Chapter 5 — Lines And Angles

Overview
This topic introduces the fundamental building blocks of geometry: points, lines, line segments, rays and angles, together with relationships such as parallel, perpendicular and intersecting. These definitions and properties are used to describe and measure shapes and figures.

Basic terms

• Point: A location in space with no size. Denoted by a capital letter (for example, A).
• Line: A straight one-dimensional figure that extends infinitely in both directions. Usually named by a lowercase letter (line l) or by two points on it (line AB).
• Line segment: Part of a line bounded by two end points A and B. Denoted AB (or BA).
• Ray: A part of a line that begins at a point and extends infinitely in one direction. Ray AB starts at A and passes through B.
• Collinear points: Points that lie on the same straight line.
• Intersecting lines: Two lines that cross each other at exactly one point.
• Parallel lines: Two lines in the same plane that never meet. Marked l || m.
• Perpendicular lines: Two lines that meet at a right angle (90°). Marked l ⟂ m.

Angle and its parts

• Angle: Formed by two rays with a common end point. The common end point is the vertex, the rays are the arms or sides. An angle with vertex B and arms BA and BC is denoted ∠ABC or ∠CBA.
• Measure of an angle: The amount of rotation from one arm to the other, measured in degrees (°).

Types of angles

• Acute angle: > 0° and < 90°.
• Right angle: Exactly 90°.
• Obtuse angle: > 90° and < 180°.
• Straight angle: Exactly 180° (forms a straight line).
• Reflex angle: > 180° and < 360°.
• Complete angle: Exactly 360°.

Angle relationships

• Adjacent angles: Two angles that have a common vertex and a common arm and do not overlap.
• Linear pair: A pair of adjacent angles whose non-common arms form a straight line. Their measures add to 180°.
• Vertically opposite angles: When two lines intersect, the opposite angles formed are equal.
• Complementary angles: Two angles whose measures add to 90°.
• Supplementary angles: Two angles whose measures add to 180°.

Notation and conventions
Use three letters to name an angle (middle letter is the vertex), e.g., ∠ABC. Use a single capital letter for a point, a pair of capital letters for a ray or segment, and a lowercase letter or two-point notation for a line.

How these are used
These definitions let us classify figures, measure angles with a protractor, reason about equalities (for example, equal corresponding angles when a transversal cuts parallel lines), and solve geometric problems.

An angle is formed when two rays (or line segments) meet at a common endpoint called the vertex. Types of angles are classified by their measures (in degrees) and by their positional relationships. Below are the common types taught in Class 7 with brief descriptions and properties.

• Zero angle (0°): Both rays coincide. Measure = 0°.
• Acute angle: Greater than 0° but less than 90°. (0° < angle < 90°)
• Right angle: Exactly 90°. Often shown with a small square at the vertex.
• Obtuse angle: Greater than 90° but less than 180°. (90° < angle < 180°)
• Straight angle: Exactly 180°. The two rays form a straight line.
• Reflex angle: Greater than 180° but less than 360°. (180° < angle < 360°)
• Full (complete) angle: Exactly 360°. The ray makes a full rotation about the vertex.

Important angle relationships (often studied with types):

• Complementary angles: Two angles whose measures add up to 90°.
• Supplementary angles: Two angles whose measures add up to 180°.
• Linear pair: A pair of adjacent angles formed when two lines intersect; they are supplementary.
• Vertically opposite angles: When two lines intersect, the opposite (vertical) angles are equal.
• Adjacent angles: Two angles that share a common side and vertex and do not overlap.

How to identify: measure an angle with a protractor from one ray to the other. Compare the measured value with the ranges above to classify the angle.

Tips for students: practise sketching each type, use a protractor to measure angles, and recognise angle types in shapes (e.g., right angles in rectangles, acute angles in equilateral triangles).

What is an angle?
Angle is the figure formed by two rays (sides) with a common endpoint called the vertex. It measures the amount of rotation one ray must be turned about the vertex to coincide with the other ray.

Units of measurement
Angles are measured in degrees (°). A full rotation is 360°, a straight angle is 180°, and a right angle is 90°. (For higher precision we can use minutes and seconds: 1° = 60 minutes (′), 1′ = 60 seconds (″)).

Types of angles

• Acute angle: between 0° and 90° (e.g., 30°)
• Right angle: exactly 90°
• Obtuse angle: between 90° and 180°
• Straight angle: exactly 180°
• Reflex angle: between 180° and 360°
• Complete angle: exactly 360°

Measuring an angle with a protractor

1. Place the protractor so its origin (center hole or midpoint mark) is on the vertex of the angle.
2. Align the baseline of the protractor with one side (ray) of the angle.
3. Read the number on the curved edge where the other side crosses. Use the inner or outer scale depending on how the baseline is aligned.

