Question

a triangle shares a single side with an irregular pentagon, as shown.

work out the size of angle (q).

not drawn accurately

Explanation:

⚡ Using what you learned: properties of polygons · properties of triangles

Step 1: Find the interior angles of the pentagon

The sum of the interior angles of a pentagon is:

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

Let's find each of the five interior angles of the pentagon:

1. Bottom-left angle: It forms a straight line with the \(114^\circ\) exterior angle:

$$ 180^\circ - 114^\circ = 66^\circ $$

1. Bottom angle: It forms a straight line with the \(27^\circ\) exterior angle:

$$ 180^\circ - 27^\circ = 153^\circ $$

1. Bottom-right angle: It forms a straight line with the \(103^\circ\) exterior angle:

$$ 180^\circ - 103^\circ = 77^\circ $$

1. Top-left angle: It forms a straight line with the \(63^\circ\) exterior angle:

$$ 180^\circ - 63^\circ = 117^\circ $$

1. Middle-right angle (shared vertex): Let this interior angle be \(x\).

Summing these five interior angles:

$$ 66^\circ + 153^\circ + 77^\circ + 117^\circ + x = 540^\circ $$

$$ 413^\circ + x = 540^\circ $$

$$ x = 127^\circ $$

Step 2: Find the interior angles of the triangle

At the shared vertex on the right, the straight line continues past the pentagon's side. The interior angle of the pentagon (\(x = 127^\circ\)), the blue angle of the triangle (\(79^\circ\)), and the exterior angle (\(103^\circ\)) lie along a straight line:

$$ x + 79^\circ + \text{angle adjacent to } 103^\circ \text{ inside the straight line} $$

Alternatively, looking at the straight line at the bottom-right:
The angle of the pentagon is \(77^\circ\). The exterior angle is \(103^\circ\), which sum to \(180^\circ\).
The line extending upwards from this vertex forms a straight line. The angles meeting at this vertex on the straight line are the pentagon's interior angle \(x = 127^\circ\) and the remaining angle on that side of the line.
Since the straight line has a total angle of \(180^\circ\), the angle adjacent to \(x\) on this line is:

$$ 180^\circ - 127^\circ = 53^\circ $$

This \(53^\circ\) angle combined with the blue angle (\(79^\circ\)) forms the third angle of the triangle at this vertex:

$$ \text{Triangle's bottom-right angle} = 53^\circ $$

Let's find the top-left angle of the triangle. At the top-left vertex of the pentagon, the straight line goes vertically.
The interior angle of the pentagon is \(117^\circ\). The angle of the triangle at this vertex and the \(63^\circ\) angle lie on a straight line:

$$ 180^\circ - 117^\circ = 63^\circ $$

Thus, the interior angle of the triangle at this top-left vertex is:

$$ 63^\circ $$

Step 3: Calculate angle q

The sum of angles in a triangle is \(180^\circ\):

$$ q + 63^\circ + 79^\circ = 180^\circ $$

$$ q + 142^\circ = 180^\circ $$

$$ q = 38^\circ $$

Answer:

38