It is given that OB = OC, and OA = OD. Show that AB is parallel to CD. Class 7

It is given that OB = OC, and OA = OD. Show that AB is parallel to CD. Class 7

Question 1.

It is given that OB = OC, and OA = OD. Show that AB is parallel to CD. [Hint: AD is a transversal for these two lines. Are there any equal alternate angles?] Class 7

Solution:

In ∆AOB and ∆DOC,

OA = OD (Given)

OB = OC (Given)

and ∠AOD = ∠DOC (Vertically opposite angles)

So, ∆AOB ≅ ∆DOC (SAS condition)

Hence, ∠OAB = ∠ODC

and ∠OBA = ∠AC A

As ∠OAB = ∠ODC

are alternate angles so, AB is parallel to CD.

Question 2.

ABCD is a square. Show that ∆ABC ≅ ∆ADC. IS ∆ABC also congruent to ∆CDA? Class 7

Give more examples of two triangles where one triangle is congruent to the other in two different ways, as in the case above. Can you give an example of two triangles where one is congruent to the other in six different ways?

Solution:

In ∆ABC and ∆ADC,

AB = AD

BC = DC

and AC = AC

So, ∆ABC ≅ ∆ADC (SSS condition)

Yes, ∆ABC is congruent to ∆CDA also. Other

triangles may be two isosceles triangles ABC and DEF, in which AB - AC = DE = DF and BC = EF. Here, ∆ABC = ∆DEF and also ∆ABC ≅ ∆DFE.

Examples of congruency in six ways :

Two equilateral triangles ABC and DEF, with AB = BC = CA = DF = EF = FD.

Here ∆ABC ≅ ∆DEF,

∆ABC ≅ ∆EDF,

∆ABC ≅ ∆FDE,

∆ABC ≅ ∆DFE, ∆ABC ≅ ∆EFD and

∆ABC ≅ ∆FED.