The angle between the two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 45°. Find the angles of the parallelogram.
Solution:
Given, the angle between the two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 45°.
We have to determine the angles of the parallelogram.
Consider a parallelogram ABCD
Let BE and BF be the perpendiculars through the vertex B to the sides DC and AD.
We know that the opposite angles of a parallelogram are equal.
So, ∠A = ∠C = x and ∠B = ∠D = y
We know that the adjacent angles of a parallelogram are supplementary.
So, ∠A + ∠B = 180°
From the figure,
∠B = ∠ABF + ∠FBE + ∠EBC
From triangle ABF, ∠ABF = 90° - x
From triangle EBC, ∠EBC = 90° - x
x + ∠ABF + ∠FBE + ∠EBC = 180°
x + 90° - x + 45° + 90° - x = 180°
x - 2x + 180° + 45° = 180°
-x = 180° - 180° - 45°
x = 45°
So, ∠A = ∠C = 45°
∠B = (90° - x) + 45° + (90° - x)
= 45° + 45° + 45°
= 90° + 45°
∠B = ∠D = 135°
Therefore, the angles are 45°, 135°, 45° and 135°.
✦ Try This: Find the value of x and y in the given parallelogram.
☛ Also Check: NCERT Solutions for Class 8 Maths
NCERT Exemplar Class 8 Maths Chapter 5 Problem 178
The angle between the two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 45°. Find the angles of the parallelogram.
Summary:
The angle between the two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 45°. The angles of the parallelogram are 45°, 135°, 45° and 135°.
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