A line l is parallel to line m and a transversal p interesects them at X, Y respectively. Bisectors of interior angles at X and Y interesct at P and Q. Is PXQY a rectangle? Given reason.
Solution:
Given, line l is parallel to line m and a transversal p intersects them at X and Y.
Bisectors of interior angles at X and Y intersect at P and Q.
We have to find if PXQY is a rectangle.
Since l || m, the alternate interior angles are equal.
∠DXY = ∠XYA
Dividing by 2 on both sides,
1/2 ∠DXY = 1/2 ∠XYA
Since XP and YQ are the bisectors,
∠PXY = ∠QYX
Since the alternate interior angles are equal, XP || QY ------ (1)
Similarly, ∠CYX = ∠BYX
Dividing by 2 on both sides,
1/2 ∠CYX = 1/2 ∠BYX
Since XP and YQ are the bisectors,
∠QXY = ∠PYX
Since the alternate interior angles are equal, XQ || PY ------- (2)
From (1) and (2).
The opposite sides are parallel.
Therefore, PXQY is a parallelogram.
We know that the interior angles on the same side of transversal are supplementary.
So, ∠DXY + ∠XYB = 180°
Dividing by 2 on both sides,
1/2 ∠DXY + 1/2 ∠XYB = 90°
From the figure,
∠PXY + ∠PYX = 90° ------------------------------ (3)
Consider triangle XYP,
By angle sum property of a triangle,
∠PXY + ∠PYX + ∠P = 180°
From (3),
90° + ∠P = 180°
∠P = 180° - 90°
∠P = 90°
From (1), (2) and (3),
Each angle is equal to 90 degrees.
Therefore, PXQY is a rectangle.
✦ Try This: The angles of a quadrilateral are in the raito 1 : 2 : 3 : 4. Find the angles.
☛ Also Check: NCERT Solutions for Class 8 Maths
NCERT Exemplar Class 8 Maths Chapter 5 Problem 175
A line l is parallel to line m and a transversal p interesects them at X, Y respectively. Bisectors of interior angles at X and Y interesct at P and Q. Is PXQY a rectangle? Given reason.
Summary:
A line l is parallel to line m and a transversal p interesects them at X, Y respectively. Bisectors of interior angles at X and Y interesct at P and Q. PXQY is a rectangle.
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