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The diagonals of a rectangle ABCD intersect at O, if \[\angle BOC = 70^\circ \] . Find \[\angle ODA\]

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Last updated date: 13th Jun 2024
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Answer
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Hint:
Here, we have to find the measure of a given angle. We will use the properties of angles in a rectangle and triangle. We will first find the measure of the opposite angle of the given angle using the property vertically opposite angle. Then we will use the property of isosceles triangle and sum property of a triangle to find the measure of the required angle.

Complete step by step solution:
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Let ABCD be a rectangle. The diagonals of a rectangle ABCD intersect at O.
We are given that \[\angle BOC = 70^\circ \].
We know that in a rectangle, vertically opposite angles are equal.
By using this property, we get
\[\angle BOC = 70^\circ = \angle AOD\] ………………………………………………………………………….\[\left( 1 \right)\]
We know that the diagonals of a rectangle are equal and bisect each other.
So, in \[\Delta AOD\] , we get
\[AO = OD\]
We know that angles opposite to equal sides of an isosceles triangle are equal.
By using this property, we get
\[\angle OAD = \angle ODA\] ……………………………………………………………………………………………\[\left( 2 \right)\]
We know that the sum of the angles of a triangle is \[180^\circ \].
By using this property in \[\Delta AOD\], we get
\[\angle OAD + \angle AOD + \angle ODA = 180^\circ \]
By substituting equation \[\left( 1 \right)\] in the above equation, we get
\[ \Rightarrow \angle ODA + \angle AOD + \angle ODA = 180^\circ \]
By adding the equal angles, we get
\[ \Rightarrow 2\angle ODA + \angle AOD = 180^\circ \]
By substituting equation \[\left( 1 \right)\] in the above equation, we get
\[ \Rightarrow 2\angle ODA + 70^\circ = 180^\circ \]
Subtracting \[70^\circ \] from both side, we get
\[ \Rightarrow 2\angle ODA = 180^\circ - 70^\circ \]
\[ \Rightarrow 2\angle ODA = 110^\circ \]
Dividing by 2 on both the sides, we get
\[ \Rightarrow \angle ODA = \dfrac{{110^\circ }}{2}\]
\[ \Rightarrow \angle ODA = 55^\circ \]

Therefore, \[\angle ODA\] is \[55^\circ \]

Note:
We know that the sum of all interior angles of a rectangle is 360 degrees. The diagonals of a rectangle bisect each other so that the lengths of the diagonals are equal in length. Since the diagonal is a straight angle the diagonals of a rectangle bisect each other at different angles where one angle is an acute angle and the other angle is an obtuse angle. We should also remember that if the diagonals bisect each other at right angles, then it is a square.