Pentagon Angle Calculation Explained

The problem asks us to find the value of the expression $(2x + 10^\circ)$ given the measures of the five interior angles of a pentagon ABCDE in terms of $x$. The angles are:

• $\angle A = (2x + 9^\circ)$
• $\angle B = (2x + 1^\circ)$
• $\angle C = (2x-1^\circ)$
• $\angle D = (2x +5^\circ)$
• $\angle E = (2x-4^\circ)$

Sum of Interior Angles of a Pentagon

A key property of polygons is that the sum of their interior angles depends on the number of sides. For a polygon with $n$ sides, the sum of the interior angles is given by the formula:

$$ \text{Sum of angles} = (n-2) \times 180^\circ $$

Since a pentagon has 5 sides ($n=5$), the sum of its interior angles is:

$$ (5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ $$

So, the sum of the angles $\angle A, \angle B, \angle C, \angle D,$ and $\angle E$ must equal $540^\circ$.

Setting Up and Solving the Equation for x

We can set up an equation by adding all the given angle expressions and equating the sum to $540^\circ$:

$$ (2x + 9^\circ) + (2x + 1^\circ) + (2x - 1^\circ) + (2x + 5^\circ) + (2x - 4^\circ) = 540^\circ $$

Now, let's combine the terms involving $x$ and the constant degree terms:

• Combine $x$ terms: $2x + 2x + 2x + 2x + 2x = 10x$
• Combine constant terms: $9^\circ + 1^\circ - 1^\circ + 5^\circ - 4^\circ = (10^\circ - 1^\circ + 5^\circ - 4^\circ) = (9^\circ + 5^\circ - 4^\circ) = (14^\circ - 4^\circ) = 10^\circ$

The equation simplifies to:

$$ 10x + 10^\circ = 540^\circ $$

To solve for $x$, first subtract $10^\circ$ from both sides:

$$ 10x = 540^\circ - 10^\circ $$

$$ 10x = 530^\circ $$

Now, divide both sides by 10:

$$ x = \frac{530^\circ}{10} $$

$$ x = 53^\circ $$

Calculating the Value of (2x + 10°)

The question asks for the value of the expression $(2x + 10^\circ)$. We now substitute the value of $x$ we found ($x = 53^\circ$) into this expression:

$$ 2x + 10^\circ = 2(53^\circ) + 10^\circ $$

First, calculate $2 \times 53^\circ$:

$$ 2 \times 53^\circ = 106^\circ $$

Now, add $10^\circ$:

$$ 106^\circ + 10^\circ = 116^\circ $$

Therefore, the value of $(2x + 10^\circ)$ is $116^\circ$.