{"id":10891,"date":"2026-11-02T02:03:32","date_gmt":"2026-11-02T02:03:32","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=10891"},"modified":"2026-11-02T02:03:32","modified_gmt":"2026-11-02T02:03:32","slug":"congruence-similarity-of-solid-shapes","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/congruence-similarity-of-solid-shapes\/","title":{"rendered":"Congruence & Similarity Of Solid Shapes"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Unit: <\/strong>Geometry<\/strong><\/h2>\n

Chapter: <\/strong>Congruence & Similarity Of Solid Shapes<\/strong><\/h3>\n

Reference: – What is a Quadrilateral, Trapezium (Trapezoid) Definition, Types of Trapezium (Isosceles, Right, Scalene), Properties of a Trapezium, Parallelogram Definition, Properties of a Parallelogram, Special Parallelograms (Rectangle, Rhombus, Square), Difference Between Trapezium and Parallelogram, Area of Trapezium, Area of Parallelogram, Solved Examples, Odd-One-Out Problems, Common Mistakes<\/em><\/p>\n

After studying this chapter, you should be able to understand:<\/strong><\/p>\n

    \n
  • What is a Trapezium (Trapezoid)<\/em><\/li>\n
  • What is a Parallelogram<\/em><\/li>\n
  • Properties of a Trapezium and a Parallelogram<\/em><\/li>\n
  • Difference Between Trapezium and Parallelogram<\/em><\/li>\n
  • Special Types of Parallelograms (Rectangle, Rhombus, Square)<\/em><\/li>\n<\/ul>\n

    Introduction to Trapezium & Parallelogram<\/strong><\/p>\n

    Definition<\/u><\/strong><\/p>\n

    A quadrilateral is a polygon with four sides, four vertices, and four angles. The sum of interior angles of any quadrilateral is 360°. The two special types of quadrilaterals we study in this chapter are the trapezium (called trapezoid in the US) and the parallelogram.<\/p>\n

    When we study trapezium and parallelogram, we essentially ask:<\/p>\n

    "How many pairs of parallel sides does this quadrilateral have?"<\/p>\n

    The answer determines whether it is a trapezium (one pair) or a parallelogram (two pairs).<\/p>\n

    Importance<\/u><\/strong><\/p>\n

      \n
    • Used in architecture, engineering, and design (roofs, bridges, beams)<\/li>\n
    • Foundation for understanding area and perimeter calculations<\/li>\n
    • Helps develop spatial reasoning and geometric intuition<\/li>\n
    • Appears in real-world objects (tables, windows, frames)<\/li>\n<\/ul>\n

      Example<\/u><\/strong><\/p>\n

      A trapezium (trapezoid) has exactly one pair of parallel sides, like a tabletop with one wider end. A parallelogram has two pairs of parallel sides, like a leaning bookcase or a diamond shape.<\/p>\n

       <\/p>\n

      Subtopics<\/u><\/strong><\/p>\n

      1. Trapezium (Trapezoid)<\/strong><\/p>\n

      Definition: A quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases. The non-parallel sides are called legs.<\/p>\n

      US Terminology: Trapezoid (US) = Trapezium (UK\/India)<\/p>\n

      Types of Trapezium<\/u><\/strong>:<\/strong><\/p>\n

      Isosceles Trapezium: The legs are equal in length. Base angles are equal. Diagonals are equal.<\/p>\n

      Right Trapezium: One leg is perpendicular to the bases (has two right angles).<\/p>\n

      Scalene Trapezium: All sides are of different lengths.<\/p>\n

      Properties of a Trapezium:<\/strong><\/p>\n

        \n
      • Exactly one pair of opposite sides is parallel<\/li>\n
      • The angles adjacent to each leg are supplementary (sum to 180°)<\/li>\n
      • The median (segment joining midpoints of legs) equals half the sum of the bases<\/li>\n<\/ul>\n

        Area of a Trapezium:<\/strong> Area = (1\/2) × (sum of parallel sides) × (height)<\/p>\n

        Formula: A = (1\/2) × (a + b) × h<\/p>\n

        Where a and b are the lengths of the parallel sides, and h is the perpendicular distance between them.<\/p>\n

        Example – Area of Trapezium:<\/strong> Bases = 8 cm and 12 cm, height = 5 cm
        \nArea = (1\/2) × (8 + 12) × 5 = (1\/2) × 20 × 5 = 10 × 5 = 50 cm²<\/p>\n

        2. Parallelogram<\/u><\/strong><\/p>\n

        Definition:<\/strong> A quadrilateral with both pairs of opposite sides parallel.<\/p>\n

        Properties of a Parallelogram:<\/strong><\/p>\n

          \n
        • Opposite sides are parallel<\/li>\n
        • Opposite sides are equal in length<\/li>\n
        • Opposite angles are equal<\/li>\n
        • Adjacent angles are supplementary (sum to 180°)<\/li>\n
        • Diagonals bisect each other (cut each other in half)<\/li>\n
        • Each diagonal divides the parallelogram into two congruent triangles<\/li>\n<\/ul>\n

          Area of a Parallelogram:<\/strong> Area = base × height<\/p>\n

          Formula: A = b × h<\/p>\n

          Where b is the length of any side, and h is the perpendicular distance to the opposite side (height).<\/p>\n

