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In the given figure ABCD is a parallelogram equilateral triangles ADF and ABE have been drawn on the sides AD and AB respectively show that CFE is also an equilateral triangle hint prove that triangle DFC is congruent to triangle BCE and angle FCE is equal to 60 degree?
Verified Answer
In the given figure ABCD is a parallelogram equilateral triangles ADF ...
To prove that triangle DFC is congruent to triangle BCE, we can use the fact that opposite sides of a parallelogram are congruent. Since ABCD is a parallelogram, we know that AD is congruent to BC and DF is congruent to EB. Therefore, triangle DFC is congruent to triangle BCE by the Side-Side-Side (SSS) congruence theorem.
To prove that angle FCE is equal to 60 degrees, we can use the fact that the angles in an equilateral triangle are all equal to 60 degrees. Since ADF and ABE are both equilateral triangles, we know that the angles FAD and EAB are both equal to 60 degrees. Therefore, the measure of angle FCE is equal to the sum of the measures of angles FAD and EAB, which is 60 + 60 = 120 degrees. Since the sum of the measures of the angles in a triangle is 180 degrees, we can conclude that the measure of angle FCE is equal to 60 degrees.
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Most Upvoted Answer
In the given figure ABCD is a parallelogram equilateral triangles ADF ...
Given:
- ABCD is a parallelogram.
- Equilateral triangles ADF and ABE have been drawn on the sides AD and AB respectively.
To Prove:
CFE is an equilateral triangle.
Proof:
Step 1: Proving Triangle DFC Congruent to Triangle BCE
- In parallelogram ABCD, opposite sides are parallel and equal in length. Therefore, AB = CD and AD = BC.
- In equilateral triangle ADF, all sides are equal in length. Therefore, AD = DF.
- In equilateral triangle ABE, all sides are equal in length. Therefore, AB = BE.
- Combining the above two equations, we have AD = DF = AB = BE.
- Since AB = CD and AD = BC, we can conclude that ABDC is a parallelogram with equal sides.
- Therefore, we can say that triangle DFC and triangle BCE are congruent by Side-Side-Side (SSS) congruence criterion.
Step 2: Proving Angle FCE = 60 degrees
- Since triangle DFC and triangle BCE are congruent, corresponding angles are also congruent. Therefore, angle DFC = angle BCE.
- In parallelogram ABCD, opposite angles are congruent. Therefore, angle C = angle DFC.
- Combining the above two equations, we have angle C = angle BCE = angle DFC.
- In an equilateral triangle, all angles are equal to 60 degrees.
- Therefore, angle C = angle BCE = angle DFC = 60 degrees.
Step 3: Proving Triangle CFE is an Equilateral Triangle
- In triangle CFE, angle FCE = angle BCE = 60 degrees (proved in Step 2).
- In an equilateral triangle, all angles are equal to 60 degrees.
- Therefore, angle CFE = 60 degrees.
- In triangle CFE, all angles are equal to 60 degrees (angle FCE = 60 degrees and angle CEF = 60 degrees).
- Therefore, triangle CFE is an equilateral triangle.
Conclusion:
From the above proof, we can conclude that triangle CFE is an equilateral triangle.
- ABCD is a parallelogram.
- Equilateral triangles ADF and ABE have been drawn on the sides AD and AB respectively.
To Prove:
CFE is an equilateral triangle.
Proof:
Step 1: Proving Triangle DFC Congruent to Triangle BCE
- In parallelogram ABCD, opposite sides are parallel and equal in length. Therefore, AB = CD and AD = BC.
- In equilateral triangle ADF, all sides are equal in length. Therefore, AD = DF.
- In equilateral triangle ABE, all sides are equal in length. Therefore, AB = BE.
- Combining the above two equations, we have AD = DF = AB = BE.
- Since AB = CD and AD = BC, we can conclude that ABDC is a parallelogram with equal sides.
- Therefore, we can say that triangle DFC and triangle BCE are congruent by Side-Side-Side (SSS) congruence criterion.
Step 2: Proving Angle FCE = 60 degrees
- Since triangle DFC and triangle BCE are congruent, corresponding angles are also congruent. Therefore, angle DFC = angle BCE.
- In parallelogram ABCD, opposite angles are congruent. Therefore, angle C = angle DFC.
- Combining the above two equations, we have angle C = angle BCE = angle DFC.
- In an equilateral triangle, all angles are equal to 60 degrees.
- Therefore, angle C = angle BCE = angle DFC = 60 degrees.
Step 3: Proving Triangle CFE is an Equilateral Triangle
- In triangle CFE, angle FCE = angle BCE = 60 degrees (proved in Step 2).
- In an equilateral triangle, all angles are equal to 60 degrees.
- Therefore, angle CFE = 60 degrees.
- In triangle CFE, all angles are equal to 60 degrees (angle FCE = 60 degrees and angle CEF = 60 degrees).
- Therefore, triangle CFE is an equilateral triangle.
Conclusion:
From the above proof, we can conclude that triangle CFE is an equilateral triangle.
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In the given figure ABCD is a parallelogram equilateral triangles ADF and ABE have been drawn on the sides AD and AB respectively show that CFE is also an equilateral triangle hint prove that triangle DFC is congruent to triangle BCE and angle FCE is equal to 60 degree?
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In the given figure ABCD is a parallelogram equilateral triangles ADF and ABE have been drawn on the sides AD and AB respectively show that CFE is also an equilateral triangle hint prove that triangle DFC is congruent to triangle BCE and angle FCE is equal to 60 degree? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about In the given figure ABCD is a parallelogram equilateral triangles ADF and ABE have been drawn on the sides AD and AB respectively show that CFE is also an equilateral triangle hint prove that triangle DFC is congruent to triangle BCE and angle FCE is equal to 60 degree? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the given figure ABCD is a parallelogram equilateral triangles ADF and ABE have been drawn on the sides AD and AB respectively show that CFE is also an equilateral triangle hint prove that triangle DFC is congruent to triangle BCE and angle FCE is equal to 60 degree?.
In the given figure ABCD is a parallelogram equilateral triangles ADF and ABE have been drawn on the sides AD and AB respectively show that CFE is also an equilateral triangle hint prove that triangle DFC is congruent to triangle BCE and angle FCE is equal to 60 degree? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about In the given figure ABCD is a parallelogram equilateral triangles ADF and ABE have been drawn on the sides AD and AB respectively show that CFE is also an equilateral triangle hint prove that triangle DFC is congruent to triangle BCE and angle FCE is equal to 60 degree? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the given figure ABCD is a parallelogram equilateral triangles ADF and ABE have been drawn on the sides AD and AB respectively show that CFE is also an equilateral triangle hint prove that triangle DFC is congruent to triangle BCE and angle FCE is equal to 60 degree?.
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