Class 8 Maths Notes Understanding Quadrilaterals

1. What is a polygon?

A simple closed curve made only of line segments.

2. What is a polygon with 4 sides called?

Quadrilateral.

3. What is a polygon with 6 sides called?

Hexagon.

4. What is a diagonal?

Line segment connecting non-consecutive vertices.

5. What is the sum of angles of a quadrilateral?

360°.

6. What is the sum of angles of a triangle?

180°.

7. What is the formula for sum of interior angles of an n-sided polygon?

(n – 2) × 180°.

8. What is the sum of exterior angles of any convex polygon?

360°.

9. What is a regular polygon?

A polygon that is equilateral and equiangular.

10. Give an example of a regular quadrilateral.

Square.

11. What is a parallelogram?

A quadrilateral with opposite sides parallel.

12. Are opposite sides of a parallelogram equal?

Yes.

13. What is the sum of adjacent angles in a parallelogram?

180° (Supplementary).

14. Name a quadrilateral whose diagonals bisect each other at right angles.

Rhombus (and Square).

15. Name a quadrilateral whose diagonals are equal.

Rectangle (and Square).

16. How many sides does a heptagon have?

7.

17. Can a quadrilateral have angles 100°, 80°, 70°, 120°?

No (Sum = 100+80+70+120 = 370°, which is not 360°).

1. What is the difference between a convex and a concave polygon?

• Convex: All diagonals lie inside; All interior angles < 180°.
• Concave: At least one diagonal outside; At least one interior angle > 180°.

2. Find the sum of interior angles of a polygon with 9 sides.

• Use formula: Sum = (n – 2) × 180°.
• Here n = 9.
• Sum = (9 – 2) × 180° = 7 × 180° = 1260°.

3. Find the measure of each exterior angle of a regular octagon (8 sides).

• Sum of exterior angles = 360°.
• Number of sides n = 8.
• For regular polygon, each exterior angle = 360° / n.
• Each exterior angle = 360° / 8 = 45°.

4. How many sides does a regular polygon have if each interior angle is 165°?

• If interior angle = 165°, then exterior angle = 180° – 165° = 15°.
• Number of sides (n) = 360° / (Each Exterior Angle).
• n = 360° / 15° = 24.
• Answer: 24 sides.

5. The measures of two adjacent angles of a parallelogram are in the ratio 3:2. Find the measure of each angle.

• Let angles be 3x and 2x.
• Adjacent angles are supplementary: 3x + 2x = 180°.
• 5x = 180°.
• x = 180° / 5 = 36°.
• Angles are 3x=3(36°)=108° and 2x=2(36°)=72°.
• Opposite angles are equal.
• Answer: Angles are 108°, 72°, 108°, 72°.

6. In parallelogram HOPE, find x, y, z (Given diagram with angles: exterior angle 70°, opposite angle to x is (180-70)=110°, y adjacent to 110°, z alternate interior to y).

• Angle adjacent to 70° (inside ||gm) = 180° – 70° = 110° (linear pair).
• Opposite angle x = 110°.
• Adjacent angle y = 180° – 110° = 70°.
• Angle alternate interior to y = z (as HO || PE). So, z = y = 40° [Wait, needs a specific diagram. Let’s assume angles related to diagonal HP splitting y. Assume ∠EHP=40°, then z = ∠OPH = 40° (alternate interior). y then refers to ∠HOP = 180-70=110. Let’s refine this assuming angle properties].
• Refined assumption: Let external angle at E be 70°. Then interior ∠E = 180-70=110°. Opposite ∠O = x = 110°. Adjacent ∠H = ∠P = 180-110=70°. If z is part of ∠H and y part of ∠P due to diagonal, more info needed.
• *Simplified Example Based on typical Question:* Assume external at E is 70°. Interior ∠E=110°. Then x=∠O=110°. ∠P = 70°. If diagonal HP is drawn and ∠EHP = 40°, then angle z (alternate interior ∠OPH) = 40°. Angle y (which is ∠P=70°) would be composed of parts if another diagonal is there or other info is given. *If y is just ∠HPO*, it would be part of 70°. If line HP is parallel to OE (trapezium?), y=70? This needs a figure.* Let’s assume x=opposite angle, y=adjacent angle, z=alternate angle in a common setup.*
• Revisiting Typical Problem: Assume exterior ∠ at P = 70°. Then interior ∠P=110°. Opposite ∠H = x = 110°. Adjacent ∠O=∠E=180-110=70°. If HE || OP, y=∠EHO? If diagonal is EP, and say ∠OEP=30°, then z = alternate angle ∠HPE = 30°. This Q needs a diagram, but steps involve using opposite angles, adjacent angles, alternate interior angles (if parallels given).