Basic constructions (using ruler, compass, protractor)

• To draw an angle of a given measure using a protractor: draw a ray from the vertex, place protractor, mark the required degree on paper, and join the mark to the vertex.
• To copy an angle using compass and ruler (no protractor):
  1. From the vertex of the given angle, draw an arc that cuts both rays; measure the distance between those intersection points using the compass.
  2. From the new vertex, draw a similar arc, then using the compass width, mark the second intersection on the arc; join this mark to the new vertex — the copied angle is formed.
• To bisect an angle (divide into two equal angles) with compass and ruler:
  1. From the vertex draw an arc that cuts both sides at A and B.
  2. With the same radius, draw arcs centered at A and B so they intersect at P.
  3. Join vertex to P. VP is the angle bisector.

Important relationships

• Complementary angles: two angles whose sum is 90°.
• Supplementary angles: two angles whose sum is 180°.
• Linear pair: adjacent angles formed by a straight line; they are supplementary.
• Vertically opposite angles: when two lines intersect, opposite angles are equal.
• All angles around a point sum to 360°.

Practical tips
Use the smaller protractor scale for acute angles and check alignment carefully. When constructing with compass, keep the compass width fixed during transfer steps. Label points clearly to avoid confusion.

What is an angle? An angle is formed by two rays (sides) with a common endpoint called the vertex. Angles are measured in degrees (°).

Basic angle types

• Acute angle: between 0° and 90°
• Right angle: exactly 90°
• Obtuse angle: between 90° and 180°
• Straight angle: exactly 180°
• Reflex angle: between 180° and 360°

Key angle relationships

• Adjacent angles: Two angles that share a common side and vertex and do not overlap.
• Linear pair: A pair of adjacent angles whose non-common sides form a straight line. Sum = 180° (they are supplementary).
• Supplementary angles: Two angles whose measures add up to 180°.
• Complementary angles: Two angles whose measures add up to 90°.
• Vertically opposite angles: When two lines intersect, opposite (vertical) angles are equal.
• Parallel lines cut by a transversal: When a transversal crosses two parallel lines, special equalities hold:
  • Corresponding angles are equal.
  • Alternate interior angles are equal.
  • Alternate exterior angles are equal.
  • Interior angles on the same side (consecutive interior) are supplementary (sum to 180°).

How to identify relationships

• Look for shared vertex and side — they are adjacent; if their other sides are opposite directions on a straight line, they form a linear pair.
• If two lines intersect, mark opposite angles — those are equal (vertically opposite).
• If a transversal intersects two parallel lines, label the eight formed angles. Use known equalities (corresponding, alternate) and supplementary pairs to compute unknown angles.

Reasoning and proofs (short): Most angle relationship results come from the idea that a straight line gives 180° and a point gives 360°. Equality of corresponding and alternate angles follows from the definition of parallel lines (same direction) and congruent triangles built by cutting with a transversal.

Tips for solving problems

• Label all angles; use x, y for unknowns and write equations using 90°, 180°, or 360° as needed.
• Use equal-angle facts for parallel lines to replace angles and solve for variables.
• Check answers by verifying sums (e.g., linear pairs sum to 180°).

Definition: Two lines are parallel if they lie in the same plane and never meet, no matter how far extended. A transversal is a line that crosses two or more lines at distinct points.

When a transversal intersects two lines, it forms eight angles (four at each intersection). If the two lines are parallel, these angles have special relationships. Understanding these angle pairs helps in solving many geometry problems.

Types of angle pairs (with a transversal cutting two lines):

• Corresponding angles: Angles in the same relative position at each intersection (e.g., top-left at first intersection and top-left at second). If lines are parallel, corresponding angles are equal.
• Alternate interior angles: Interior angles (between the two lines) that lie on opposite sides of the transversal. For parallel lines these are equal.
• Alternate exterior angles: Exterior angles (outside the two lines) on opposite sides of the transversal. For parallel lines these are equal.
• Consecutive interior angles (also called same-side interior): Interior angles on the same side of the transversal. For parallel lines these add up to 180° (supplementary).

Important consequences (stated simply):

• If two lines are parallel, then corresponding angles are equal.
• If two lines are parallel, then each pair of alternate interior angles is equal.
• If two lines are parallel, then consecutive interior angles are supplementary (sum to 180°).
• The converses are also true: if corresponding (or alternate interior) angles are equal, the two lines are parallel.

How to use this: When given angle measures, use these relationships to find unknown angles. For example, if a corresponding angle is 55°, the matching corresponding angle is also 55°. If one interior angle is 70°, the consecutive interior angle adjacent on the same side of the transversal is 110°.

Tips for sketching: Draw two clear parallel lines (use arrows on them) and a transversal. Label angles 1–8 around the intersections so you can refer to corresponding, alternate, and interior pairs easily. Colour equal angles or put the same number of arc marks on equal angles to visualise relationships.

Definition: Two lines in the same plane that never meet, no matter how far they are extended, are called parallel lines. We write l || m to mean line l is parallel to line m.

Set-up: When two parallel lines l and m are cut by a third line t (called a transversal), several angle relationships appear. Label the eight angles around the two intersection points with the transversal as usual (four at the first intersection and four at the second).

