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□ABCD is a trapezium with AB∥CD. E and F are on AD and DC respectively such that
DE = DF.
−→EF intersects
−−→BC at P and P F = P C. If ∠ADC = 46, find ∠B, ∠P.?
DE = DF.
−→EF intersects
−−→BC at P and P F = P C. If ∠ADC = 46, find ∠B, ∠P.?
Most Upvoted Answer
□ABCD is a trapezium with AB∥CD. E and F are on AD and DC respectively...
Understanding the Trapezium Configuration
- In trapezium ABCD, with AB parallel to CD, we know that angles ∠ADC and ∠ABC are supplementary.
- Given ∠ADC = 46°, we can find ∠ABC as follows:
- ∠ABC = 180° - ∠ADC = 180° - 46° = 134°.
Properties of Points E and F
- Points E and F are located on AD and DC respectively with DE = DF.
- This indicates that triangle DEF is isosceles, which leads to ∠DEF = ∠DFE.
Analyzing Intersection Point P
- The line segment EF intersects BC at point P.
- Given that PF = PC, point P is equidistant from F and C, making triangle PFC isosceles.
- Therefore, we have ∠PFC = ∠PCF.
Applying Angle Properties
- In triangle DEF, since DE = DF, we have ∠DEF = ∠DFE.
- Since EF is a transversal cutting through BC, we apply alternate interior angles:
- ∠PFE = ∠ABC = 134°.
Finding ∠P
- In triangle PFC:
- The angles must sum to 180°.
- Let ∠PFC = x; thus, ∠PCF = x too.
- Therefore, 2x + ∠FPC = 180°.
- Since ∠FPC = 134°:
- 2x + 134° = 180°.
- Solving gives 2x = 46° → x = 23°.
Conclusion on Angles
- Therefore, ∠B = 134° and ∠P = 23°.
This concise analysis provides clarity on how the angles within trapezium ABCD relate to each other based on given conditions.
- In trapezium ABCD, with AB parallel to CD, we know that angles ∠ADC and ∠ABC are supplementary.
- Given ∠ADC = 46°, we can find ∠ABC as follows:
- ∠ABC = 180° - ∠ADC = 180° - 46° = 134°.
Properties of Points E and F
- Points E and F are located on AD and DC respectively with DE = DF.
- This indicates that triangle DEF is isosceles, which leads to ∠DEF = ∠DFE.
Analyzing Intersection Point P
- The line segment EF intersects BC at point P.
- Given that PF = PC, point P is equidistant from F and C, making triangle PFC isosceles.
- Therefore, we have ∠PFC = ∠PCF.
Applying Angle Properties
- In triangle DEF, since DE = DF, we have ∠DEF = ∠DFE.
- Since EF is a transversal cutting through BC, we apply alternate interior angles:
- ∠PFE = ∠ABC = 134°.
Finding ∠P
- In triangle PFC:
- The angles must sum to 180°.
- Let ∠PFC = x; thus, ∠PCF = x too.
- Therefore, 2x + ∠FPC = 180°.
- Since ∠FPC = 134°:
- 2x + 134° = 180°.
- Solving gives 2x = 46° → x = 23°.
Conclusion on Angles
- Therefore, ∠B = 134° and ∠P = 23°.
This concise analysis provides clarity on how the angles within trapezium ABCD relate to each other based on given conditions.
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□ABCD is a trapezium with AB∥CD. E and F are on AD and DC respectively such that DE = DF. −→EF intersects −−→BC at P and P F = P C. If ∠ADC = 46, find ∠B, ∠P.?
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□ABCD is a trapezium with AB∥CD. E and F are on AD and DC respectively such that DE = DF. −→EF intersects −−→BC at P and P F = P C. If ∠ADC = 46, find ∠B, ∠P.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about □ABCD is a trapezium with AB∥CD. E and F are on AD and DC respectively such that DE = DF. −→EF intersects −−→BC at P and P F = P C. If ∠ADC = 46, find ∠B, ∠P.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for □ABCD is a trapezium with AB∥CD. E and F are on AD and DC respectively such that DE = DF. −→EF intersects −−→BC at P and P F = P C. If ∠ADC = 46, find ∠B, ∠P.?.
□ABCD is a trapezium with AB∥CD. E and F are on AD and DC respectively such that DE = DF. −→EF intersects −−→BC at P and P F = P C. If ∠ADC = 46, find ∠B, ∠P.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about □ABCD is a trapezium with AB∥CD. E and F are on AD and DC respectively such that DE = DF. −→EF intersects −−→BC at P and P F = P C. If ∠ADC = 46, find ∠B, ∠P.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for □ABCD is a trapezium with AB∥CD. E and F are on AD and DC respectively such that DE = DF. −→EF intersects −−→BC at P and P F = P C. If ∠ADC = 46, find ∠B, ∠P.?.
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