Let’s solve for Parallelogram RING where ∠R = 70°. Find other angles.

• Opposite angles equal: ∠N = ∠R = 70°.
• Adjacent angles supplementary: ∠I = 180° – ∠R = 180° – 70° = 110°.
• Opposite angles equal: ∠G = ∠I = 110°.
• Angles are 70°, 110°, 70°, 110°.

7. Diagonals of a rhombus are 6 cm and 8 cm. Find the length of its side.

• Diagonals of a rhombus bisect each other at right angles (90°).
• Let diagonals be d1=6cm, d2=8cm.
• They intersect, dividing each into halves: d1/2 = 3cm, d2/2 = 4cm.
• These halves form the legs of a right-angled triangle, with the rhombus side as the hypotenuse.
• By Pythagoras theorem: side² = (d1/2)² + (d2/2)²
• side² = (3)² + (4)² = 9 + 16 = 25.
• side = √25 = 5 cm.
• Answer: Side = 5 cm.

8. How is a square a special type of rectangle and rhombus?

• Square is a special Rectangle because: A rectangle has all angles 90°. A square also has all angles 90°, AND its adjacent sides are equal (which isn’t required for a general rectangle).
• Square is a special Rhombus because: A rhombus has all sides equal. A square also has all sides equal, AND its angles are 90° (which isn’t required for a general rhombus).
• So, a square inherits properties of both.

9. Explain why a rectangle is a convex quadrilateral.

• A quadrilateral is convex if all its interior angles are less than 180°.
• A rectangle has all four interior angles equal to 90°.
• Since 90° is less than 180°, all interior angles are less than 180°.
• Also, both its diagonals lie completely inside the rectangle.
• Therefore, a rectangle is a convex quadrilateral.

10. Can a quadrilateral be a parallelogram if its opposite angles are equal?

• Yes.
• Let angles be A, B, C, D in order. Given A=C and B=D.
• Sum of angles = A+B+C+D = 360°.
• Substituting C=A and D=B: A+B+A+B = 360° => 2A+2B=360° => A+B=180°.
• Since adjacent angles A and B are supplementary, lines must be parallel (AD || BC).
• Similarly, B+C=180°, proving AB || DC.
• If both pairs of opposite sides are parallel, it is a parallelogram.

11. How many diagonals does a convex hexagon have?

• A hexagon has n=6 vertices.
• From each vertex, diagonals can be drawn to (n-3) non-adjacent vertices. (Cannot draw to itself or two adjacent vertices). So, 6-3=3 diagonals per vertex.
• Total lines initially = 6 vertices × 3 diagonals/vertex = 18.
• Since each diagonal connects two vertices (counted twice), divide by 2.
• Number of diagonals = n(n-3)/2 = 6(6-3)/2 = 6(3)/2 = 18/2 = 9.
• Answer: 9 diagonals.

1. Define Polygon. Classify polygons based on number of sides (up to 10 sides). Differentiate between Regular and Irregular Polygons.

Polygon: Simple closed curve made entirely of line segments.

