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In Fig. 7.8, O A\ =\ O Ba n d\ O D\ =\ O...

In Fig. 7.8, `O A\ =\ O B``a n d\ O D\ =\ O C`. Show that `(i) DeltaA O D~=DeltaB O C ` and `(ii) A D\ ||\ B C `.

Text Solution

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(1) Given :`OA = OB`And `OD = OC`
To Prove: `triangleAOD cong triangleBOC`
Proof : Lines `CD` and `AB` intersect. `angleAOD = angleBOC` (Vertically Opposite Angles)
In `triangleAOD` and `triangleBOC`,`OA = OB` (Given)
`angleAOD = angleBOC` (Vertically Opposite Angles)
`OD = OC ` (Given)
Therefore, `triangleAOD cong triangleBOC` (By SAS Congruence)----- (1)
(2) To prove`AD ║ BC`
Proof :Since `triangleAOD cong triangleBOC` {From (1)}
`angleOAD = angleOBC` { CPCT }
`angleOAD and angleOBC` for the pair of Alternate angles.
If a transversal intersects two lines such that the pair of Alternate interiors Angles are equal, then the Lines are Parallel.
Therefore, `AD ║ BC` .
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