Class 7 | Maths

Chapter 7 (Pt 1) | A Tale of Three Intersecting Lines

Aligned with CBSE | NCERT | School & Olympiad Standards
Focus on What Matters Most
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Must KnowNCERT Core Syllabus
Triangle Inequality
  • Sum of any two sides must be strictly greater than the third side (a+b>c) — this is why 3, 4, 8 cm sticks can never meet at a vertex.
  • Direct path between two points is always shorter than any roundabout path via a third point.
Angle Rules
  • Angle Sum Property: the three interior angles always add up to exactly 180°.
  • Exterior Angle Property: an exterior angle equals the sum of the two opposite interior angles.
Constructions
  • SSS: compass arcs locate the third vertex using two fixed distances.
  • SAS: needs two sides and the angle strictly included between them.
  • ASA: needs one side and two angles whose sum is strictly less than 180°.
  • Altitudes: perpendicular from a vertex to the opposite side, drawn with a set-square.
Classification Snapshot
By sides: Equilateral (all equal) · Isosceles (two equal) · Scalene (none equal).
By angles: Acute (all <90°) · Right (one =90°) · Obtuse (one >90°).
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Three intersecting lines forming a triangle
Quick Recall — Sprint Facts
a + b > cGolden rule for triangle existence
∠A+∠B+∠C = 180°Angle Sum formula
Equilateral = 60°Every angle locked at 60°
Included AngleRequired for SAS construction
Base Angles < 180°Required for ASA construction
Max 1 Right/ObtuseA triangle can have at most one
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🔑 Key Terms to Know
Collinear
Points on the exact same straight line — no triangle can form.
Altitude
Perpendicular height from a vertex to the opposite side.
Equilateral
Three equal sides, three 60° angles.
Isosceles
Exactly two sides equal.
Scalene
All three sides of different lengths.
Included Angle/Side
The angle between two given sides, or side between two given angles.
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🏆 Olympiad & Challenger Edge
Four Triangle Centres
Orthocenter (altitudes), Centroid (medians, 2:1 ratio), Incenter (angle bisectors), Circumcenter (perpendicular bisectors).
Euler Line
Orthocenter, Centroid, and Circumcenter lie on one straight line in every non-equilateral triangle (HG:GO = 2:1).
Cryptarithm Integration
Digit puzzles like X4 + Y = Z11 combine algebra with angle deduction.
Vocabulary Note
Orthogonality (perpendicularity) shows up here only as bonus vocabulary alongside the centres above — not a core NCERT idea this chapter.
The67Sprint
Altitude outside the triangle
For an obtuse triangle the altitude falls outside — extend the base with a dotted line to drop the perpendicular.
Check the longest side first
Sort the three sides, then confirm the two smaller ones sum to more than the longest — don't add lengths randomly.
SAS needs the included angle
The angle must sit sandwiched exactly between the two known sides, not any angle in the figure.
Right angle IS the altitude
In a right-angled triangle, the two perpendicular sides forming the 90° angle already act as the altitudes.
Tent to Tree
Walking straight from tent to mango tree beats going via a flagpole first — the Triangle Inequality in daily life.
Bridges & Roofs
Engineers build with triangles because they are rigid and cannot be deformed without breaking a beam.
Compass as Radar
A compass arc plots every point at one fixed distance — a physical locus, not just a drawing habit.
The Spider-Box Puzzle
Unfolding a 3D box into a flat net turns a spider's shortest crawl into a straight-line hypotenuse problem — a core NCERT activity, not just a HOTS extra.
Missing Units
Leaving out "cm" or "°" on a constructed diagram costs a half mark.
Erasing Compass Arcs
Examiners look for visible intersection arcs as proof of an SSS construction — never erase them.
Solid Line for Extended Base
The extended base for an obtuse altitude must be dotted, not solid, or marks are deducted.
Free-Handing the Altitude
Always show the set-square aligned against the ruler for the 90° drop — don't free-hand it.
Logic Chain — How We Prove Two Lines Are Parallel
Step 1
Draw a line through the triangle's apex parallel to the base — this connects to the alternate interior angles it creates.
Step 2
Those alternate angles equal the triangle's base angles, which connects to rewriting the straight line at the apex as three angles.
Step 3
The three angles on that straight line sum to 180°, which connects directly to the triangle's own three interior angles.
Step 4
This proves the Angle Sum Property logically — and it leads to the ASA construction rule that base angles must sum to less than 180° to ever meet.
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Mentor Byte
A triangle refuses to be told what shape to take — three lengths either lock into exactly one form, or they don't meet at all. Let students fold a paper triangle so its three corners touch the base in one straight line; that flat 180° they see with their own hands will outlast any formula written on a board.
Parent Tip
Hand your child three uncooked spaghetti strands of different lengths and ask them to form a triangle. Snap one shorter and try again — they'll feel the Triangle Inequality before they ever read it.