A, B, C and D are four points on a circle - Edexcel - GCSE Maths - Question 1 - 2019 - Paper 2
Question 1
A, B, C and D are four points on a circle.
AEC and DEB are straight lines.
Triangle AED is an equilateral triangle.
Prove that triangle ABC is congruent to triang... show full transcript
Worked Solution & Example Answer:A, B, C and D are four points on a circle - Edexcel - GCSE Maths - Question 1 - 2019 - Paper 2
Step 1
Prove that angle ABC = angle DCB
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Answer
Since triangle AED is equilateral, we know that each angle in triangle AED measures 60 degrees. Therefore, angle DAB is equal to 60 degrees. By the inscribed angle theorem, angle ADB will be equal to half the measure of arc AB, causing angle ADB to also be 60 degrees. This means angle ABC = angle DCB, both equal to 60 degrees.
Step 2
Prove that angle ACB = angle DAB
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Answer
As angle DAB is equal to 60 degrees, and both angles ACB and DAB subtend the same arc AC, we can conclude that angle ACB = angle DAB = 60 degrees.
Step 3
Prove that side AB = side DC
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Answer
Since all sides of triangle AED are equal and triangle ABC is inscribed in the circle subtended by the same arc, we have that side AB is equal to side DC.
Step 4
Conclusion
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Answer
We have shown that:
Angle ABC = angle DCB (60 degrees)
Angle ACB = angle DAB (60 degrees)
Side AB = side DC
By the Angle-Side-Angle (ASA) criterion for triangle congruence, triangle ABC is congruent to triangle DCB.
