The problem states that the diagonals of a quadrilateral ABCD bisect each other. This is a key property that defines a parallelogram.
Parallelogram Property Identification
A quadrilateral whose diagonals bisect each other is always a parallelogram. Therefore, ABCD is a parallelogram.
Adjacent Angles in a Parallelogram
In any parallelogram, consecutive (adjacent) angles are supplementary. This means their sum is equal to $180^{\circ}$.
So, for parallelogram ABCD:
- $\angle A + \angle B = 180^{\circ}$
- $\angle B + \angle C = 180^{\circ}$
- $\angle C + \angle D = 180^{\circ}$
- $\angle D + \angle A = 180^{\circ}$
Calculating Angle B
We are given that $\angle A = 45^{\circ}$. Using the property that adjacent angles are supplementary:
$ \angle A + \angle B = 180^{\circ} $
Substitute the value of $\angle A$:
$ 45^{\circ} + \angle B = 180^{\circ} $
To find $\angle B$, subtract $45^{\circ}$ from both sides:
$ \angle B = 180^{\circ} - 45^{\circ} $
$ \angle B = 135^{\circ} $
Therefore, the measure of angle B is $135^{\circ}$.
Similar Questions
- If one angle of a parallelogram is $18^\circ$ less than twice the smallest angle of the same parallelogram, then the largest angle of the parallelogram is:
- Find the area of a trapezium whose parallel sides are 10 cm and 20 cm and non-parallel sides are equal to 10 cm.
- The angles of a quadrilateral are in the ratios 2 : 4 : 6 : 8. The smallest of these angles is:
- The angles of a quadrilateral are in the ratio of 1 : 3 : 4 : 7. What is the difference between the largest and the smallest angle?
- If ABCD is a cyclic quadrilateral and ABC is an equilateral triangle find the angle of CDA.
- If one angle of a parallelogram is $39^\circ$ less than twice the smallest angle, then the smallest angle of the parallelogram is:
- If one angle of a parallelogram is $30^\circ$ less than twice the measure of the smallest angle, then the measure of the largest angle of the parallelogram is:
- In a rhombus ABCD, if $\angle ACB = 40^\circ, \text{then } \angle ADB = ?$
- P is the mid-point of side BC of a parallelogram ABCD such that $\angle BAP = \angle DAP$.
If AD = 10 cm, then CD = ? - If the angles of a quadrilateral are in the ratio of 4 : 9 : 11 : 12, then the largest of these angles is:
Important Questions from Quadrilaterals
What is the value of AC 2– BD 2
What is the point of intersection of the diagonals?
What is the area of the parallelogram?
ABCD is a cyclic quadrilateral. Diagonals BD and AC intersect each other at E. If ∠BEC = 138° and ∠ECD = 35°, then what is the measure of ∠BAC?
A circle is inscribed in a quadrilateral ABCD, touching sides AB, BC CD and DA at P, Q, R and S, respectively. If AS = 6 cm, BC = 12 cm, and CR = 5 cm, then the length of AB (in cm) is:
