If an angle of a parallelogram is two-third of its adjacent angle, find the angles of the parallelogram .
Let the measure of the angle be x
∴ The measure of the angle adjacent is `(2x)/3`
We know that the adjacent angle of a parallelogram is supplementary
Hence x + `(2x) / 3` = 180°
2x + 3x = 540°
⇒ 5x = 540°
⇒ x = 108°
Adjacent angles are supplementary
⇒ x +108° = 180°
⇒ x =180° -108° = 72°
⇒ x = 72°
Hence, four angles are : 180°, 72°,108°, 72°
Concept: Angle Sum Property of a Quadrilateral
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If an angle of a parallelogram is two third of its adjacent angle, find the angles of the parallelogram.
Solution
Let in a parallelogram ABCD,
∠A=x
Then its adjacent angle , ∠B=23x
But, ∠A+∠B=180∘ [Since, sum of the consecutive angles of a parallelogram is 180∘.
⇒x+23x=180∘
⇒3x+2x3=180∘
⇒5x3=180∘
⇒x=180∘×35
⇒x=36∘×3
⇒x=108∘
⇒∠A=108∘---(1)
⇒∠B=180∘−108∘=72∘---(2)
Since, opposite angles of a parallelogram are equal.
⇒∠A=∠C=108∘ ---(3)
and ⇒∠B=∠D=72∘---(4)
Hence angles are 108∘,72∘,108∘,72∘.
Mathematics
RD Sharma
Standard IX
28