Angle Sums and Properties of Polygons

Together with the top interior angle, these three angles sit perfectly on the straight parallel line. Because angles on a straight line always sum to $180^\circ$, the three interior angles of the triangle must also sum to $180^\circ$. Furthermore, because an exterior angle and its adjacent interior angle also sum to $180^\circ$ on a straight line, we can substitute this to prove that the exterior angle of a triangle is always equal to the sum of the two opposite interior angles.

Calculating Missing Angles in a Triangle

Find the missing angle $R$ in $\triangle PQR$ where $\angle P = 52^\circ$ and $\angle Q = 63^\circ$.

Step 1: State the calculation formula.

$$\text{Missing Angle} = 180^\circ - (\text{Angle}_1 + \text{Angle}_2)$$

Step 2: Substitute the known values.

$$\angle R = 180^\circ - (52^\circ + 63^\circ)$$

Step 3: Calculate the final answer.

$$\angle R = 180^\circ - 115^\circ = 65^\circ$$

Deduce the Angle Sum in Any Polygon

Every straight-sided shape can be broken down into triangles. This process, called triangulation, is the key to finding the sum of the interior angles for any irregular polygon or regular shape. For example, a quadrilateral ($4$ sides) can be split into exactly $2$ triangles, giving a total interior angle sum of $360^\circ$.

To do this for any shape, pick a single vertex and draw diagonals to all other non-adjacent vertices. You will find that any polygon with $n$ sides can be split into exactly $(n - 2)$ non-overlapping triangles.

Since each triangle contributes $180^\circ$ to the total, the overall interior angle sum is found using the formula:

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

Calculating the Sum of Interior Angles

A pentagon has interior angles of $95^\circ$, $121^\circ$, $90^\circ$, $104^\circ$, and $x$. Calculate the value of $x$.

Step 1: State the formula for the sum of interior angles.

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

Step 2: Substitute $n = 5$ for a pentagon to find the total sum.

$$\text{Sum} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ$$

Step 3: Set up an equation with the known angles.

$$95^\circ + 121^\circ + 90^\circ + 104^\circ + x = 540^\circ$$

Step 4: Simplify and solve for $x$.

$$410^\circ + x = 540^\circ$$ $$x = 130^\circ$$

Deriving Properties of Regular Polygons

A regular polygon (such as an equilateral triangle or a square) has sides of equal length and equal interior angles. If you walk around the entire perimeter of any polygon and return to your starting point, you have made a full turn, meaning the sum of the exterior angles is always $360^\circ$.

To find a single exterior angle of a regular polygon, you simply share this full turn equally among its $n$ vertices:

$$\text{Exterior angle} = \frac{360^\circ}{n}$$

It is important to note that the exterior angle is formed by extending a side; it is not the reflex angle outside the shape. At any vertex, the interior angle and the exterior angle sit on a straight line, meaning they are supplementary angles. This means that:

$$\text{Interior Angle} + \text{Exterior Angle} = 180^\circ$$

By combining these two facts, we can explicitly derive the algebraic formula for a single interior angle. Since the exterior angle is $\frac{360^\circ}{n}$, and the interior and exterior angles sum to $180^\circ$, substituting the exterior angle formula gives:

$$\text{Interior Angle} = 180^\circ - \frac{360^\circ}{n}$$

We can also prove the $360^\circ$ exterior sum algebraically. First, the total sum of all interior and exterior pairs at $n$ vertices is $180n^\circ$. Second, we know the sum of interior angles is $180^\circ(n - 2)$, which expands to $180n^\circ - 360^\circ$. Finally, subtracting the interior sum from the total sum leaves exactly $360^\circ$ for the exterior angles.