Question

Solve the problems. Write the complete proof in your paper homework and for online (only) complete the probing statement (if any) that is a part of your proof or related to it. Given: ABCD is a trapezoid, AB = CD, m∠1=m∠2, m∠3=m∠4, AC ∩ BD = O. Prove: m∠AOD = m∠BCD

Answer 1

Hence proved that m∠AOD = m∠BCD.

Explanation:

Given:

A Trapezoid with AC and BD as diagonals with O as intersection point to it.

Four angles formed at A and D are equal.

To Prove:

m∠AOD = m∠BCD

Proof:

Consider a Quadrilateral ABCD as trapezoid with diagonals as AC and BD

(Refer the Attachment).

And AB=CD which means the Quadrilateral ABCD isosceles in nature

i.e. angle A=angle D

Now

Consider triangle AOD and triangle BCD

Apply sum of all angles of triangle is 180.

And also

m∠2+∠AOD+m∠3=180

∠AOD=180-m∠2-m∠3..............equation(1)

And for Triangle BCD

m∠BCD=180-m∠3-m∠4,.....................Equation(2)

∠A=∠D................ as Quadrilateral is Isosceles.

i.e.∠2=∠3

That means by transitive property,

∠1=∠4=∠3=∠2

Using this values in above equation(1) we get ,

∠AOD=180-∠3-∠4.......................(3)

Now comparing equation 2 and 3 we get

RHS is equal for both equation that means LHS is also equal .

Therefore ∠AOD=∠BCD

Hence proved.