CBSE Class 7 Maths Chapter 5: Lines And Angles Revision Notes

Essential Geometric Definitions

Line Segment

A part of a line with two fixed endpoints. It has a definite length.

Ray

A part of a line that has one starting point (vertex) and extends infinitely in the other direction.

Complementary Angles

A pair of angles whose measures sum up to exactly $90^\circ$.

Supplementary Angles

A pair of angles whose measures sum up to exactly $180^\circ$.

Adjacent Angles

Two angles that share a common vertex, a common arm, and have non-common arms on opposite sides of the common arm.

Linear Pair

A pair of adjacent angles whose non-common arms are opposite rays, always summing up to $180^\circ$.

Vertically Opposite Angles

Angles formed opposite to each other when two lines intersect. They are always equal in measure.

Transversal

A line that intersects two or more other lines at distinct points.

Understanding Angle Pairs in Detail

In geometry, angles rarely exist in isolation. Understanding angle pairs is essential to solving complex geometric proofs. When two angles sum up to exactly $90^\circ$, they are known as complementary angles. For instance, $30^\circ$ and $60^\circ$ complement each other. If the sum of two angles is $180^\circ$, they are called supplementary angles, such as $110^\circ$ and $70^\circ$.

Another critical relationship is adjacent angles. These share a common vertex and a common arm, but do not overlap. When adjacent angles are supplementary, their non-common arms form a straight line, creating a linear pair. Additionally, when two straight lines intersect, they form vertically opposite angles directly across from each other. A key theorem to remember for your CBSE exams is that vertically opposite angles are always equal in measure.

Comparison: Complementary vs. Supplementary Angles

Identifying Angles Made by a Transversal

1. — Find the single straight line that intersects two other coplanar lines at distinct points.
2. — Angles lying between the two lines are interior angles; angles lying outside are exterior angles.
3. — Look for angles in corresponding positions at each intersection (forming an 'F' shape). If lines are parallel, these are equal.
4. — Look for pairs of angles on opposite sides of the transversal. Interior alternate angles form a 'Z' shape and are equal if the lines are parallel.
5. — Identify interior angles on the same side of the transversal (forming a 'C' or 'U' shape). If lines are parallel, they sum up to $180^\circ$ (supplementary).

Solved Quick-Revision Examples

• {"title":"Example 1: Finding Complementary Angles","content":"Find the angle which is equal to its complement.\n\nSolution:\nLet the required angle be $x$. Its complement will also be $x$.\nSince they are complementary, their sum is $90^\\circ$:\n$x + x = 90^\\circ$\n$2x = 90^\\circ \\implies x = 45^\\circ$\nTherefore, the angle equal to its complement is $45^\\circ$."}
• {"title":"Example 2: Applying Linear Pair Concept","content":"In a linear pair, one angle is $115^\\circ$. Find the measure of the other angle.\n\nSolution:\nWe know that the sum of angles in a linear pair is $180^\\circ$.\nLet the unknown angle be $y$.\n$y + 115^\\circ = 180^\\circ$\n$y = 180^\\circ - 115^\\circ = 65^\\circ$\nTherefore, the other angle is $65^\\circ$."}

Must Remember Points for Exams

• A line segment has two fixed end points, whereas a line extends infinitely in both directions without any endpoints.
• A ray starts from one fixed point and extends infinitely in one direction.
• Two angles are complementary if their sum is exactly $90^\circ$.
• Two angles are supplementary if their sum is exactly $180^\circ$.
• Adjacent angles must have a common vertex, a common arm, and no overlapping interior points.
• The angles of a linear pair always lie on a straight line and are supplementary ($180^\circ$).
• When two lines intersect, the vertically opposite angles formed are always equal.
• If two parallel lines are cut by a transversal, each pair of corresponding angles and alternate angles are equal.
• If two parallel lines are cut by a transversal, interior angles on the same side of the transversal are supplementary ($180^\circ$).

Exam Traps & Smart Tricks

Exam Warning: Students often make the mistake of assuming alternate interior angles or corresponding angles are always equal. Remember, this is only true if and only if the two lines intersected by the transversal are parallel ($l \parallel m$). Always look for the parallel notation or given statements in your question paper before equating them! Also, verify if a pair of angles is adjacent by checking all three criteria (vertex, arm, non-overlapping interior).

Quick Check: Class 7 Lines and Angles Quiz

• Can two obtuse angles be supplementary to each other? No. An obtuse angle is strictly greater than $90^\circ$. If we add two angles greater than $90^\circ$, their sum will always exceed $180^\circ$, making it impossible for them to be supplementary.
• Find the supplement of $105^\circ$. Since supplementary angles sum to $180^\circ$, the supplement of $105^\circ$ is $180^\circ - 105^\circ = 75^\circ$.
• If two lines intersect and one pair of vertically opposite angles is acute, what will the other pair be? The other pair of vertically opposite angles must be obtuse. This is because any adjacent pair of angles on a straight line must form a linear pair (summing up to $180^\circ$ with the acute angle).
• Two parallel lines are cut by a transversal. If one interior angle is $80^\circ$, find its co-interior angle. Interior angles on the same side of a transversal (co-interior angles) are supplementary when lines are parallel. Therefore, the co-interior angle is $180^\circ - 80^\circ = 100^\circ$.

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