Geometry Revision Notes

What is an Angle Bisector?

An angle bisector is a line, ray or segment that passes through the vertex of an angle and divides the angle into two angles of equal measure. For example, if an angle measures 100 degrees, its bisector will divide it into two 50-degree angles. This equal division is the defining characteristic of an angle bisector. It's a fundamental concept in geometry used in various constructions and proofs.

Properties of an Angle Bisector

Every point on the angle bisector is equidistant from the two arms (or rays) of the angle. This means if you pick any point on the bisector and measure its perpendicular distance to each arm, those distances will be identical. This property is important for understanding the locus of points that are equidistant from two intersecting lines, which is precisely what an angle bisector represents in geometric terms.

Constructing an Angle Bisector (Foundation Method)

To construct an angle bisector using a compass and straightedge:

1. Place the compass point at the vertex of the angle and draw an arc that intersects both arms of the angle. Label these intersection points, say, A and B.
2. With the compass point at A, draw another arc within the angle.
3. Without changing the compass width, place the compass point at B and draw another arc that intersects the previous arc. Label this intersection point C.
4. Draw a straight line from the vertex of the angle through point C. This line is the angle bisector. This method ensures accuracy and is a key skill for geometric constructions.

Applying Angle Bisectors in Problems

Angle bisectors are often used to solve problems involving angles in polygons, especially triangles. For instance, in a triangle, the intersection point of the three angle bisectors is called the incenter, which is the center of the triangle's inscribed circle. When solving problems, look for situations where an angle is explicitly stated to be bisected, or where you need to find a line that creates two equal angles. Knowledge of angle properties (e.g., angles on a straight line, angles in a triangle) combined with the concept of bisection will be vital.

Understanding Perpendicular Lines

Two lines are perpendicular if they intersect at a right angle (90 degrees). The symbol for perpendicular is $\perp$. When you construct a perpendicular from a point to a line, you are creating a line segment that represents the shortest possible distance between that point and the given line. This concept is crucial in geometry and has applications in various areas, such as finding distances and determining geometric properties of shapes.

Method 1: Point OFF the Line (Compass and Straightedge)

To construct a perpendicular from a point P that is not on line L:

1. Set your compass to a radius large enough to intersect line L at two points. Place the compass point on P and draw an arc that crosses line L at two distinct points, A and B.
2. Now, from point A, draw an arc below (or above) line L. Keep the same compass radius.
3. From point B, draw another arc with the same radius that intersects the arc from step 2. Label the intersection point C.
4. Use a straightedge to draw a line connecting P and C. This line PC is perpendicular to line L and passes through P.

Method 2: Point ON the Line (Compass and Straightedge)

To construct a perpendicular to line L at a point P that is on the line L:

1. Place your compass point on P and draw two arcs of the same radius along line L, one on each side of P. Label these points D and E.
2. Open your compass to a radius larger than the distance PD (or PE).
3. From point D, draw an arc above (or below) line L.
4. From point E, draw another arc with the same radius that intersects the arc from step 3. Label the intersection point F.
5. Use a straightedge to draw a line connecting P and F. This line PF is perpendicular to line L at P.

Key Principles and Applications

The construction of a perpendicular relies on the properties of isosceles triangles and circles. In Method 1, points A and B are equidistant from P. Points A and B are also equidistant from C (by construction). Thus, PC is the perpendicular bisector of AB. In Method 2, points D and E are equidistant from P. Points D and E are also equidistant from F, making PF the perpendicular bisector of DE. This skill is foundational for constructing squares, rectangles, and finding the shortest distance from a point to a line.

Locus from a Fixed Point (Circle)

The locus of points equidistant from a single fixed point is a circle. The fixed point serves as the centre of the circle, and the fixed distance represents the radius. To construct this, place the compass point on the fixed point (centre), open the compass to the specified fixed distance (radius), and draw a complete curve. Ensure the compass remains steady to maintain a consistent radius for an accurate circle. This is a fundamental type of locus often used to represent 'within' or 'at a certain distance from' a location.

Locus from Two Fixed Points (Perpendicular Bisector)

The locus of points equidistant from two fixed points, A and B, is the perpendicular bisector of the line segment AB. This line not only cuts the segment AB exactly in half, but it also intersects at a 90-degree angle. To construct this: 1. Set your compass to a radius greater than half the length of AB. 2. Place the compass point on A and draw an arc above and below AB. 3. Without changing the compass setting, place the compass point on B and draw intersecting arcs above and below AB. 4. Draw a straight line connecting the two points where the arcs intersect. This line is the perpendicular bisector.

Locus from Two Intersecting Lines (Angle Bisector)

The locus of points equidistant from two intersecting lines is the angle bisector of the angle formed by these lines. This line divides the original angle into two equal, smaller angles. To construct this: 1. Place the compass point on the vertex (the point where the lines intersect). 2. Draw an arc that intersects both lines. 3. From each of these intersection points on the lines, draw two more arcs that intersect inside the angle. Ensure the compass opening for these arcs is the same. 4. Draw a straight line from the vertex through the point where these two arcs intersect. This line is the angle bisector.

Recognising Angles at a Point

Angles at a point are angles that share a common vertex (the point where lines or rays meet) and collectively form a complete turn. Visually, if you can trace a path around the vertex through each angle and end up facing the same direction you started, then these are angles at a point. This principle applies regardless of how many angles meet at that central point.

The Sum of Angles at a Point

A complete turn or rotation measures 360 degrees. Therefore, the sum of all angles that meet at a single point must always be 360 degrees. This is a fundamental rule in geometry and is used to solve for unknown angles when other angles around the point are known. For instance, if angles a, b, and c meet at a point, then a + b + c = 360°.

Calculating Unknown Angles

To find an unknown angle at a point, you need to add together all the known angles. Once you have this sum, subtract it from 360 degrees. The result will be the value of the missing angle. For example, if three angles around a point are 100°, 120°, and x, then 100° + 120° + x = 360°. So, 220° + x = 360°, meaning x = 360° - 220° = 140°.