ABCD is a trapezium in which AB is parallel to DC. The vertices A, B, C and D pass through a circle. Which of the following are correct? 1. AD = BC 2. ∠A + ∠ C = 180° 3. ∠ A + ∠ D = 180° Select the correct answer using the code given below :
Understanding Cyclic Trapezium Properties
Let's analyze the properties of a cyclic trapezium where ABCD is a trapezium with AB parallel to DC, and all its vertices A, B, C, and D lie on a circle. A quadrilateral whose vertices lie on a circle is called a cyclic quadrilateral. Thus, ABCD is a cyclic quadrilateral which is also a trapezium.
Analyzing Statement 1: AD = BC
A cyclic trapezium is a trapezium whose non-parallel sides are equal in length. Since ABCD is a trapezium with AB parallel to DC, AD and BC are the non-parallel sides. A key property of a cyclic trapezium is that it must be an isosceles trapezium. In an isosceles trapezium, the non-parallel sides are equal.
Therefore, for a cyclic trapezium ABCD with AB || DC, the non-parallel sides AD and BC must be equal.
Statement 1: AD = BC is correct.
Analyzing Statement 2: \(\angle\)A + \(\angle\)C = 180\(^\circ\)
Since the vertices A, B, C, and D lie on a circle, ABCD is a cyclic quadrilateral. A fundamental property of any cyclic quadrilateral is that the sum of its opposite angles is 180\(^\circ\). The pairs of opposite angles in quadrilateral ABCD are (\(\angle\)A, \(\angle\)C) and (\(\angle\)B, \(\angle\)D).
According to the property of cyclic quadrilaterals, the sum of opposite angles \(\angle\)A and \(\angle\)C is 180\(^\circ\).
Statement 2: \(\angle\)A + \(\angle\)C = 180\(^\circ\) is correct.
Analyzing Statement 3: \(\angle\)A + \(\angle\)D = 180\(^\circ\)
ABCD is a trapezium with AB parallel to DC (AB || DC). When a transversal line intersects two parallel lines, the consecutive interior angles on the same side of the transversal are supplementary (sum to 180\(^\circ\)).
In trapezium ABCD, AB || DC. AD is a transversal line intersecting the parallel lines AB and DC. The angles \(\angle\)A and \(\angle\)D are consecutive interior angles on the same side of the transversal AD.
Therefore, the sum of \(\angle\)A and \(\angle\)D must be 180\(^\circ\).
Similarly, BC is also a transversal intersecting the parallel lines AB and DC. The angles \(\angle\)B and \(\angle\)C are consecutive interior angles on the same side of the transversal BC, so \(\angle\)B + \(\angle\)C = 180\(^\circ\).
Statement 3: \(\angle\)A + \(\angle\)D = 180\(^\circ\) is correct.
Conclusion on Cyclic Trapezium Properties
Based on the analysis of the properties of a cyclic trapezium and parallel lines:
- Statement 1 (AD = BC) is correct because a cyclic trapezium is an isosceles trapezium.
- Statement 2 (\(\angle\)A + \(\angle\)C = 180\(^\circ\)) is correct because opposite angles of a cyclic quadrilateral are supplementary.
- Statement 3 (\(\angle\)A + \(\angle\)D = 180\(^\circ\)) is correct because consecutive interior angles between parallel lines are supplementary.
All three statements are correct.
Revision Table: Key Geometric Properties
| Property | Description | Application in this Problem |
|---|---|---|
| Cyclic Quadrilateral | A quadrilateral whose vertices lie on a circle. | ABCD is a cyclic quadrilateral. |
| Opposite Angles of Cyclic Quad | Sum of opposite angles is \(180^\circ\) (supplementary). | \(\angle\)A + \(\angle\)C = \(180^\circ\) \(\angle\)B + \(\angle\)D = \(180^\circ\) |
| Trapezium | A quadrilateral with at least one pair of parallel sides. | ABCD is a trapezium with AB || DC. |
| Consecutive Interior Angles | Angles on the same side of a transversal, between parallel lines; sum is \(180^\circ\). | \(\angle\)A + \(\angle\)D = \(180^\circ\) \(\angle\)B + \(\angle\)C = \(180^\circ\) |
| Cyclic Trapezium | A trapezium whose vertices lie on a circle. Always isosceles. | ABCD is a cyclic trapezium. |
| Isosceles Trapezium | Non-parallel sides are equal; base angles are equal. | AD = BC \(\angle\)D = \(\angle\)C \(\angle\)A = \(\angle\)B |
Additional Information on Quadrilateral Geometry
Understanding the properties of different quadrilaterals is essential in geometry. Here's a bit more detail:
- Trapezium: The defining feature is having exactly one pair of parallel sides (in some definitions, at least one pair). The parallel sides are called bases.
- Cyclic Quadrilateral: This property relates the quadrilateral to a circle. The most important property is that opposite angles are supplementary.
- Isosceles Trapezium: This is a special type of trapezium where the non-parallel sides are equal. As a consequence, the base angles are equal in pairs (angles along each parallel base are equal). For example, in trapezium ABCD with AB || DC, if AD = BC, then \(\angle\)D = \(\angle\)C and \(\angle\)A = \(\angle\)B.
- Cyclic Trapezium and Isosceles Trapezium: A remarkable theorem states that a trapezium is cyclic if and only if it is an isosceles trapezium. This means if a trapezium can be inscribed in a circle, its non-parallel sides must be equal, and vice versa. This theorem directly supports why Statement 1 (AD=BC) is true.
Combining these properties helps explain why all three statements are correct for the given cyclic trapezium ABCD with AB || DC.
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Select the correct answer using the code given below :
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Select the correct answer using the code given below :
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Important Questions from Quadrilaterals
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