A, B, C and D are four points on the circumference of a circle - Edexcel - GCSE Maths - Question 15 - 2017 - Paper 2
Question 15
A, B, C and D are four points on the circumference of a circle.
AEC and BED are straight lines.
Prove that triangle ABE and triangle DCE are similar.
You must give... show full transcript
Worked Solution & Example Answer:A, B, C and D are four points on the circumference of a circle - Edexcel - GCSE Maths - Question 15 - 2017 - Paper 2
Step 1
Show that angle ABE = angle DCE
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Answer
Since A, B, C, and D are points on the circumference of the circle, angle ABE subtends arc AE while angle DCE subtends arc DC. According to the Inscribed Angle Theorem, angles subtending the same arc are equal. Thus, we have:
angle ABE=angle DCE
Step 2
Show that angle BAE = angle CDE
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Answer
Similarly, angle BAE subtends arc BD and angle CDE subtends arc CA. Using the same reasoning, these angles are also equal:
angle BAE=angle CDE
Step 3
State that angles ABE, BAE, CDE, and DCE are corresponding angles
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Answer
With the above two pairs of angles established as equal, we can conclude that triangles ABE and DCE have two pairs of angles which are equal. Therefore, by the Angle-Angle (AA) similarity criterion, triangles ABE and DCE are similar:
