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## Answers

Constrution. Join OA.

Given:-

O is midpoint of AC

this imply OA=OC (Midpint divides lig segment in equal halfs)

now,In Trianle ABC abd OBC

Angle "A" = Angle "A"( common)

Anglle "C" = Angle "C"(common)

OA = OC ( equal).

Now,

__BC__=

__AB__= BC

__OB BC__

BC

BC

__now,__

__AB * Bc = Bc* Ob__

or

AB = OB

or

AB = OB

......

now draw a line parallel to BC.

now draw another line parallel to AB.

let us name the point at which these two lines intersect each other as D.

join A and D.

now the quadrilateral ABCD becomes a rectangle since its opposite sides(AD,BC and AB,DC) are equal and are parallel to each other.

now let us consider the two triangles,BCD and BAD.

here,

angle BAD = angle BCD.

AD = BC.

CD = BA.

by SAS congruency rule, we can conclude that the triangles BCD and BAD are congruent to each other.

we know that for any two congruent triangles all the corresponding parts are equal to each other.

so now,

angle DBC = angle ABD.

angle CDB = angle ADB.

here the magnitudes of the four angles is half of 90 degrees as they are equal to each other. since half of 90 degrees is 45 degrees, the magnitude of the four angles are equal to 45 degrees.

now let us consider the triangle BAD.

here,

angle BAD = 90 degrees.

angle ADB = angle DBA = 45 degrees.

we know that in a triangle if two angles are equal, then their corresponding opposite angles are equal to each other.

here,

AB and AD are corresponding opposite sides of the two equal angles. so AB = AD.

as a result all the sides of the rectangle become equal.

therefore, the quadrilateral becomes square.

now consider the two triangles ADC and ABC.

here,

AB = AD.

BC = CD.

angle ABC = angle ADC.

by SAS congruency rule, we can conclude that triangle ADC is congruent to triangle ABC.

we know that for any two congruent triangles, all the corresponding parts are equal to each other. so angle BAC is equal to angle DAC.

we know that in a triangle if two angles are equal to each other, then their corresponding sides are also equal to each other. by applying this rule in triangle AOB, we can conclude that OA = OB.