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📖 Mathematics

Std 7
Curriculum Book Guide
1 2 3 4 5 6 7 8
Chapters
1 Large Numbers Around Us 2 Arithmetic Expressions 3 A Peek Beyond the Point 4 Expressions Using Letter-Numbers 5 Parallel and Intersecting Lines 6 Number Play 7 A Tale of Three Intersecting Lines 8 Working with Fractions
7
Chapter 7
Skill: 30%

A Tale of Three Intersecting Lines

A Tale of Three Intersecting Lines

Chapter at a Glance

This chapter covers the properties, constructions, and classifications of triangles. It introduces the Triangle Inequality Theorem, demonstrating why certain side combinations cannot form closed three-sided shapes. Using paper folding, set squares, and compasses, the text details how to construct triangles under SSS (Side-Side-Side), SAS (Side-Angle-Side), and ASA (Angle-Side-Angle) conditions. It proves the Angle Sum Property ($180^\circ$) using alternate interior angles on parallel lines and explains the Exterior Angle Theorem. Finally, it details altitudes (heights) of triangles and classifies triangles based on their side lengths and angle measures.

Key Definitions & Terminology

  • Triangle: A closed, three-sided geometric shape formed by three intersecting lines, consisting of three vertices and three interior angles.
  • Equilateral Triangle: A triangle with all three side lengths equal.
  • Isosceles Triangle: A triangle with at least two side lengths equal.
  • Scalene Triangle: A triangle with three different side lengths.
  • Acute-Angled Triangle: A triangle in which all three interior angles are acute ($<90^\circ$).
  • Right-Angled Triangle (Right Triangle): A triangle containing exactly one right angle ($90^\circ$).
  • Obtuse-Angled Triangle: A triangle containing exactly one obtuse angle ($>90^\circ$).
  • Altitude: A perpendicular line segment drawn from a vertex of a triangle to its opposite side (or an extension of it).
  • Exterior Angle: An angle formed between one side of a triangle and the extension of an adjacent side.

Formulas, Rules & Properties

  • Triangle Inequality Theorem:
  • In any triangle with sides $a, b, c$:
    $$a < b + c, \quad b < a + c, \quad c < a + b$$
  • Short Test: A triangle exists if and only if the sum of the two shorter sides is strictly greater than the longest side:
    $$a_{\text{short1}} + a_{\text{short2}} > a_{\text{longest}}$$
  • feaseability of Arcs Intersection (Base $c$ and Radii $a, b$):
  • Case 1 (Touch at a point): $a + b = c \implies$ No triangle (forms a straight line).
  • Case 2 (No intersection): $a + b < c \implies$ No triangle.
  • Case 3 (Internal intersection): $a + b > c \implies$ Triangle exists.
  • Angle Sum Property of Triangles:
  • The sum of the interior angles of any triangle is always $180^\circ$:
    $$\angle A + \angle B + \angle C = 180^\circ$$
  • Exterior Angle Theorem:
  • The measure of an exterior angle of a triangle equals the sum of its two interior opposite angles:
    $$\angle ACD = \angle A + \angle B$$
  • SAS Included Angle Boundary:
  • The included angle $\theta$ between two given sides must satisfy $0^\circ < \theta < 180^\circ$.
  • ASA Feasibility Rule:
  • A triangle exists if and only if the sum of the two given base angles $A$ and $B$ satisfies:
    $$0^\circ < A + B < 180^\circ$$

Core Concepts & Topics

  • Efficient SSS Construction with Compass:
  • Constructing a triangle by drawing base $AB$, creating an arc of radius $r_1$ from vertex $A$, creating a second arc of radius $r_2$ from vertex $B$, and setting the intersection point of the arcs as vertex $C$.
  • Euclid's Proof of the Angle Sum Property:
  • Constructing a line $XY$ parallel to base $BC$ through vertex $A$.
  • Using transversals $AB$ and $AC$ to identify alternate interior angles: $\angle XAB = \angle B$ and $\angle YAC = \angle C$.
  • Since $XY$ is a straight line, $\angle XAB + \angle BAC + \angle YAC = 180^\circ$, which proves $\angle B + \angle A + \angle C = 180^\circ$.
  • Altitudes in Obtuse Triangles:
  • The altitude from an acute vertex to its opposite side in an obtuse triangle falls outside the shape. The base line must be extended to drop the perpendicular from the vertex.
  • Set Square Altitude Construction:
  • Aligning a ruler to the base, sliding a set square along the ruler until its vertical edge aligns with the opposite vertex, and drawing the perpendicular line segment.
  • Triangles with Sides as Altitudes:
  • In a right-angled triangle, the two perpendicular sides serve as altitudes to each other.

