Question

In a rhombus, an altitude from the vertex of an obtuse angle bisects the opposite side. Find the measures of the angles of the rhombus.

Answer 1

The obtuse angles will measure 120° and the acute angles will measure 60°.  

Drawing the altitude forms a right triangle.  Let the side of the rhombus be x.  The altitude bisects the opposite side, so the base of the triangle formed would be 1/2x.  We have an expression for the side opposite this portion of the obtuse angle (cut by the altitude) and for the hypotenuse (the side length of the rhombus, x).  opposite/hypotenuse is the ratio for sine.  Our equation then looks like this:
$\sin x=\frac{\frac{1}{2}x}{x} \\ \sin x=\frac{1}{2}x \div x \\ \sin x=\frac{1}{2}x \div \frac{x}{1} \\ \sin x=\frac{1x}{2} * \frac{1}{x}=\frac{1x}{2x}=\frac{1}{2}$
Now we take the inverse sine of both sides:
sin⁻¹(sin x)=sin⁻¹(1/2)
x=30

Since this portion of the triangle is 30, and the right angle is 90, the missing angle (an acute angle of the rhombus) is 180-30-90=60°.  Since the acute angles and obtuse angles of a rhombus are supplementary, the obtuse angles must be 180-60=120°.

Answer 2

The angles of the rhombus determines the inclination of the sides of the

rhombus.

The correct response;

• The measures of the four angles of the rhombus are 60°, 60°, 120°, and 120°.

Method used to arrive at the above response;

Given;

The altitude of the rhombus bisects the opposite side.

The angle of the rhombus at the vertex point of the altitude = An obtuse angle

Required:

The measure of the angles of the rhombus

Solution:

The length of the sides of a rhombus are equal.

The opposite angles of a rhombus are equal.

Let x represent the length of a side of the rhombus, we have;

The bisector from the vertex of the obtuse angle = An altitude

The angle the altitude forms with the side it bisects = 90°

Therefore;

The triangle formed by the altitude, half the bisected side and the side of the rhombus adjacent to the bisected side = A right triangle

The hypotenuse side to the right triangle = The side of the rhombus

Let θ represent the angle opposite the altitude, and between the sides of

the rhombus, in the right triangle by trigonometric ratios, we have;

• $\displaystyle cos(\theta) = \mathbf{\frac{Adjacent}{Hypotenuse}}$

$\displaystyle cos(\theta) = \frac{0.5 \cdot x}{x} = 0.5$

θ = arccos(0.5) = 60°

Therefore, two acute (opposite) angles of the rhombus are 60°

The two obtuse angles of the rhombus are each = (360° - 2×60°) ÷ 2 = 120°

• The angles of the rhombus are 60°, 60°, 120°, 120°.

Learn more about trigonometric ratios here:

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