Angles, parallel lines, polygons, circles, constructions

Geometry: Angles, Parallel Lines, Polygons, Circles, Constructions

1. Angles

An angle is the figure formed by two rays sharing a common endpoint called the vertex. Think of it like a slice of pizza 🍕 – the tip of the slice is the vertex.

• Acute angle: < $90^\circ$
• Right angle: $90^\circ$ (like a corner of a book)
• Obtuse angle: $>90^\circ$ and $<180^\circ$
• Straight angle: $180^\circ$ (a straight line)
• Reflex angle: $>180^\circ$ and $<360^\circ$

In a triangle, the sum of the interior angles is always $180^\circ$:

$$\alpha + \beta + \gamma = 180^\circ$$

1. Draw the triangle.
2. Mark the known angles.
3. Subtract the sum of the known angles from $180^\circ$ to find the missing angle.

2. Parallel Lines & Transversals

Two lines are parallel if they never meet, no matter how far they are extended. Imagine two train tracks running side‑by‑side.

When a third line, called a transversal, cuts across two parallel lines, several pairs of angles are formed:

If any pair of these angles is equal (or supplementary where required), we can conclude the two lines are parallel.

3. Polygons

A polygon is a closed figure with straight sides. Key terms:

• Regular polygon: all sides and angles equal.
• Irregular polygon: sides/angles not all equal.
• Convex: all interior angles < $180^\circ$; no indentations.
• Concave: at least one interior angle > $180^\circ$.

The sum of interior angles for an n‑sided polygon is:

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

The sum of exterior angles (one at each vertex) is always $360^\circ$, regardless of the number of sides.

1. Count the number of sides (n).
2. Apply the formula to find the total interior angle sum.
3. Divide by n to find each interior angle if the polygon is regular.

4. Circles

A circle is the set of all points equidistant from a centre point O. Key measurements:

• Radius (r): distance from centre to any point on the circle.
• Diameter (d): the longest chord, passing through the centre. $d = 2r$.
• Circumference (C): the perimeter. $C = 2\pi r$.
• Area (A): $A = \pi r^2$.

Other terms:

• Chord: a straight line segment whose endpoints lie on the circle.
• Tangent: a line that touches the circle at exactly one point.
• Secant: a line that cuts the circle at two points.
• Arc: a portion of the circle’s circumference.
• Sector: the region bounded by two radii and the arc between them.

The central angle subtended by an arc is the angle whose vertex is the centre O and whose sides pass through the arc’s endpoints.

$$\text{Arc length } s = r \theta$$
$$\text{Sector area } A = \frac{1}{2} r^2 \theta$$
(θ in radians)

5. Compass & Straightedge Constructions

With only a compass and straightedge, we can perform precise constructions. Here are some common ones:

1. Perpendicular bisector of a segment AB:
   1. Draw circles with centres A and B, radius > AB/2.
   2. The intersection points of the circles are C and D.
   3. Draw line CD; it bisects AB at right angles.
2. Angle bisector of ∠BAC:
   1. Draw an arc from A intersecting AB at E and AC at F.
   2. Draw circles with centres E and F, radius EF.
   3. The intersection point G lies on the bisector; draw AG.
3. Constructing a square from a given side:
   1. Use the segment as one side.
   2. Construct perpendicular bisectors at each endpoint to get the other sides.
   3. Connect the new points to complete the square.

Remember: the compass must keep the same radius when moving from one point to another – this is the key to accurate construction! 🎯