ∆ABD , we have 

 AC=AD    [Given ]

 ∠BAC=∠BAD  [Given]

 AB=AB [Common side]

∆ABC≅∆ABD  [SAS  rule]

 BC=BD  [CPCT]

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 : Given, ABCD is a quadrilateral , AD=BC and ∠DAB=∠CBA .

    To prove  (i) ∆ABD≅∆BAC

                       (ii) BD=AC

                      (iii)  ∠ABD=∠BAC

Proof : (i) In ∆ABD and ∆BAC , we have

AB=AB  [ Common side]

 ∠DAB=∠CBA  [Given]

 AD=BC [Given]

 ∴  ∆ABD≅∆BAC [SAS rule]   Proved

(ii)  In ∆ABD and ∆BAC , we have

AB=AB  [ Common side]

 ∠DAB=∠CBA  [Given]

 AD=BC [Given]

 ∴  ∆ABD≅∆BAC [SAS rule]

      BD=AC  [CPCT]   Proved

(iii)  In ∆ABD and ∆BAC , we have

AB=AB  [ Common side]

 ∠DAB=∠CBA  [Given]

 AD=BC [Given]

 ∴  ∆ABD≅∆BAC [SAS rule]

 ∠ABD=∠BAC  [CPCT] Proved

3. AD and BC are equal perpendicular to a line segment AB (see Fig. 7.18) . Show that CD bisects AB .

Solution : Given, AD and BC are equal perpendicular to a line segment AB . Then we show that CD bisects AB .

Proof : In ∆BOC and ∆DOA , we have

BC=AD  [Given]

 ∠OBC=∠OAD=90° [ given]

 ∠BOC=∠DOA [Vertically opposite angle]

∆BOC≅∆DOA [ASA rule]

  OB=OA  [CPCT]

   So,  CD bisects AB .    Proved