In a trapezium ABCD, sides AB and DC are parallel to each other. E is mid-point of AD and F is mid-point of BC. Prove that : AB + DC = 2EF.

Join BE and produce to meet CD produced at point P. In △ PDE and △ BAE, ⇒ ∠PED = ∠BEF (Vertically opposite angles are equal) ⇒ AE = ED (Since, E is the mid-point of AD) ⇒ ∠EDP = ∠EAB (Alternate angles are equal) ∴ △ PDE ≅ △ BAE (By A.S.A. axiom) We know that, Corresponding parts of congruent triangle are equal. ∴ BE = EP and AB = PD. In △ BPC, Since, E and F are mid-points of sides BP and BC respectively. ∴ EF = . To prove : AB + CD = 2EF ........(1) Substituting value in L.H.S. of equation (1), we get : ⇒ AB + CD = PD + CD = PC. Substituting value in R.H.S. of equation (2), we get : ⇒ 2EF = = PC. Since, L.H.S. = R.H.S. Hence, proved that AB + CD = 2EF.

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In parallelogram ABCD, E is the mid-point of AB and AP is parallel to EC which meets DC at point O and BC produced at P. Prove that :
(i) BP = 2AD
(ii) O is mid-point of AP.

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In △ ABC, AD is the median and DE is parallel to BA, where E is a point in AC. Prove that BE is also a median.

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Mathematics

In a trapezium ABCD, sides AB and DC are parallel to each other. E is mid-point of AD and F is mid-point of BC.
Prove that :
AB + DC = 2EF.

Mid-point Theorem

Related Questions

In parallelogram ABCD, E is the mid-point of AB and AP is parallel to EC which meets DC at point O and BC produced at P. Prove that :
(i) BP = 2AD
(ii) O is mid-point of AP.

In △ ABC, AD is the median and DE is parallel to BA, where E is a point in AC. Prove that BE is also a median.

Mathematics

In a trapezium ABCD, sides AB and DC are parallel to each other. E is mid-point of AD and F is mid-point of BC.Prove that :AB + DC = 2EF.

Mid-point Theorem

Related Questions

In parallelogram ABCD, E is the mid-point of AB and AP is parallel to EC which meets DC at point O and BC produced at P. Prove that :(i) BP = 2AD(ii) O is mid-point of AP.

In △ ABC, AD is the median and DE is parallel to BA, where E is a point in AC. Prove that BE is also a median.

In a trapezium ABCD, sides AB and DC are parallel to each other. E is mid-point of AD and F is mid-point of BC.
Prove that :
AB + DC = 2EF.

In parallelogram ABCD, E is the mid-point of AB and AP is parallel to EC which meets DC at point O and BC produced at P. Prove that :
(i) BP = 2AD
(ii) O is mid-point of AP.