SOLUTION: A parallelogram whose vertices are A(4,1),B(1,2), C(-5,-1)and D(-2,-2).find the angles of the vertices of the parallelogram.

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Question 1009696: A parallelogram whose vertices are A(4,1),B(1,2), C(-5,-1)and D(-2,-2).find the angles of the vertices of the parallelogram.
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!


We find the slope of AB using the slope formula:

m%22%22=%22%22%28%282%29-%281%29%29%2F%28%281%29-%284%29%29%22%22=%22%221%2F%28-3%29%22%22=%22%22-1%2F3

We find the slope of AD using the slope formula:

m%22%22=%22%22%28%28-2%29-%281%29%29%2F%28%28-2%29-%284%29%29%22%22=%22%22%28-3%29%2F%28-6%29%22%22=%22%221%2F2

Then we use the formula for the tangent of the angle between two lines
AB and AD:

tan%28theta%29%22%22=%22%22%28m%5B1%5D-m%5B2%5D%29%2F%281%2Bm%5B1%5Dm%5B2%5D%29

tan%28theta%29%22%22=%22%22%28+%28-1%2F3%29-%281%2F2%29+%29%2F%28+1%2B%28-1%2F3%29%281%2F2%29+%29

tan%28theta%29%22%22=%22%22%28+%28-1%2F3%29-%281%2F2%29+%29%2F%281-1%2F6%29

Multiply every term in the top and bottom by LCD=6

tan%28theta%29%22%22=%22%22%28+%28-1%2F3%29%2A6-%281%2F2%29%2A6+%29%2F%28+1%2A6-expr%281%2F6%29%2A6+%29

tan%28theta%29%22%22=%22%22%28-2-3%29%2F%286-1%29

tan%28theta%29%22%22=%22%22%28-5%29%2F5

tan%28theta%29%22%22=%22%22-1

This formula will give either the tangent of the acute or obtuse angle
between two lines, which are supplementary, depending on our choice for
m1 and m2.  The tangents of the acute and obtuse angles between two lines
are different only by their signs.

Since A is acute, tan(A) = 1 and m∠A = 45°

Since the opposite interior angles of a parallelogram are equal in measure,

m∠C = 45°

Since adjacent interior angles of a parallelogram are supplementary,

m∠B = 180°-45° = 135° and since ∠D is opposite ∠B, m∠D also = 135°.

Edwin