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ABC is an right angle triangle and PRQ is also right angle triangle and 90 degree angle on C,R. now question starts if angle B and angle Q are acute angle such that sin B= sinQ then prove that angle B= angle Q?
Most Upvoted Answer
ABC is an right angle triangle and PRQ is also right angle triangle an...
Proof:
Given: ABC is a right angle triangle with angle C as 90 degrees.
PRQ is also a right angle triangle with angle R as 90 degrees.
Angle B and angle Q are acute angles.
sin B = sin Q.
To prove: angle B = angle Q.
Proof by contradiction:
Assume that angle B is not equal to angle Q.
Case 1: angle B > angle Q
Let's assume angle B is greater than angle Q.
Since ABC is a right angle triangle, the sum of all angles in a triangle is 180 degrees. Therefore, angle B + angle C + angle A = 180 degrees.
In triangle ABC, angle C = 90 degrees. Substituting the value of angle C, we get angle B + 90 degrees + angle A = 180 degrees.
Simplifying the equation, we have angle B + angle A = 90 degrees.
In triangle PRQ, angle R = 90 degrees. Therefore, angle Q + 90 degrees + angle P = 180 degrees.
Simplifying the equation, we have angle Q + angle P = 90 degrees.
Given that sin B = sin Q, we can write the equation as sin B = sin (90 - angle P).
Using the identity sin (90 - x) = cos x, we have sin B = cos angle P.
Since angle B > angle Q, sin B > sin Q. But from our assumption, sin B = sin Q.
This is a contradiction, as sin B cannot be greater than sin Q when angle B > angle Q.
Therefore, our assumption that angle B > angle Q is false.
Case 2: angle B < angle="" />
Let's assume angle B is less than angle Q.
Using similar logic as in Case 1, we can arrive at a contradiction.
Therefore, our assumption that angle B < angle="" q="" is="" />
Conclusion:
Since both assumptions, angle B > angle Q and angle B < angle="" q,="" lead="" to="" contradictions,="" we="" can="" conclude="" that="" angle="" b="" must="" be="" equal="" to="" angle="" />
Hence, angle B = angle Q.
This proves that if sin B = sin Q, then angle B = angle Q.
Given: ABC is a right angle triangle with angle C as 90 degrees.
PRQ is also a right angle triangle with angle R as 90 degrees.
Angle B and angle Q are acute angles.
sin B = sin Q.
To prove: angle B = angle Q.
Proof by contradiction:
Assume that angle B is not equal to angle Q.
Case 1: angle B > angle Q
Let's assume angle B is greater than angle Q.
Since ABC is a right angle triangle, the sum of all angles in a triangle is 180 degrees. Therefore, angle B + angle C + angle A = 180 degrees.
In triangle ABC, angle C = 90 degrees. Substituting the value of angle C, we get angle B + 90 degrees + angle A = 180 degrees.
Simplifying the equation, we have angle B + angle A = 90 degrees.
In triangle PRQ, angle R = 90 degrees. Therefore, angle Q + 90 degrees + angle P = 180 degrees.
Simplifying the equation, we have angle Q + angle P = 90 degrees.
Given that sin B = sin Q, we can write the equation as sin B = sin (90 - angle P).
Using the identity sin (90 - x) = cos x, we have sin B = cos angle P.
Since angle B > angle Q, sin B > sin Q. But from our assumption, sin B = sin Q.
This is a contradiction, as sin B cannot be greater than sin Q when angle B > angle Q.
Therefore, our assumption that angle B > angle Q is false.
Case 2: angle B < angle="" />
Let's assume angle B is less than angle Q.
Using similar logic as in Case 1, we can arrive at a contradiction.
Therefore, our assumption that angle B < angle="" q="" is="" />
Conclusion:
Since both assumptions, angle B > angle Q and angle B < angle="" q,="" lead="" to="" contradictions,="" we="" can="" conclude="" that="" angle="" b="" must="" be="" equal="" to="" angle="" />
Hence, angle B = angle Q.
This proves that if sin B = sin Q, then angle B = angle Q.
Community Answer
ABC is an right angle triangle and PRQ is also right angle triangle an...
Send me in this only. And with explanation of it .I know whole solution but not get understand let see give in this only my WhatsApp is off yet.
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ABC is an right angle triangle and PRQ is also right angle triangle and 90 degree angle on C,R. now question starts if angle B and angle Q are acute angle such that sin B= sinQ then prove that angle B= angle Q? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about ABC is an right angle triangle and PRQ is also right angle triangle and 90 degree angle on C,R. now question starts if angle B and angle Q are acute angle such that sin B= sinQ then prove that angle B= angle Q? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ABC is an right angle triangle and PRQ is also right angle triangle and 90 degree angle on C,R. now question starts if angle B and angle Q are acute angle such that sin B= sinQ then prove that angle B= angle Q?.
ABC is an right angle triangle and PRQ is also right angle triangle and 90 degree angle on C,R. now question starts if angle B and angle Q are acute angle such that sin B= sinQ then prove that angle B= angle Q? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about ABC is an right angle triangle and PRQ is also right angle triangle and 90 degree angle on C,R. now question starts if angle B and angle Q are acute angle such that sin B= sinQ then prove that angle B= angle Q? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ABC is an right angle triangle and PRQ is also right angle triangle and 90 degree angle on C,R. now question starts if angle B and angle Q are acute angle such that sin B= sinQ then prove that angle B= angle Q?.
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