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In a parallelogram, one angle is three times its adjacent angle. Find all the angles of the parallelogram.
Answer (2)
The angles in the parallelogram are 45 degrees and 135 degrees. There are two angles of each measure. This is because one angle is three times its adjacent angle, leading to the relationship of those angle measures.
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In order to solve this problem, we need to understand the properties of a parallelogram and how its angles relate to each other.
In a parallelogram, opposite angles are equal, and adjacent angles are supplementary, meaning they add up to 18 0 ∘ .
Let's denote one of the angles in the parallelogram as x . According to the problem, this angle is three times its adjacent angle. We can say that if the adjacent angle is y , then:
x = 3 y
Since x and y are adjacent angles in a parallelogram, they must add up to 18 0 ∘ :
x + y = 18 0 ∘
Substituting the first equation into the second equation gives:
3 y + y = 18 0 ∘
4 y = 18 0 ∘
Now, solve for y :
y = 4 18 0 ∘
y = 4 5 ∘
With y known, we can find x using the first equation:
x = 3 × 4 5 ∘ = 13 5 ∘
Now, recall that opposite angles in a parallelogram are equal. Hence, the angles opposite to the ones we calculated are also 135° and 45° respectively. Therefore, the angles of the parallelogram are:
13 5 ∘
4 5 ∘
13 5 ∘
4 5 ∘
To summarize, in the given parallelogram, one angle is three times its adjacent angle, and all the angles are 135° and 45°, with each occurring twice due to the properties of a parallelogram.
