A, B, C and D are four points on a circle - Edexcel - GCSE Maths - Question 2 - 2019 - Paper 1
Question 2
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 2 - 2019 - Paper 1
Step 1
Prove that angle AED = angle ADB
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Answer
Since triangle AED is equilateral, all its angles are equal to 60°.
Thus, angle AED = 60°.
Step 2
Prove that angle ADB = angle DBC
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Answer
Since A, B, C, and D are points on the circle, the angles subtended at the circumference by the same arc are equal.
Therefore, angle ADB = angle DBC.
Step 3
Prove that angle ABC = angle DCB
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Answer
The angles at the center of the circle subtending the same arc also have the same measure.
Since DEB is a straight line, angle ABC + angle ADB = 180°.
From the previous point, angle ADB = angle DBC, hence angle ABC = angle DCB.
Step 4
Show that triangles ABC and DCB are congruent
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Answer
We have shown:
Angle ABC = angle DCB.
Angle ADB = angle DBC.
Side AC is common to both triangles.
By the Angle-Side-Angle (ASA) congruence criterion, triangle ABC is congruent to triangle DCB.