          Example – Area of Parallelogram:<\/strong> Base = 10 cm, height = 6 cm
          \nArea = 10 × 6 = 60 cm²<\/p>\n

          3. Special Types of Parallelograms<\/strong><\/p>\n

          Rectangle:<\/strong> A parallelogram with all angles 90° (right angles).
          \nProperties: All properties of parallelogram + diagonals are equal<\/p>\n

          Rhombus:<\/strong> A parallelogram with all sides equal.
          \nProperties: All properties of parallelogram + diagonals are perpendicular bisectors of each other + diagonals bisect the interior angles<\/p>\n

          Square:<\/strong> A parallelogram with all sides equal AND all angles 90°.
          \nProperties: All properties of rectangle + all properties of rhombus (diagonals equal, perpendicular, bisect angles)<\/p>\n

          4. Venn Diagram Relationship<\/strong><\/p>\n

          Square ⊂ Rhombus ⊂ Parallelogram
          \nSquare ⊂ Rectangle ⊂ Parallelogram
          \nRectangle and Rhombus are both Parallelograms, but their intersection is Square<\/p>\n

           <\/p>\n

          Solved Examples<\/strong><\/p>\n

          Example 1 – Area of Trapezium:<\/strong> Find the area of a trapezium with parallel sides 15 cm and 25 cm and height 8 cm.<\/p>\n

          Solution:<\/strong> A = (1\/2) × (15 + 25) × 8 = (1\/2) × 40 × 8 = 20 × 8 = 160 cm²<\/p>\n

          Answer:<\/strong> 160 cm²<\/p>\n

           <\/p>\n

          Example 2 – Area of Parallelogram:<\/strong> Find the area of a parallelogram with base 12 m and height 5 m.<\/p>\n

          Solution:<\/strong> A = 12 × 5 = 60 m²<\/p>\n

          Answer:<\/strong> 60 m²<\/p>\n

           <\/p>\n

          Example 3 – Parallelogram Angle:<\/strong> In parallelogram ABCD, angle A = 70°. Find angle B.<\/p>\n

          Solution:<\/strong> Adjacent angles in a parallelogram are supplementary.
          \nAngle A + angle B = 180° → 70° + angle B = 180° → angle B = 110°<\/p>\n

          Answer:<\/strong> 110°<\/p>\n

           <\/p>\n

          Example 4 – Isosceles Trapezium:<\/strong> In an isosceles trapezium, the legs are 6 cm each, bases are 10 cm and 14 cm. Find the perimeter.<\/p>\n

          Solution:<\/strong> Perimeter = sum of all sides = 10 + 14 + 6 + 6 = 36 cm<\/p>\n

          Answer:<\/strong> 36 cm<\/p>\n

           <\/p>\n

          Common Mistakes to Avoid<\/u><\/strong><\/p>\n

          Mistake 1 – Confusing trapezium with parallelogram<\/strong>
          \nThinking a trapezium has two pairs of parallel sides is wrong.
          \nCorrect understanding: Trapezium has exactly one pair; parallelogram has two pairs.<\/p>\n

          Mistake 2 – Using the wrong height for area<\/strong>
          \nHeight must be perpendicular distance between bases (trapezium) or between base and opposite side (parallelogram).
          \nCorrect understanding: Height is NOT the length of the slanted side.<\/p>\n

          Mistake 3 – Assuming all parallelograms are rectangles<\/strong>
          \nA parallelogram can have slanted sides. Only rectangles have 90° angles.
          \nCorrect understanding: Parallelogram = opposite sides parallel; rectangle = parallelogram with right angles.<\/p>\n

          Mistake 4 – Forgetting that a square is both a rectangle and a rhombus<\/strong>
          \nA square satisfies all properties of both rectangles and rhombuses.
          \nCorrect understanding: Square ⊂ Rectangle and Square ⊂ Rhombus.<\/p>\n

          Mistake 5 – Thinking all trapeziums are isosceles<\/strong>
          \nOnly isosceles trapeziums have equal legs and equal base angles.
          \nCorrect understanding: Trapeziums can be scalene or right-angled too.<\/p>\n

          Mistake 6 – Misidentifying the median formula for trapezium<\/strong>
          \nThe median (mid-segment) = (a + b)\/2, not (a × b)\/2.
          \nCorrect understanding: Median is the average of the two bases.<\/p>\n

           <\/p>\n

          Quick Reference Summary<\/u><\/strong><\/p>\n

          Trapezium (Trapezoid):<\/strong> Exactly 1 pair of parallel sides<\/p>\n

          Parallelogram:<\/strong> 2 pairs of parallel sides<\/p>\n

          Trapezium Area:<\/strong> A = (1\/2) × (a + b) × h (a, b = parallel sides, h = height)<\/p>\n

          Parallelogram Area:<\/strong> A = b × h (b = base, h = height)<\/p>\n

          Rectangle:<\/strong> Parallelogram with all angles 90°<\/p>\n

          Rhombus:<\/strong> Parallelogram with all sides equal<\/p>\n

          Square:<\/strong> Parallelogram with all sides equal AND all angles 90°<\/p>\n

          Properties Shared:<\/strong> Opposite sides parallel (parallelogram family), diagonals bisect each other (parallelogram family)<\/p>\n

          Key Difference:<\/strong> Trapezium has 1 pair of parallel sides; Parallelogram has 2 pairs<\/p>\n

           <\/p>\n

          <\/p>\n