Solving Rhombus Angles using Ratio

The question asks us to find the smallest angle of a rhombus where the adjacent angles are in the ratio 3:6.

A key property of a rhombus is that its adjacent angles are supplementary. This means that the sum of any two adjacent angles in a rhombus is always \(180^\circ\).

Let the two adjacent angles of the rhombus be represented by \(3x\) and \(6x\), based on the given ratio of 3:6.

Since the adjacent angles are supplementary, we can write the equation:

\(3x + 6x = 180^\circ\)

Now, we solve for \(x\):

• Combine like terms: \(9x = 180^\circ\)
• Divide both sides by 9: \(x = \frac{180^\circ}{9}\)
• Calculate the value of x: \(x = 20^\circ\)

Now that we have the value of \(x\), we can find the measure of each adjacent angle:

• First angle: \(3x = 3 \times 20^\circ = 60^\circ\)
• Second angle: \(6x = 6 \times 20^\circ = 120^\circ\)

So, the two adjacent angles of the rhombus are \(60^\circ\) and \(120^\circ\).

The angles of the rhombus are alternating between these two values because opposite angles in a rhombus are equal. Therefore, the four angles of the rhombus are \(60^\circ, 120^\circ, 60^\circ, 120^\circ\).

We are asked to find the smallest angle of the rhombus. Comparing the two angle measures we found, \(60^\circ\) and \(120^\circ\), the smallest angle is \(60^\circ\).

Let's verify that these angles are indeed adjacent and supplementary:

• Adjacent angles: \(60^\circ\) and \(120^\circ\). Their sum is \(60^\circ + 120^\circ = 180^\circ\). This confirms they are supplementary.
• The ratio of adjacent angles: \(60^\circ : 120^\circ\) simplifies to \(1 : 2\). Wait, the ratio given is 3:6, which also simplifies to 1:2. So the calculation is consistent with the simplified ratio. If we use 3x and 6x directly: \(3x : 6x\) is indeed \(3:6\). Our angles \(60^\circ\) and \(120^\circ\) are in the ratio \(60:120\), which simplifies to \(6:12\), and further to \(1:2\). The ratio 3:6 also simplifies to 1:2. So the ratio is correctly represented.

The smallest angle is \(60^\circ\).

Revision Table: Rhombus Properties

Additional Information: Understanding Rhombus Angles

A rhombus is a special type of parallelogram. This means it inherits all the properties of a parallelogram, such as opposite sides being parallel and equal, and diagonals bisecting each other. However, a rhombus has additional unique properties, primarily concerning its sides and angles.

In any quadrilateral, the sum of all interior angles is \(360^\circ\). For a rhombus with angles \(A, B, C, D\), if angles A and B are adjacent, then \(A + B = 180^\circ\). Since opposite angles are equal, \(A=C\) and \(B=D\). So the angles are \(A, B, A, B\). The sum is \(A+B+A+B = 2(A+B)\). Since \(A+B=180^\circ\), the sum is \(2 \times 180^\circ = 360^\circ\), which holds true for all quadrilaterals.

In our case, the angles are \(60^\circ, 120^\circ, 60^\circ, 120^\circ\). The sum is \(60 + 120 + 60 + 120 = 360^\circ\).

Understanding the supplementary nature of adjacent angles is crucial for solving problems involving rhombus angles and ratios.