Angle Types and Perpendicularity

Types of Angles (Acute, Obtuse, Right, Straight, Reflex, Zero, Complete)

Angles are fundamental geometric shapes whose measure quantifies the amount of rotation between two rays sharing a common vertex. Based on this measure, angles are classified into various types. Understanding these classifications is essential for describing and working with geometric figures and relationships. The measure of an angle is most commonly expressed in degrees ($^\circ$).

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Classification of Angles by Measure

Here, we explore the primary types of angles, defined by their specific range or exact degree measure:

1. Zero Angle

A Zero angle is the angle formed when the two arms of the angle lie exactly on top of each other, pointing in the same direction, with their vertex coinciding. There is no rotation between the two arms. Its measure is precisely $0^\circ$.

Visually, the two rays appear as a single ray.

2. Acute Angle

An Acute angle is any angle whose measure is greater than zero degrees ($0^\circ$) but strictly less than ninety degrees ($90^\circ$). Acute angles are "sharp" or "pointed".

If $\theta$ represents the measure of an angle, then an acute angle satisfies the condition: $0^\circ < \theta < 90^\circ$.

Examples of acute angles include $15^\circ$, $45^\circ$, $78^\circ$, $89.9^\circ$.

3. Right Angle

A Right angle is an angle whose measure is exactly $90^\circ$ (ninety degrees). The arms of a right angle are perpendicular to each other. Right angles are very common and are typically found at the corners of squares, rectangles, and many other structures.

Right angles are often indicated in diagrams by a small square drawn at the vertex, rather than a curved arc.

Example: The angle formed by the adjacent sides of a book or a table corner.

If $\theta$ is the measure of a right angle, then $\theta = 90^\circ$.

4. Obtuse Angle

An Obtuse angle is an angle whose measure is greater than ninety degrees ($90^\circ$) but strictly less than one hundred eighty degrees ($180^\circ$). Obtuse angles are "wide" or "blunt".

If $\theta$ represents the measure of an angle, then an obtuse angle satisfies the condition: $90^\circ < \theta < 180^\circ$.

Examples of obtuse angles include $95^\circ$, $130^\circ$, $175^\circ$, $179.9^\circ$.

5. Straight Angle

A Straight angle is an angle whose measure is exactly $180^\circ$ (one hundred eighty degrees). When the two arms of an angle form a straight line, extending in exactly opposite directions from the vertex, the angle is a straight angle. The vertex lies somewhere on the straight line formed by the arms.

Note that the two rays forming a straight angle are called opposite rays because they share a common endpoint (the vertex) and extend in perfectly opposite directions along the same line.

6. Reflex Angle

A Reflex angle is an angle whose measure is greater than one hundred eighty degrees ($180^\circ$) but strictly less than three hundred sixty degrees ($360^\circ$). When two rays form an angle, there are usually two angles created (unless it's a straight, zero, or complete angle): the interior angle (usually less than or equal to $180^\circ$) and the reflex angle (the larger angle traced around the outside).

If $\theta$ represents the measure of a reflex angle, then $180^\circ < \theta < 360^\circ$.

Examples of reflex angles include $210^\circ$, $270^\circ$, $300^\circ$, $350^\circ$.

For any angle $\alpha$ that is less than $180^\circ$ (acute, right, or obtuse), the measure of the corresponding reflex angle is $360^\circ - \alpha$.

7. Complete Angle (or Full Angle)

A Complete angle, also known as a Full angle, is formed when a ray completes one full rotation around its endpoint and returns to its original position. The measure of a complete angle is exactly $360^\circ$ (three hundred sixty degrees). Graphically, a complete angle looks identical to a zero angle (the arms overlap), but it represents a full turn.

It represents a full revolution.

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Summary of Angle Types

Here is a table summarising the classification based on measure:

Type of Angle	Measure ($\theta$)
Zero Angle	$\theta = 0^\circ$
Acute Angle	$0^\circ < \theta < 90^\circ$
Right Angle	$\theta = 90^\circ$
Obtuse Angle	$90^\circ < \theta < 180^\circ$
Straight Angle	$\theta = 180^\circ$
Reflex Angle	$180^\circ < \theta < 360^\circ$
Complete Angle / Full Angle	$\theta = 360^\circ$

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Example

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Perpendicular Lines: Definition and Properties

We have discussed intersecting lines, which are lines that cross at a single point. A special and highly important case of intersecting lines occurs when they cross each other in a specific manner, forming a right angle. Such lines are known as perpendicular lines. This concept is fundamental in geometry, appearing in various shapes (like squares, rectangles, right triangles) and having numerous applications in real-world constructions, architecture, and coordinate geometry.

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Definition

Two distinct lines, or a line and a ray, or two rays, or two line segments, are said to be perpendicular to each other if they intersect (meet) at a point and form a right angle ($90^\circ$) at their point of intersection.

The symbol used to denote perpendicularity is $\perp$. If a line $l$ is perpendicular to a line $m$, we write this relationship as $l \perp m$. This notation is concise and universally understood in mathematics.

Consider two lines, say line AB and line CD, intersecting at a point O. If the angle formed at O by any arm of line AB and any arm of line CD is $90^\circ$, then line AB is perpendicular to line CD.

