Q1. What is a quadrilateral?
Answer:
A quadrilateral is a plane figure formed by four distinct vertices and four line segments joining them in order, such that every point other than the vertices lies on exactly one of the four segments.
Q2. What are the vertices, sides, diagonals, and angles of a quadrilateral ABCD?
Answer:
- Vertices: A, B, C and D.
- Sides: AB, BC, CD and DA.
- Diagonals: AC and BD.
- Internal angles: ∠A, ∠B, ∠C and ∠D.
Q3. What is a convex quadrilateral?
Answer:
A quadrilateral is called convex when all its internal angles are less than 180°. Equivalently, its diagonals intersect each other.
Q4. What is a non-convex quadrilateral?
Answer:
A non-convex quadrilateral is a quadrilateral that has a dent, where one of its internal angles is greater than 180°. Its diagonals do not intersect each other.
Q5. What is a parallelogram? Name some special types of parallelograms.
Answer:
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. A rectangle, a rhombus and a square are special types of parallelograms.
Q6. State the three important properties of a parallelogram.
Answer:
The three important properties of a parallelogram are:
1. Its opposite sides are equal.
2. Its opposite angles are equal.
3. Its diagonals bisect each other.
Q7. Prove that the opposite sides of a parallelogram are equal.
Answer:
Let ABCD be a parallelogram and draw diagonal AC. Since AB is parallel to DC and AD is parallel to BC, alternate interior angles are equal. Also, AC is common to triangles ABC and CDA. Therefore, the triangles are congruent by the ASA test. Hence, AB = CD and AD = BC.
Q8. What is the relationship between the opposite angles of a parallelogram?
Answer:
The opposite angles of a parallelogram are equal. Thus, in parallelogram ABCD,
∠A=∠C
∠B=∠D
Also, any two adjacent angles add up to 180°.
Q9. What happens to the diagonals of a parallelogram when they intersect?
Answer:
The diagonals of a parallelogram bisect each other. If diagonals AC and BD intersect at O, then
AO = OC, BO = OD
Therefore, O is the midpoint of both diagonals.
Q10. How can we prove that a quadrilateral is a parallelogram if its opposite sides are equal?
Answer:
Let ABCD be a quadrilateral in which AB = CD and AD = BC. Draw diagonal AC. The two triangles ABC and CDA have three corresponding equal sides, so they are congruent by the SSS test. The corresponding angles are equal, which establishes that AB is parallel to CD and AD is parallel to BC. Hence, ABCD is a parallelogram.
Q11. How can we prove that a quadrilateral is a parallelogram if its opposite angles are equal?
Answer:
Let the opposite angles of quadrilateral ABCD be equal. Then ∠A = ∠C and ∠B = ∠D. Since the sum of the four angles is 360°,
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