ABC is a triangle - Leaving Cert Mathematics - Question 6B - 2012
Question 6B
ABC is a triangle.
D is the point on BC such that AD ⊥ BC.
E is the point on AC such that BE ⊥ AC.
AD and BE intersect at O.
Prove that |∠DOC| = |∠DEC|.
Worked Solution & Example Answer:ABC is a triangle - Leaving Cert Mathematics - Question 6B - 2012
Step 1
Consider the quadrilateral DOEC.
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Answer
We have
∣∠CDO∣+∣∠OECD∣=90°+90°=180°.
This implies that DOEC is a cyclic quadrilateral (by the converse of Corollary 5).
Step 2
Using the properties of cyclic quadrilaterals.
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Answer
Therefore,
∣∠DOC∣=∣∠DEC∣
because angles subtended by the same arc in a cyclic quadrilateral are equal.
Step 3
Conclusion.
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Answer
Thus, by Theorem 19, we have proven that
∣∠DOC∣=∣∠DEC∣
as required.
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