Q3 of 23 Page 187

In the adjacent figure ABCD is a parallelogram and E is the midpoint of the side BC. If DE and AB are produced to meet at F, show that AF = 2AB.


Given :- ABCD is a parallelogram


And CE=EB


Formula used:- ASA congruency property


If 2 angles and one side between them in two


triangles are equal then both triangle are congruent


Solutions :-


In Δ DCE and Δ FBE


∠ DEC=∠ BEF [Vertically opposite angles]


As DC || BF


∠ ECD=∠ EBF [Alternate angles]


And;


CE=EB [Given]


∴ Δ DCE ≅ Δ FBE


As Δ DCE ≅ Δ FBE


Then


DC=BF


And DC=AB [opposite sides of parallelogram are equal]


Hence ;


AB=BF


AF=AB+BF


AF=2AB.


Hence Proved;


More from this chapter

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1

The opposite angles of a parallelogram are and Find the measure of each angle of the parallelogram.

 2

Find the measure of all the angles of a parallelogram, if one angle is 24° less than the twice of the smallest angle.

 4

In the adjacent figure ABCD is a parallelogram P, Q are the midpoints of sides AB and DC respectively. Show that PBCQ is also a parallelogram.


 5

ABC is an isosceles triangle in which AB = AC. AD bisects exterior angle QAC and CD || BA as shown in the figure. Show that

(i) ∠DAC = ∠BCA


(ii) ABCD is a parallelogram