Observe the following diagram illustrating perpendicular lines:

In the diagram, the lines AB and CD intersect at point O. The small square at O indicates that the angle formed, specifically $\angle \text{AOC}$, is a right angle, i.e., $m\angle \text{AOC} = 90^\circ$. Because one angle is $90^\circ$, all the angles formed at the intersection are $90^\circ$ (due to properties of intersecting lines, like vertically opposite angles and angles on a straight line). Therefore, line AB is perpendicular to line CD, which can be written as $\overleftrightarrow{\text{AB}} \perp \overleftrightarrow{\text{CD}}$.

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Properties of Perpendicular Lines

Perpendicular lines have several key properties that are useful in geometric problems and proofs:

• All Angles are Right Angles: When two lines intersect perpendicularly, they form four angles at the point of intersection. All four of these angles are right angles, each measuring $90^\circ$.
  

  If $\overleftrightarrow{\text{AB}} \perp \overleftrightarrow{\text{CD}}$ at O, then $\angle \text{AOC}$, $\angle \text{COB}$, $\angle \text{BOD}$, and $\angle \text{DOA}$ are all $90^\circ$ angles.

• A Special Case of Intersecting Lines: Perpendicular lines are always intersecting lines. Parallel lines, which do not intersect, cannot be perpendicular.
• Uniqueness in a Plane (Through a point on the line): In a given plane, through any point on a given line, there exists exactly one line that is perpendicular to the given line and passes through that specific point. This property is fundamental for constructions involving perpendiculars.
• Uniqueness in a Plane (Through a point not on the line): In a given plane, through any point not on a given line, there exists exactly one line that is perpendicular to the given line and passes through that external point. This allows us to drop a perpendicular from a point to a line.
• Shortest Distance: The shortest distance from a point to a line is measured along the perpendicular line segment from the point to the line. Any other segment connecting the point to the line will be longer than the perpendicular segment.
• Parallel Lines and Perpendicularity: If two lines are both perpendicular to the same line in a plane, then the two lines are parallel to each other. (e.g., if $l \perp n$ and $m \perp n$, then $l || m$). Conversely, if a transversal is perpendicular to one of two parallel lines, it is perpendicular to the other as well.

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Answer:

Perpendicular Bisector

Building on the concepts of line segments and perpendicularity, we encounter the notion of a perpendicular bisector. This is a geometric entity (a line, ray, or line segment) that performs two specific actions simultaneously on a given line segment: it cuts the segment exactly in half, and it does so at a right angle. The perpendicular bisector holds significant importance in geometric constructions and theorems.

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Definition

A line (or a ray or a line segment) is defined as the perpendicular bisector of a given line segment if it satisfies the following two conditions:

1. It intersects the given line segment at its midpoint. A midpoint is a point that divides a line segment into two equal halves. If M is the midpoint of segment $\overline{\text{AB}}$, then the length of $\overline{\text{AM}}$ is equal to the length of $\overline{\text{MB}}$.
2. It is perpendicular to the given line segment at the point of intersection. This means the angle formed between the bisector and the line segment at their intersection point is exactly $90^\circ$.

Let $\overline{\text{AB}}$ be a line segment and let $l$ be a line. If line $l$ is the perpendicular bisector of $\overline{\text{AB}}$, it means there is a point M on $\overline{\text{AB}}$ such that M is the midpoint of $\overline{\text{AB}}$ (so $\text{AM} = \text{MB}$), and line $l$ intersects $\overline{\text{AB}}$ at M such that $l \perp \overline{\text{AB}}$ at M.

The perpendicular bisector is a line, meaning it extends infinitely in both directions, even though it is defined in relation to a finite segment.

Consider the following diagram:

In this diagram, line $l$ passes through point M on the line segment $\overline{\text{AB}}$. The marks on $\overline{\text{AM}}$ and $\overline{\text{MB}}$ indicate that their lengths are equal, signifying that M is the midpoint of $\overline{\text{AB}}$. The small square symbol at M indicates that the angle between line $l$ and line segment $\overline{\text{AB}}$ is $90^\circ$. Therefore, line $l$ is the perpendicular bisector of segment $\overline{\text{AB}}$.

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Properties of a Perpendicular Bisector

The perpendicular bisector has several important properties, particularly related to the distance of points from the endpoints of the segment:

• Equidistance Property (Forward): Any point that lies on the perpendicular bisector of a line segment is equidistant from the two endpoints of the segment. That is, if P is any point on the perpendicular bisector of $\overline{\text{AB}}$, then the distance from P to A ($\text{PA}$) is equal to the distance from P to B ($\text{PB}$).
  

  In the figure above, if line $l$ is the perpendicular bisector of $\overline{\text{AB}}$ and P is any point on $l$, then $\text{PA} = \text{PB}$. This forms the basis for many constructions and proofs.

• Equidistance Property (Converse): Conversely, any point in the plane that is equidistant from the two endpoints of a line segment must lie on the perpendicular bisector of that segment. If $\text{PA} = \text{PB}$, then point P lies on the perpendicular bisector of $\overline{\text{AB}}$. This is the converse of the first property and is equally important.
• Uniqueness: For any given line segment, there exists one and only one perpendicular bisector. This line is unique.

These properties are fundamental and can be formally proven using congruent triangles (for the forward property) or other geometric principles.

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