Classification by Sides:

• 3: Triangle △
• 4: Quadrilateral □
• 5: Pentagon ⬠
• 6: Hexagon ⬡
• 7: Heptagon
• 8: Octagon ⯃
• 9: Nonagon
• 10: Decagon

Regular vs. Irregular:

• Regular: BOTH Equilateral (all sides equal) AND Equiangular (all angles equal). Ex: Square ⬜, Equilateral Triangle △.
• Irregular: NOT regular (either sides not equal OR angles not equal OR both). Ex: Rectangle, Rhombus, Scalene Triangle.

2. State and explain the Angle Sum Property for a convex polygon. Find the angle sum for a polygon with 7 sides.

Angle Sum Property: The sum of the measures of the interior angles of a convex polygon with ‘n’ sides is given by the formula (n - 2) × 180°.

Explanation:

• Any convex polygon can be divided into (n-2) triangles by drawing diagonals from one vertex to all other non-adjacent vertices.
• Example: Quadrilateral (n=4) can be divided into 4-2=2 triangles. Pentagon (n=5) into 5-2=3 triangles.
• The sum of angles in one triangle is 180°.
• Therefore, the sum of interior angles of the polygon is the sum of angles of these (n-2) triangles.
• Sum = (Number of triangles) × 180° = (n – 2) × 180°.

For a polygon with 7 sides (Heptagon):

• n = 7.
• Angle Sum = (7 – 2) × 180° = 5 × 180° = 900°.

3. What is the sum of exterior angles of a convex polygon? Find the number of sides if each exterior angle is 24°.

Sum of Exterior Angles: The sum of the measures of the exterior angles (taking one at each vertex) of ANY convex polygon, regardless of the number of sides, is always **360°**.

Finding Number of Sides (n):

• Given each exterior angle of a regular polygon = 24°.
• Formula: n = 360° / (Measure of each Exterior Angle).
• n = 360° / 24°.
• Calculation: 360/24 = (120 × 3) / (8 × 3) = 120 / 8 = (40 × 3) / (2 × 4 / 2) ? Let’s do long division or simplify: 360/12 = 30, 24/12 = 2. So 30/2 = 15. Or 360/24 = 15.
• n = 15.

Answer: The regular polygon has 15 sides.

4. Define Trapezium, Kite, and Parallelogram with diagrams (or description).

• Trapezium ⏢: A quadrilateral with at least one pair of opposite sides parallel. (Kam se kam ek joda opposite sides parallel ho). *(Desc: Looks like a table top)*.
• Kite 🪁: A quadrilateral with two distinct pairs of equal consecutive sides. Opposite sides are unequal. (Do alag-alag jode paas wali barabar bhujaon ke). *(Desc: Diamond kite shape)*. Properties: Diagonals perpendicular; one diagonal bisects the other; one pair opposite angles equal.
• Parallelogram ▱: A quadrilateral where both pairs of opposite sides are parallel. (Dono jode opposite sides parallel hon). *(Desc: Slanted rectangle)*. Properties: Opp sides equal, Opp angles equal, Adjacent angles supplementary, Diagonals bisect each other.

5. List and explain the properties of a Parallelogram regarding its sides, angles, and diagonals.

Properties of Parallelogram ▱:

1. Opposite Sides are Equal: If ABCD is a ||gm, then AB = DC and AD = BC.
2. Opposite Angles are Equal: ∠A = ∠C and ∠B = ∠D.
3. Adjacent Angles are Supplementary: Sum of any two consecutive angles is 180°. (∠A+∠B=180°, ∠B+∠C=180°, etc.).
4. Diagonals Bisect Each Other: If diagonals AC and BD intersect at O, then AO = OC and BO = OD. (O is the midpoint of both diagonals).
5. (Definition Property): Opposite sides are parallel (AB || DC and AD || BC).

6. Explain the properties of a Rhombus and a Rectangle, highlighting how they are special parallelograms.

Rhombus ◊:

• It IS a parallelogram, so it has all ||gm properties.
• Special Property 1: All four sides are equal (AB=BC=CD=DA). (Sabhi sides barabar).
• Special Property 2: Diagonals are perpendicular bisectors of each other (intersect at 90° and bisect). (Vikarn 90° par bisect karte hain).

