Finding the Largest Angle of a Pentagon

Let's find the largest angle of the pentagon. We are given that one angle is \(140^\circ\) and the remaining four angles are in the ratio \(1 : 2 : 3 : 4\).

Calculate the Sum of Interior Angles of a Pentagon

The sum of the interior angles of a polygon with \(n\) sides is given by the formula \((n-2) \times 180^\circ\).

For a pentagon, the number of sides is \(n=5\).

Sum of interior angles of a pentagon = \((5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ\).

Calculate the Sum of the Remaining Angles

We know one angle is \(140^\circ\). The sum of the other four angles is the total sum minus this known angle.

Sum of the remaining four angles = \(540^\circ - 140^\circ = 400^\circ\).

Determine the Individual Remaining Angles

The remaining four angles are in the ratio \(1 : 2 : 3 : 4\). Let these angles be \(x\), \(2x\), \(3x\), and \(4x\), where \(x\) is a common multiplier.

The sum of these angles is \(x + 2x + 3x + 4x\).

So, \(x + 2x + 3x + 4x = 10x\).

We know the sum is \(400^\circ\), so:

\(10x = 400^\circ\)

To find \(x\), we divide the sum by 10:

\(x = \frac{400^\circ}{10} = 40^\circ\)

Now we can find the measure of each of the remaining four angles:

• First angle = \(x = 40^\circ\)
• Second angle = \(2x = 2 \times 40^\circ = 80^\circ\)
• Third angle = \(3x = 3 \times 40^\circ = 120^\circ\)
• Fourth angle = \(4x = 4 \times 40^\circ = 160^\circ\)

Identify the Largest Angle

The five angles of the pentagon are \(140^\circ\) and the four angles we just calculated: \(40^\circ, 80^\circ, 120^\circ, 160^\circ\).

Let's list all five angles:

\(140^\circ\), \(40^\circ\), \(80^\circ\), \(120^\circ\), \(160^\circ\)

Comparing these values, the largest angle is \(160^\circ\).

The largest among these angles is \(160^\circ\).