Prove that the bisectors of any two consecutive anglesof a parallelogram intersect at right angle.

in a parallelogram, the bisectors of any two consecutive angles intersect at right angle. AR and DC respectively and AL-CN

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Since one of the properties of parallelogram say the sum of the consecutive angle of a parallelogram is a supplementary angle hence, we can write,\[\angle ABC + \angle DCB = {180^ \circ }\]Now draw a line OB bisecting \[\angle B\]and a line OC bisecting \[\angle C\]joining at O since line OB and OC bisects the angle; hence we can say\[\angle OBC = \dfrac{{\angle ABC}}{2}\]\[\angle OCB = \dfrac{{\angle DCB}}{2}\]Hence in the \[\vartriangle OBC\]\[ \angle OBC + \angle OCB + \angle BOC = 180 \\ x + y + \angle BOC = 180 \\ \]This can be written as\[\dfrac{{\angle ABC}}{2} + \dfrac{{\angle DCB}}{2} + \angle BOC = 180\]Hence by solving\[\dfrac{1}{2}\left( {\angle ABC + \angle DCB} \right) + \angle BOC = 180\]Since\[\angle ABC + \angle DCB = {180^ \circ }\]hence we can write\[ \dfrac{1}{2}\left( {\angle ABC + \angle DCB} \right) + \angle BOC = 180 \\ \dfrac{1}{2} \times 180 + \angle BOC = 180 \\ 90 + \angle BOC = 180 \\ \angle BOC = {90^ \circ } \\ \]So the value of\[\angle BOC = {90^ \circ }\],Therefore we can say the bisector of any two consecutive angles intersect at the right angle.Hence proved.

Note: A property of that parallelogram says that if a parallelogram has all sides equal, then their diagonal bisector intersects perpendicularly. An angle bisector is a ray that splits an angle into two consecutive congruent, smaller angles.

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The Bisectors of any Two Consecutive Angles of a Parallelogram Intersect at Right Angle

Concept Explanation

The Bisectors of Any Two Consecutive Angles Intersect at Right Angle: We know that parallelogram is a quadrilateral in which both the opposite pair of sides are parallel and equal to each other, The bisectors of any two consecutive angles intersect at right angle.
Theorem 5: In a parallelogram, the bisectors of any two consecutive angles intersect at right angle. GIVEN A parallelogram ABCD such that the bisectors of consecutive angles A and B intersect at P. To prove <APB = 90 Proof Since ABCD is a parallelogram. Therefore, AD BC Now, AD BC and transversal AB intersects them. <A + <B = 180 [ Sum of consecutive interior angles is 180] ...(i) [ AO and BO are bisectors of <A and <B] In APB, we have <1 + <APB + <2 = 180 90 + <APB = 180 [ From (i), <1 + <2 = 90] <APB = 90
ILLUSTRATION: In the given figure, AP and BP are the bisectors of A and B which meet at P inside the parallelogram ABCD. Prove that 2APB = C + D.
Solution: In a parallelogram the angle bisectors of a parallelogram meet at right angles. ...........(1) As ABCD is a parallelogram. Therefore, AD BC Now, AD BC and transversal CD intersects them. [ Sum of consecutive interior angles is 180] [From Eq. 1] Hence Proved