Worked Examples

  • SSS Sidelength Feasibility Checks:
  • Sidelengths $3\text{ cm}, 4\text{ cm}, 8\text{ cm}$: $3 + 4 = 7 < 8 \implies$ Triangle does not exist.
  • Sidelengths $2\text{ cm}, 3\text{ cm}, 6\text{ cm}$: $2 + 3 = 5 < 6 \implies$ Triangle does not exist.
  • Sidelengths $10\text{ km}, 10\text{ km}, 25\text{ km}$: $10 + 10 = 20 < 25 \implies$ Triangle does not exist.
  • Sidelengths $5\text{ mm}, 10\text{ mm}, 20\text{ mm}$: $5 + 10 = 15 < 20 \implies$ Triangle does not exist.
  • Sidelengths $12\text{ cm}, 20\text{ cm}, 40\text{ cm}$: $12 + 20 = 32 < 40 \implies$ Triangle does not exist.
  • Triangle Inequality Checks:
  • Can $2, 2, 5$ form a triangle? No, $2+2 < 5$.
  • Can $3, 4, 6$ form a triangle? Yes, $3 < 4+6$, $4 < 6+3$, $6 < 3+4$.
  • Can $5, 5, 8$ form a triangle? Yes, $5 < 5+8$, $8 < 5+5$.
  • Can $10, 20, 25$ form a triangle? Yes, $10 < 20+25$, $20 < 25+10$, $25 < 10+20$.
  • Can $10, 20, 35$ form a triangle? No, $35 > 10+20$.
  • Can $24, 26, 28$ form a triangle? Yes, $24 < 26+28$, $26 < 28+24$, $28 < 24+26$.
  • Can $3, 6, 9$ form a triangle? No, $3+6=9$ (Case 1: circles touch).
  • Determining the Third Side Range:
  • Given sides $1, 100$: The third side $x$ must satisfy $100 - 1 < x < 100 + 1 \implies 99 < x < 101$. Possible values: $99.5, 100, 100.7, 100.5, 99.4$.
  • Given sides $5, 5$: The third side $x$ must satisfy $5-5 < x < 5+5 \implies 0 < x < 10$. Possible values: $5, 4, 3, 4.9, 1$.
  • Given sides $3, 7$: The third side $x$ must satisfy $7-3 < x < 7+3 \implies 4 < x < 10$. Possible values: $5, 8, 7, 6.4, 6$.
  • Base Angle Feasibility Ranges:
  • Problem: An angle of $40^\circ$ is at base vertex $A$. Find the values of base angle $B$ for which a triangle cannot be formed.
  • Solution: If the arms from $A$ and $B$ are parallel, they never meet. Transversal $AB \implies \angle A + \angle B = 180^\circ \implies \angle B = 140^\circ$. For any angle $\angle B \ge 140^\circ$, a triangle cannot be formed.
  • Feasible Angle Pairs:
  • Can $35^\circ, 150^\circ$ be angles of a triangle? No, $35^\circ + 150^\circ = 185^\circ > 180^\circ$.
  • Can $70^\circ, 30^\circ$ be angles of a triangle? Yes, $70^\circ + 30^\circ = 100^\circ < 180^\circ$.
  • Can $90^\circ, 85^\circ$ be angles of a triangle? Yes, $90^\circ + 85^\circ = 175^\circ < 180^\circ$.
  • Can $50^\circ, 150^\circ$ be angles of a triangle? No, $50^\circ + 150^\circ = 200^\circ > 180^\circ$.
  • Finding the Third Angle using Alternate Angles:
  • Problem: In $\triangle ABC$, find the third angle if two angles are $36^\circ$ and $72^\circ$.
  • Solution: Draw line $XY \parallel BC$ through $A$. Alternate interior angles: $\angle XAB = \angle B = 36^\circ$ and $\angle YAC = \angle C = 72^\circ$. Straight line sum: $36^\circ + \angle BAC + 72^\circ = 180^\circ \implies \angle BAC = 72^\circ$.
  • Same process:
    • $150^\circ, 15^\circ \implies 180^\circ - (150^\circ + 15^\circ) = 15^\circ$.
    • $90^\circ, 30^\circ \implies 180^\circ - (90^\circ + 30^\circ) = 60^\circ$.
    • $75^\circ, 45^\circ \implies 180^\circ - (75^\circ + 45^\circ) = 60^\circ$.
  • Two Equal Angles:
  • Problem: In $\triangle ABC$, $\angle B = \angle C$ and $\angle A = 50^\circ$. Find $\angle B$ and $\angle C$.
  • Solution: $50^\circ + 2\angle B = 180^\circ \implies 2\angle B = 130^\circ \implies \angle B = \angle C = 65^\circ$.
  • Exterior Angle Calculation:
  • Problem: In $\triangle ABC$, $\angle A = 50^\circ, \angle B = 60^\circ$. Find exterior angle $\angle ACD$.
  • Solution: $\angle ACD = \angle A + \angle B = 50^\circ + 60^\circ = 110^\circ$.

Practical Activities & Experiments

  • Compass SSS Triangles: Constructing equilateral and scalene triangles by setting a compass to given side lengths and drawing intersecting arcs.
  • Overlapping Circles Construction: Drawing two equal circles centered at $A$ and $B$ that pass through each other's centers, and using their intersection points to construct equilateral and isosceles triangles.
  • Euclidean Paper folding sum proof: Folding a triangular paper cutout so that the three corners meet at a single point on the base line, demonstrating they form a straight line ($180^\circ$).
  • Dropping Altitudes via set square sliding: Placing a set square against a ruler along the base of a triangle and sliding it until the vertical edge touches the opposite vertex to draw a perpendicular line.
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