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In a triangles ABC, D is a mistake point of BC; AD is produced upto E so that DE=AD. Prove that: 1) ∆ABD is congruent to ∆ ECD 2) AB=EC 3) AB is parallel to EC?
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Proof:
To prove the given statements, we will use the properties of congruent triangles and the fact that the opposite sides of a parallelogram are equal.
1) ∆ABD is congruent to ∆ECD:
Since DE = AD, we can conclude that ∆ADE is an isosceles triangle with DE = AD.
Now, in ∆ABD and ∆ECD, we can see that ∠ABD = ∠ECD (corresponding angles) and ∠BAD = ∠CDE (vertically opposite angles).
Since ∠ABD = ∠ECD and ∠BAD = ∠CDE, we can conclude that ∆ABD ≅ ∆ECD by the angle-side-angle (ASA) congruence criterion.
2) AB = EC:
Since ∆ABD ≅ ∆ECD, we can conclude that AD = ED (corresponding sides of congruent triangles).
Given that DE = AD, we can substitute AD for DE.
So, AD = AD, which implies that AD = AD.
Since AD = AD, we can conclude that AB = EC (opposite sides of a parallelogram are equal).
3) AB is parallel to EC:
Since AB = EC (proved in the previous step), and opposite sides of a parallelogram are parallel, we can conclude that AB is parallel to EC.
To summarize:
1) ∆ABD is congruent to ∆ECD.
2) AB = EC.
3) AB is parallel to EC.
Hence, the given statements have been proved using the properties of congruent triangles and the fact that opposite sides of a parallelogram are equal.
To prove the given statements, we will use the properties of congruent triangles and the fact that the opposite sides of a parallelogram are equal.
1) ∆ABD is congruent to ∆ECD:
Since DE = AD, we can conclude that ∆ADE is an isosceles triangle with DE = AD.
Now, in ∆ABD and ∆ECD, we can see that ∠ABD = ∠ECD (corresponding angles) and ∠BAD = ∠CDE (vertically opposite angles).
Since ∠ABD = ∠ECD and ∠BAD = ∠CDE, we can conclude that ∆ABD ≅ ∆ECD by the angle-side-angle (ASA) congruence criterion.
2) AB = EC:
Since ∆ABD ≅ ∆ECD, we can conclude that AD = ED (corresponding sides of congruent triangles).
Given that DE = AD, we can substitute AD for DE.
So, AD = AD, which implies that AD = AD.
Since AD = AD, we can conclude that AB = EC (opposite sides of a parallelogram are equal).
3) AB is parallel to EC:
Since AB = EC (proved in the previous step), and opposite sides of a parallelogram are parallel, we can conclude that AB is parallel to EC.
To summarize:
1) ∆ABD is congruent to ∆ECD.
2) AB = EC.
3) AB is parallel to EC.
Hence, the given statements have been proved using the properties of congruent triangles and the fact that opposite sides of a parallelogram are equal.
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In a triangles ABC, D is a mistake point of BC; AD is produced upto E so that DE=AD. Prove that: 1) ∆ABD is congruent to ∆ ECD 2) AB=EC 3) AB is parallel to EC?
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In a triangles ABC, D is a mistake point of BC; AD is produced upto E so that DE=AD. Prove that: 1) ∆ABD is congruent to ∆ ECD 2) AB=EC 3) AB is parallel to EC? 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 a triangles ABC, D is a mistake point of BC; AD is produced upto E so that DE=AD. Prove that: 1) ∆ABD is congruent to ∆ ECD 2) AB=EC 3) AB is parallel to EC? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a triangles ABC, D is a mistake point of BC; AD is produced upto E so that DE=AD. Prove that: 1) ∆ABD is congruent to ∆ ECD 2) AB=EC 3) AB is parallel to EC?.
In a triangles ABC, D is a mistake point of BC; AD is produced upto E so that DE=AD. Prove that: 1) ∆ABD is congruent to ∆ ECD 2) AB=EC 3) AB is parallel to EC? 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 a triangles ABC, D is a mistake point of BC; AD is produced upto E so that DE=AD. Prove that: 1) ∆ABD is congruent to ∆ ECD 2) AB=EC 3) AB is parallel to EC? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a triangles ABC, D is a mistake point of BC; AD is produced upto E so that DE=AD. Prove that: 1) ∆ABD is congruent to ∆ ECD 2) AB=EC 3) AB is parallel to EC?.
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