Hard Level Worksheet
Very Short Answer Questions (1 Mark Each)
(1) Write the condition for a quadrilateral to be a parallelogram using diagonals. If the diagonals
Perfect! When diagonals bisect each other, the quadrilateral is always a parallelogram.
(2) If the diagonals of a quadrilateral bisect each other, name the type of quadrilateral.
Excellent! Diagonals bisecting each other is the defining property of a parallelogram.
(3) Write one property of the diagonals of a rhombus.
Diagonals are
Correct! Rhombus diagonals bisect each other at right angles.
(4) In a parallelogram, if one angle is x, write the measure of its adjacent angle in terms of x.
Measure of adjacent angle =
Great! Adjacent angles in a parallelogram are supplementary.
(5) Write the sum of exterior angles of any polygon.
Perfect! The sum of exterior angles of any polygon is always 360°.
Short Answer Questions (2 Marks Each)
Note: Answer each question with complete proof and detailed calculations. Write down the answers on sheet and submit to the school subject teacher.
(1) Prove that the diagonals of a parallelogram bisect each other.
(2) In a rectangle, the length is twice its breadth. If the diagonal is 10 cm, find the length and breadth.
Breadth =
Perfect! Breadth =
(3) Prove that the opposite angles of a rhombus are equal.
(4) In a trapezium, the non-parallel sides are equal. Show that the trapezium is isosceles.
(5) In a kite, prove that the longer diagonal bisects the shorter diagonal at right angles.
Long Answer Questions (4 Marks Each)
Note: Answer each question with complete proof and detailed calculations. Write down the answers on sheet and submit to the school subject teacher.
(1) Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.
(2) In a parallelogram ABCD, E and F are midpoints of sides AB and CD respectively. Show that EF is parallel to AD and BC, and equal to their half.
(3) The diagonals of a rectangle are equal and bisect each other. Prove that it is a parallelogram.
(4) In a parallelogram, prove that the line joining the midpoints of opposite sides bisects the parallelogram into two congruent rectangles.
(5) In a quadrilateral, the bisectors of opposite angles meet at a point on the diagonal. Prove that the quadrilateral is a kite.
Part B: Objective Questions (1 Mark Each)
Choose the correct answer and write the option (a/b/c/d)
(1) If the diagonals of a parallelogram are perpendicular, it is:
(a) Square (b) Rectangle (c) Rhombus (d) Kite
Correct! A parallelogram with perpendicular diagonals is a rhombus.
(2) In a rectangle, the diagonals are:
(a) Equal and perpendicular (b) Equal and bisect each other (c) Unequal and perpendicular (d) None of these
Correct! Rectangle diagonals are equal and bisect each other but not perpendicular.
(3) In a square, each diagonal divides the square into:
(a) Two rectangles (b) Two congruent triangles (c) Two parallelograms (d) Two kites
Correct! Each diagonal divides a square into two congruent right triangles.
(4) The property "diagonals bisect each other" is true for:
(a) Parallelogram (b) Rhombus (c) Rectangle (d) All of these
Correct! All parallelograms (including rhombus and rectangle) have diagonals that bisect each other.
(5) A quadrilateral with exactly one pair of parallel sides is:
(a) Parallelogram (b) Trapezium (c) Rhombus (d) Rectangle
Correct! A trapezium has exactly one pair of parallel sides.
(6) If one pair of opposite sides of a quadrilateral is equal and parallel, then it is a:
(a) Trapezium (b) Parallelogram (c) Rhombus (d) Square
Correct! One pair of opposite sides equal and parallel makes it a parallelogram.
(7) The diagonals of a kite:
(a) Are equal (b) Bisect each other (c) Are perpendicular (d) None
Correct! Kite diagonals are perpendicular (one bisects the other at right angles).
(8) In a parallelogram, the sum of any two adjacent angles is:
(a) 180° (b) 90° (c) 120° (d) 360°
Correct! Adjacent angles in a parallelogram are supplementary (sum = 180°).
(9) Which of these is not always true for a rhombus?
(a) All sides are equal (b) All angles are equal (c) Diagonals bisect each other at right angles (d) Opposite angles are equal
Correct! A rhombus doesn't always have all angles equal (that's only true for a square).
(10) The sum of all exterior angles of a polygon is:
(a) 90° (b) 180° (c) 270° (d) 360°
Correct! The sum of exterior angles of any polygon is always 360°.
Let's classify advanced quadrilateral theorems and properties!!!
Excellent! You understand advanced quadrilateral theorems and their applications.
True or False: Advanced Quadrilateral Theorems
Determine whether these advanced statements are True or False:
Comprehensive Hard Quiz
🎉 You Did It! What You've Mastered:
By completing this advanced worksheet, you now have expert knowledge of:
(1) Advanced Parallelogram Theorems: Diagonal bisection properties and their converses
(2) Rhombus Advanced Properties: Sum of squares relationship between sides and diagonals
(3) Complex Geometric Proofs: Multi-step proofs using congruence and similarity
(4) Coordinate Geometry Applications: Using algebraic methods for geometric proofs
(5) Kite Properties: Advanced diagonal relationships and angle bisector theorems
(6) Trapezium Theorems: Midpoint properties and isosceles trapezium characteristics
(7) Rectangle Calculations: Advanced applications of Pythagorean theorem
(8) Quadrilateral Classification: Using properties to identify and prove quadrilateral types
(9) Midpoint Theorems: Understanding how midpoints create special relationships
(10) Angle Bisector Properties: Advanced applications in quadrilateral identification
(11) Exterior Angle Theorems: Understanding polygon angle relationships
(12) Advanced Problem Solving: Multi-step geometric reasoning and proof construction
(13) Congruence Applications: Using SSS, SAS, ASA, RHS in complex proofs
(14) Perpendicular Relationships: Understanding when and why diagonals are perpendicular
(15) Integration of Concepts: Connecting multiple geometric principles in single proofs
Outstanding mastery of advanced quadrilateral geometry - ready for higher-level mathematics!