Geometry and Angle Properties for the ESAT

Properties of angles around a point, lines and vertices

To solve complex geometry problems, you must first master the basic properties of angles at intersections and on lines. These foundational rules allow you to determine unknown values by creating equations based on known sums.

1. The sum of all angles around a single point is $360^\circ$ ($a + b + c + d = 360^\circ$).
2. The sum of angles that lie on one side of a straight line is $180^\circ$ ($a + b = 180^\circ$, $b + c = 180^\circ$, etc.).
3. Perpendicular lines intersect at a right angle, which is exactly $90^\circ$.
4. Vertically opposite angles, formed at the vertex where two straight lines cross, are equal ($a = c$ and $b = d$).

Angle properties of parallel lines

When a transversal line crosses a pair of parallel lines, specific angle relationships are established that are constant regardless of the angle of the transversal.

If a line crosses parallel lines:

• Alternate angles are equal ($d = f$ and $c = e$). These are often found in Z shaped configurations.
• Corresponding angles are equal ($a = e$, $b = f$, $c = g$, and $d = h$). These are found in F shaped configurations.
• Allied or co-interior angles are supplementary, meaning they add up to $180^\circ$ ($c + f = 180^\circ$ and $d + e = 180^\circ$). These are found in C shaped configurations.

Angle properties of triangles and quadrilaterals

Triangles and quadrilaterals follow strict internal sum rules that form the basis for polygon geometry.

The sum of the interior angles of any triangle is $180^\circ$ ($a + b + c = 180^\circ$). Furthermore, the exterior angle of a triangle is equal to the sum of the two interior opposite angles ($d = a + b$, $e = a + c$, and $f = b + c$).

The sum of the interior angles of any quadrilateral is $360^\circ$ ($a + b + c + d = 360^\circ$).

Interior and exterior angle sum of polygons

For any polygon with $n$ sides, the total sum of the interior angles in degrees is calculated using the formula $180(n - 2)$. This is because any $n$-sided polygon can be split into $n - 2$ triangles.

The sum of the exterior angles of any polygon, regardless of the number of sides, is always $360^\circ$.

Worked Examples: Basic Properties

Finding unknown angles at a point

Find the value of $a$ in the following diagram.

Since the sum of angles at a point is $360^\circ$, we set up the equation: $90^\circ + a + 2a + 129^\circ = 360^\circ$ $3a + 219^\circ = 360^\circ$ $3a = 141^\circ$ $a = 47^\circ$

Finding unknown angles on a straight line

Find the value of $a$ in the diagram below.

The sum of angles on a straight line is $180^\circ$: $a + 2a + 3a = 180^\circ$ $6a = 180^\circ$ $a = 30^\circ$

Finding vertically opposite angles

Find the value of $a$ using the diagram of three straight lines.

As shown by the red highlighted lines, $a$ and $42^\circ$ are vertically opposite angles. Therefore, $a = 42^\circ$.

Worked Examples: Parallel Lines

Alternate angles

Find the value of $a$.

Angle $a$ is alternate to the $56^\circ$ angle, as they are contained within the Z shape formed by the transversal crossing parallel lines. Thus, $a = 56^\circ$.

Co-interior or allied angles

Find the size of angle $a$.

Angle $a$ and $56^\circ$ are co-interior (allied) angles, forming a C shape. They must sum to $180^\circ$: $a + 56^\circ = 180^\circ$ $a = 180^\circ - 56^\circ = 124^\circ$

Finding multiple angles in parallel lines

Find $a$, $b$, $c$, and $d$.

1. $a + 125^\circ = 180^\circ$ (angles on a straight line). So $a = 55^\circ$.
2. $b$ and $a$ are vertically opposite, so $b = 55^\circ$.
3. $c$ is a corresponding angle to the given $125^\circ$, so $c = 125^\circ$.
4. $d$ is vertically opposite to $c$, so $d = 125^\circ$.

Combining angle facts

Find angle $a$.

Method 1: Let $b$ be the angle on the straight line with $35^\circ$ ($b = 180 - 35 = 145^\circ$). Let $c$ be the corresponding angle to $b$ on the middle parallel line ($c = 145^\circ$). Finally, $a$ is corresponding to $c$, so $a = 145^\circ$.

Method 2: Identify corresponding angles relative to the $35^\circ$ angle. Let $f = 35^\circ$ and $h = 35^\circ$. Angle $a$ sits on a straight line with $h$, so $a = 180 - 35 = 145^\circ$.

Worked Examples: Polygons

Triangle and Quadrilateral calculations

Find angles $a$ and $b$ in the triangle diagram.

First, $3a + 42^\circ = 180^\circ$ (straight line), so $3a = 138^\circ$ and $a = 46^\circ$. To find $b$, we can use the exterior angle property: $a + b = 3a$, which means $b = 2a = 92^\circ$. Alternatively, using the sum of triangle angles: $a + b + 42^  = 180^$, so $b = 180 - 42 - 46 = 92^\circ$.

What is the size of angle $a$ in the quadrilateral?

The interior sum is $360^\circ$. The missing interior angle is $360 - (75 + 105 + 100) = 80^\circ$. Since $a$ and $80^\circ$ are on a straight line, $a = 180 - 80 = 100^\circ$.

Complex Polygon sums

Example 1: Three angles of a pentagon are $100^\circ$, $110^\circ$, and $82^\circ$. The other two are equal. Find the size of the two equal angles.

A pentagon has 5 sides. Sum $= 180(5 - 2) = 540^\circ$. Subtracting known angles: $540 - (100 + 110 + 82) = 248^\circ$. Dividing by 2 gives $124^\circ$ per angle.

Example 2: Find the interior angle of a regular dodecagon (12 sides).

Method 1: Exterior angle sum is $360^\circ$. Each exterior angle $= 360 / 12 = 30^\circ$. The interior angle is $180 - 30 = 150^\circ$. Method 2: Interior sum $= 180(12 - 2) = 1800^\circ$. Each interior angle $= 1800 / 12 = 150^\circ$.

Example 3: Exterior angles of a pentagon are $45^\circ$, $88^\circ$, $70^\circ$, $30^\circ$, and $x^\circ$. Find $x$.

The sum of exterior angles is $360^\circ$: $45 + 88 + 70 + 30 + x = 360$. Thus $x = 360 - 233 = 127^\circ$.