Question 197170: What are the measures of the angles of a rhombus ABCD of side 5m and whose longest diagonal is 8m? Answer by J2R2R(94) (Show Source):
You can put this solution on YOUR website! You have a triangle made from of two sides of the rhombus AB and BC equal to 5cm and the diagonal AC equal to 8cm.
The Cosine Rule for triangle ABC states
AC^2 = AB^2 + BC^2 - 2 AB BC cos ABC
Therefore
64 = 25 + 25 - 50 cos ABC
64 = 50 - 50 cos ABC
14 = - 50 cos ABC
-7/25 = cos ABC
Angle ABC = approx 106.3 degrees as is the opposite angle CDA of the rhombus.
The other two angles BCA and CAB of the triangle (isosceles) are equal and half the angle made in the corner by the rhombus since a triangle on one side of a diagonal of a rhombus is a mirror image of the triangle on the other side of the diagonal.
Alternatively you could say that since the opposite sides of a rhombus are parallel then adjacent angles add up to 180 degrees and opposite angles are equal.
Thus angles DAB and BCD are approximately 180 – 106.3 = 73.7 degrees.