Properties (when l || m):

• Corresponding angles are equal. Each angle in one intersection has a matching angle in the other intersection that lies in the same relative position. Example: ∠1 = ∠5.
• Alternate interior angles are equal. Angles on opposite sides of the transversal and between the two lines are equal. Example: ∠3 = ∠5.
• Alternate exterior angles are equal. Angles on opposite sides of the transversal and outside the two lines are equal. Example: ∠1 = ∠7.
• Consecutive interior (co-interior) angles are supplementary. Interior angles on the same side of the transversal add to 180°. Example: ∠3 + ∠6 = 180°.
• Perpendicular-to-same-line rule: If two lines are each perpendicular to the same line, they are parallel to each other.
• Equidistance: Parallel lines are at a constant distance from each other (they are equidistant everywhere).

Why these are true (short proofs):

• Corresponding angles: Consider straight lines: the angle at the first intersection plus the adjacent interior angle form a linear pair (sum 180°). Using the fact that corresponding positions mirror each other across the transversal, you can use congruent triangle arguments or angle addition to show equality.
• Alternate interior/exterior: By reflecting positions across the transversal or using triangle congruence, alternate angles match because parallel lines maintain the same direction (slope) so the transversal cuts them in the same way.
• Co-interior supplementary: Each co-interior angle is supplementary to the same corresponding angle at the other intersection; hence they sum to 180°.

Converses: Each property can be used in reverse to prove two lines are parallel. These are important for solving geometry problems:

• If a transversal makes a pair of corresponding angles equal, then the two lines are parallel.
• If a transversal makes a pair of alternate interior angles equal, then the two lines are parallel.
• If a transversal makes a pair of alternate exterior angles equal, then the two lines are parallel.
• If a transversal makes a pair of co-interior angles supplementary (sum to 180°), then the two lines are parallel.
• If two lines are each perpendicular to a third line, then they are parallel to each other.

How to use these in problems: When you are given angle measures created by a transversal, match the angle relationship (corresponding, alternate, co-interior) and either conclude equality/supplementarity (if lines are given parallel) or conclude the lines are parallel (using the converse) to find missing angles or prove statements.

Tips for students: Always mark known parallel signs and label angles. Use the simplest matching rule first (corresponding or alternate) and remember that linear pairs sum to 180°.

What are perpendicular lines? Two lines are perpendicular if they meet to form a right angle (90°). We write this as l ⟂ m. Perpendicularity is a fundamental relation in geometry used to create right angles, squares, and many constructions.

Basic properties

• The angle between two perpendicular lines equals 90°.
• If a line is perpendicular to one of two parallel lines, it is perpendicular to the other.
• The perpendicular bisector of a segment is the line of points equidistant from the segment's endpoints.

Three common constructions (using ruler and compass)

1. Perpendicular at a point on a line (to draw a line through a point P on line AB so that it is perpendicular to AB):
   Step 1: With center P, draw arcs of equal radius cutting AB at two points, X and Y.
   Step 2: With centers X and Y and a radius greater than half XY, draw two arcs that intersect at Q (above) and R (below).
   Step 3: Draw line QR; it passes through P and is perpendicular to AB.
2. Perpendicular from a point outside a line (drop a perpendicular from P to line AB):
   Step 1: With center P draw an arc that cuts AB at two points X and Y.
   Step 2: With centers X and Y and radius > XY/2 draw arcs that meet at Q on the side of P.
   Step 3: Draw PQ; it meets AB at the foot of the perpendicular.
3. Perpendicular bisector of a segment AB (to find midpoint and draw a perpendicular):
   Step 1: With centers A and B and the same radius (> half AB) draw two arcs on both sides of AB that intersect at points Q and R.
   Step 2: Draw QR; it is the perpendicular bisector of AB, meeting AB at its midpoint and at right angle.

Why these methods work: The intersecting arcs create equal chords and triangles; by congruence or equal distances, the constructed line is equidistant from endpoints or creates right angles. The perpendicular bisector property follows because every point on it has equal distance to A and B.

When and where used: Perpendicular constructions are used in drawing plans, carpentry (right-angle corners), map making, engineering designs, constructing grids, and solving many geometry problems.

What this topic covers: "Applications and problem-solving" in Lines and Angles asks you to use angle relationships (like vertically opposite, linear pairs, corresponding and alternate angles) to answer questions, often by setting up and solving simple equations. Problems typically involve intersecting lines, a transversal cutting parallel lines, or practical diagrams that model real situations.

Key ideas and approach:

• Identify the type of lines and angles in the figure (intersecting lines, parallel lines with a transversal, perpendicular lines).
• Mark equal or supplementary angles using known relationships (vertically opposite are equal, linear pair sum = 180°, complementary sum = 90°, corresponding/alternate interior equal when lines are parallel, interior on same side supplementary).
• If angles are given algebraically, translate relationships into an equation, solve for the unknown, then find required angles.
• Check your answer by using more than one relationship where possible (consistency check).

Common problem types: finding missing angles formed by intersecting lines; using a transversal with parallel lines to find labelled angles; solving for variables in angle expressions; verifying whether lines are parallel from angle information.

Problem-solving tips:

• Always label the diagram and mark equal or supplementary angles with matching symbols.
• Use different colours or small ticks to group equal angles visually.
• Write the equation that comes from the angle relationship clearly (for example, x + 2y = 180) and solve step by step.
• Remember sum around a point is 360° and straight line gives 180° — these often provide quick checks.