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In triangle ABCD , C is any point on BD such that AB = BC and AC = CD. Prove that angle BAD : angle ADB = 3:1?
Most Upvoted Answer
In triangle ABCD , C is any point on BD such that AB = BC and AC = CD....
In isosceles triangle, the two angles opposite to the equal sides are also equal. So, for ΔABC, ∠BAC = ∠ACB and for ΔACD, ∠CAD = ∠ADC
As ∠ACB is outside angle of ΔACD ,
so ∠ACB = ∠CAD + ∠ADC
⇒ ∠ACB = 2* ∠ADC (As, ∠CAD = ∠ADC )
⇒ ∠BAC = 2* ∠ADC (As, ∠BAC = ∠ACB )
Now, according to the question,
∠BAD - ∠CAD = ∠BAC
⇒ ∠BAD - ∠ADC = 2* ∠ADC [As, ∠CAD = ∠ADC and ∠BAC = 2* ∠ADC]
⇒ ∠BAD = 3* ∠ADC
⇒ ∠BAD = 3* ∠ADB [As, ∠ADC and ∠ADB are same angles]
⇒ ∠BAD/∠ADB = 3/1
So, ∠BAD: ∠ADB =3:1 is the solution.
Community Answer
In triangle ABCD , C is any point on BD such that AB = BC and AC = CD....
Given:
Triangle ABCD, where AB = BC and AC = CD.
To Prove:
Angle BAD : Angle ADB = 3:1
Proof:
1. Draw a Diagram:
Draw a diagram of Triangle ABCD and label the points as given in the problem.
2. Let's Analyze the Given Information:
From the given information, we know that AB = BC and AC = CD.
3. Mark Additional Points:
Let E be the point on AC such that AE = EC. Join BE.
4. Apply Angle Bisector Theorem:
Since AE = EC, the line segment BE bisects angle ABC. Therefore, angle ABE = angle EBC.
5. Apply Angle Sum Property:
In triangle ABC, angle ABE + angle ABC + angle EBC = 180 degrees.
6. Substitute the Known Values:
Since angle ABE = angle EBC, we can rewrite the equation as angle ABE + angle ABE + angle ABC = 180 degrees.
7. Simplify the Equation:
2(angle ABE) + angle ABC = 180 degrees.
8. Divide Both Sides by 2:
(angle ABE) + (angle ABC)/2 = 90 degrees.
9. Angle ABE = Angle ABC/2:
We can rewrite the equation as angle ABE = angle ABC/2.
10. Apply Angle Bisector Theorem Again:
Since AE = EC, the line segment BD bisects angle ABC. Therefore, angle ABD = angle DBC.
11. Apply Angle Sum Property:
In triangle ABD, angle ABD + angle BAD + angle DAB = 180 degrees.
12. Substitute the Known Values:
Since angle ABD = angle DBC, we can rewrite the equation as angle ABD + angle ABD + angle DAB = 180 degrees.
13. Simplify the Equation:
2(angle ABD) + angle DAB = 180 degrees.
14. Divide Both Sides by 2:
(angle ABD) + (angle DAB)/2 = 90 degrees.
15. Angle ABD = Angle DAB/2:
We can rewrite the equation as angle ABD = angle DAB/2.
16. Relationship between Angle ABD and Angle BAD:
Since angle ABD = angle DAB/2, we can substitute this value into the equation from step 11:
angle DAB/2 + angle BAD + angle DAB = 180 degrees.
17. Simplify the Equation:
3(angle DAB/2) + angle BAD = 180 degrees.
18. Divide Both Sides by 3:
(angle DAB/2) + angle BAD/3 = 60 degrees.
19. Angle BAD/3 = Angle DAB/2:
We can rewrite the equation as angle BAD/3 = angle DAB/2.
20. Relationship between Angle BAD and Angle ADB:
Since angle BAD/3 = angle DAB/2, we can substitute this value into the equation
Triangle ABCD, where AB = BC and AC = CD.
To Prove:
Angle BAD : Angle ADB = 3:1
Proof:
1. Draw a Diagram:
Draw a diagram of Triangle ABCD and label the points as given in the problem.
2. Let's Analyze the Given Information:
From the given information, we know that AB = BC and AC = CD.
3. Mark Additional Points:
Let E be the point on AC such that AE = EC. Join BE.
4. Apply Angle Bisector Theorem:
Since AE = EC, the line segment BE bisects angle ABC. Therefore, angle ABE = angle EBC.
5. Apply Angle Sum Property:
In triangle ABC, angle ABE + angle ABC + angle EBC = 180 degrees.
6. Substitute the Known Values:
Since angle ABE = angle EBC, we can rewrite the equation as angle ABE + angle ABE + angle ABC = 180 degrees.
7. Simplify the Equation:
2(angle ABE) + angle ABC = 180 degrees.
8. Divide Both Sides by 2:
(angle ABE) + (angle ABC)/2 = 90 degrees.
9. Angle ABE = Angle ABC/2:
We can rewrite the equation as angle ABE = angle ABC/2.
10. Apply Angle Bisector Theorem Again:
Since AE = EC, the line segment BD bisects angle ABC. Therefore, angle ABD = angle DBC.
11. Apply Angle Sum Property:
In triangle ABD, angle ABD + angle BAD + angle DAB = 180 degrees.
12. Substitute the Known Values:
Since angle ABD = angle DBC, we can rewrite the equation as angle ABD + angle ABD + angle DAB = 180 degrees.
13. Simplify the Equation:
2(angle ABD) + angle DAB = 180 degrees.
14. Divide Both Sides by 2:
(angle ABD) + (angle DAB)/2 = 90 degrees.
15. Angle ABD = Angle DAB/2:
We can rewrite the equation as angle ABD = angle DAB/2.
16. Relationship between Angle ABD and Angle BAD:
Since angle ABD = angle DAB/2, we can substitute this value into the equation from step 11:
angle DAB/2 + angle BAD + angle DAB = 180 degrees.
17. Simplify the Equation:
3(angle DAB/2) + angle BAD = 180 degrees.
18. Divide Both Sides by 3:
(angle DAB/2) + angle BAD/3 = 60 degrees.
19. Angle BAD/3 = Angle DAB/2:
We can rewrite the equation as angle BAD/3 = angle DAB/2.
20. Relationship between Angle BAD and Angle ADB:
Since angle BAD/3 = angle DAB/2, we can substitute this value into the equation
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In triangle ABCD , C is any point on BD such that AB = BC and AC = CD. Prove that angle BAD : angle ADB = 3:1?
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In triangle ABCD , C is any point on BD such that AB = BC and AC = CD. Prove that angle BAD : angle ADB = 3:1? 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 triangle ABCD , C is any point on BD such that AB = BC and AC = CD. Prove that angle BAD : angle ADB = 3:1? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In triangle ABCD , C is any point on BD such that AB = BC and AC = CD. Prove that angle BAD : angle ADB = 3:1?.
In triangle ABCD , C is any point on BD such that AB = BC and AC = CD. Prove that angle BAD : angle ADB = 3:1? 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 triangle ABCD , C is any point on BD such that AB = BC and AC = CD. Prove that angle BAD : angle ADB = 3:1? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In triangle ABCD , C is any point on BD such that AB = BC and AC = CD. Prove that angle BAD : angle ADB = 3:1?.
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