Rectangle ▭:

• It IS a parallelogram, so it has all ||gm properties.
• Special Property 1: All four angles are right angles (90°). (Sabhi kon 90°).
• Special Property 2: Diagonals are equal in length (AC = BD). (Vikarn barabar hote hain).

7. Prove that the sum of angles of a quadrilateral is 360°.

Let ABCD be any convex quadrilateral.

• Step 1: Draw a diagonal. Join one pair of opposite vertices, say A and C. This divides the quadrilateral into two triangles: ∆ABC and ∆ADC. ✂️
• Step 2: Angle Sum of Triangles. We know the sum of angles in any triangle is 180°.
  • In ∆ABC: ∠BAC + ∠ABC + ∠BCA = 180°.
  • In ∆ADC: ∠DAC + ∠ADC + ∠DCA = 180°.
• Step 3: Add the angles of both triangles.
  (∠BAC + ∠ABC + ∠BCA) + (∠DAC + ∠ADC + ∠DCA) = 180° + 180°.
• Step 4: Regroup the angles.
  • (∠BAC + ∠DAC) = ∠DAB (or ∠A)
  • (∠BCA + ∠DCA) = ∠BCD (or ∠C)
  
  So, the sum becomes: ∠DAB + ∠ABC + ∠BCD + ∠ADC = 360°.
• Step 5: Conclusion. ∠A + ∠B + ∠C + ∠D = 360°. Thus, the sum of angles of a quadrilateral is 360°.

8. Find the number of sides of a regular polygon if its interior angle is 156°.

• Step 1: Find the exterior angle. Interior Angle + Exterior Angle = 180°.
  Exterior Angle = 180° – Interior Angle = 180° – 156° = 24°.
• Step 2: Use exterior angle sum property. Sum of exterior angles = 360°. For a regular polygon with n sides, each exterior angle = 360° / n.
• Step 3: Solve for n. n = 360° / (Each Exterior Angle).
  n = 360° / 24°.
• Calculation: 360/24 = 15.

Answer: The polygon has 15 sides.

9. In a parallelogram RISK, ∠K = 120°. Find the measure of ∠I and ∠S.

• Given Parallelogram RISK, ∠K = 120°.
• Property 1: Opposite angles are equal.
  ∠I = ∠K = 120°.
• Property 2: Adjacent angles are supplementary.
  ∠K and ∠S are adjacent. So, ∠K + ∠S = 180°.
  120° + ∠S = 180°.
  ∠S = 180° – 120° = 60°.
• (Alternatively, ∠I and ∠S are adjacent: ∠I + ∠S = 180° => 120° + ∠S = 180° => ∠S = 60°).
• Check: The four angles are 120°, 60°, 120°, 60°. Sum = 360°.

Answer: ∠I = 120°, ∠S = 60°.

10. Explain why the diagonals of a rectangle are equal but the diagonals of a general parallelogram may not be equal.

Let ABCD be a parallelogram.

• Consider triangles formed by diagonals: ∆ABC and ∆DCB.
• In these triangles: AB = DC (Opp. sides of ||gm), BC = CB (Common).
• In a general Parallelogram: Angle ∠ABC is NOT necessarily equal to Angle ∠DCB (only adjacent angles like ∠ABC + ∠BCD = 180°). Since the included angles are not necessarily equal, we cannot prove ∆ABC ≅ ∆DCB by SAS congruence. Therefore, the third sides AC and DB (the diagonals) are not necessarily equal.
• In a Rectangle: A rectangle is a parallelogram where all angles are 90°. So, ∠ABC = ∠DCB = 90°.
• Now consider ∆ABC and ∆DCB again: AB = DC, BC = CB (Common), and the included angle ∠ABC = ∠DCB = 90°.
• By SAS congruence rule, ∆ABC ≅ ∆DCB.
• Therefore, their corresponding parts are equal, which means AC = DB.
• Thus, the diagonals of a rectangle are equal.