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Two circles APQC and PBDQ intersect each other at the points P and Q and APB and CQD are two parallel straight lines. Then only one of the following statements is always true. Which one is it?
- a)ABDC is a cyclic quadrilateral
- b)AC is parallel to BD
- c)ABDC is a rectangle
- d)AACQ is a right angle
Correct answer is option 'B'. Can you explain this answer?
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Two circles APQC and PBDQ intersect each other at the points P and Q a...
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Two circles APQC and PBDQ intersect each other at the points P and Q a...
Given Information:
- Two circles APQC and PBDQ intersect each other at the points P and Q.
- APB and CQD are two parallel straight lines.
To Prove:
AC is parallel to BD.
Proof:
Step 1: Draw the diagram
Let's start by drawing the diagram. We have two circles APQC and PBDQ intersecting at points P and Q. We also have parallel lines APB and CQD.
[INSERT DIAGRAM HERE]
Step 2: Identify the angles
Let's label the angles in the diagram for easier reference.
[INSERT DIAGRAM WITH ANGLES LABELED HERE]
Step 3: Consider the angles formed by intersecting chords
Since APB and CQD are parallel lines, we have several pairs of corresponding angles:
1. ∠APB and ∠CQD are corresponding angles formed by the intersecting chords AP and CQ.
2. ∠AQB and ∠CPD are corresponding angles formed by the intersecting chords AQ and CP.
3. ∠APQ and ∠CQD are corresponding angles formed by the intersecting chords AP and CQ.
Step 4: Apply the properties of corresponding angles
Since we know that corresponding angles formed by intersecting chords are congruent, we can conclude the following:
1. ∠APB = ∠CQD (corresponding angles formed by intersecting chords AP and CQ)
2. ∠AQB = ∠CPD (corresponding angles formed by intersecting chords AQ and CP)
3. ∠APQ = ∠CQD (corresponding angles formed by intersecting chords AP and CQ)
Step 5: Consider the triangle APQ
Since APQC is a cyclic quadrilateral, we know that opposite angles are supplementary. Therefore, ∠APQ + ∠ACQ = 180°.
Since ∠APQ = ∠CQD (by corresponding angles), we can rewrite the equation as ∠CQD + ∠ACQ = 180°.
Step 6: Identify the corresponding angles
From Step 4, we know that ∠APB = ∠CQD. Therefore, we can rewrite the equation from Step 5 as ∠APB + ∠ACQ = 180°.
Step 7: Identify the alternate interior angles
Since APB and CQD are parallel lines, we know that ∠APB and ∠ACQ are alternate interior angles.
Step 8: Apply the property of alternate interior angles
Since alternate interior angles are congruent, we can rewrite the equation from Step 6 as ∠APB + ∠APB = 180°.
Step 9: Simplify the equation
Simplifying the equation from Step 8, we get 2∠APB = 180°.
Step 10: Solve for ∠APB
Dividing both sides of the equation from Step 9 by 2, we get ∠APB = 90°.
Step 11: Identify the angles in the
- Two circles APQC and PBDQ intersect each other at the points P and Q.
- APB and CQD are two parallel straight lines.
To Prove:
AC is parallel to BD.
Proof:
Step 1: Draw the diagram
Let's start by drawing the diagram. We have two circles APQC and PBDQ intersecting at points P and Q. We also have parallel lines APB and CQD.
[INSERT DIAGRAM HERE]
Step 2: Identify the angles
Let's label the angles in the diagram for easier reference.
[INSERT DIAGRAM WITH ANGLES LABELED HERE]
Step 3: Consider the angles formed by intersecting chords
Since APB and CQD are parallel lines, we have several pairs of corresponding angles:
1. ∠APB and ∠CQD are corresponding angles formed by the intersecting chords AP and CQ.
2. ∠AQB and ∠CPD are corresponding angles formed by the intersecting chords AQ and CP.
3. ∠APQ and ∠CQD are corresponding angles formed by the intersecting chords AP and CQ.
Step 4: Apply the properties of corresponding angles
Since we know that corresponding angles formed by intersecting chords are congruent, we can conclude the following:
1. ∠APB = ∠CQD (corresponding angles formed by intersecting chords AP and CQ)
2. ∠AQB = ∠CPD (corresponding angles formed by intersecting chords AQ and CP)
3. ∠APQ = ∠CQD (corresponding angles formed by intersecting chords AP and CQ)
Step 5: Consider the triangle APQ
Since APQC is a cyclic quadrilateral, we know that opposite angles are supplementary. Therefore, ∠APQ + ∠ACQ = 180°.
Since ∠APQ = ∠CQD (by corresponding angles), we can rewrite the equation as ∠CQD + ∠ACQ = 180°.
Step 6: Identify the corresponding angles
From Step 4, we know that ∠APB = ∠CQD. Therefore, we can rewrite the equation from Step 5 as ∠APB + ∠ACQ = 180°.
Step 7: Identify the alternate interior angles
Since APB and CQD are parallel lines, we know that ∠APB and ∠ACQ are alternate interior angles.
Step 8: Apply the property of alternate interior angles
Since alternate interior angles are congruent, we can rewrite the equation from Step 6 as ∠APB + ∠APB = 180°.
Step 9: Simplify the equation
Simplifying the equation from Step 8, we get 2∠APB = 180°.
Step 10: Solve for ∠APB
Dividing both sides of the equation from Step 9 by 2, we get ∠APB = 90°.
Step 11: Identify the angles in the
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Two circles APQC and PBDQ intersect each other at the points P and Q and APB and CQD are two parallel straight lines. Then only one of the following statements is always true. Which one is it?a)ABDC is a cyclic quadrilateralb)AC is parallel to BDc)ABDC is a rectangled)AACQ is a right angleCorrect answer is option 'B'. Can you explain this answer?
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Solutions for Two circles APQC and PBDQ intersect each other at the points P and Q and APB and CQD are two parallel straight lines. Then only one of the following statements is always true. Which one is it?a)ABDC is a cyclic quadrilateralb)AC is parallel to BDc)ABDC is a rectangled)AACQ is a right angleCorrect answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for Quant.
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