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In quadrilateral ABCD, AB is the shortes...

In quadrilateral ABCD, AB is the shortest side and DC is the longest side. Prove that :
(i) `angleB gt angleD`
(ii) `angleA gt angleC`

Text Solution

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(i) join BD.
In `DeltaABD`
`A gt AB" "( :' AB" is smallest side")`
`implies" "angle ABD gt angleADB` ...(1)
In `DeltaBCD`
`CD gt BC" "( :' CD" is largest side")`
`angleCBD gt angleBDC` ...(2)
Adding equations (1) and (2)
`angleABD+angleCBD gt angleADB+angleBDC`
`angleABC gt angleADC`
`implies" "angleB gt angleD`
(ii) Join AC
In `DeltaABC`
`BC gt AB" "( :' AB" is smallest side")`
`angleBAC gt angleACB` ...(1)
In `DeltaACD`
`CD gt AD" "( :' CD" is largest side")`
`implies" "angleCAD gt angleACD` ...(2)
Adding equations (1) and (2)
`angleBAC+angleCAD gt angleACB+angleACD`
`implies" "angleBAD gt angleBCD`
`implies" "angleA gt angle C`
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