ABC is a triangle - Leaving Cert Mathematics - Question 6B - 2014
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 - 2014
Step 1
Consider the quadrilateral DOEC.
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Answer
We know that the angles in quadrilateral DOEC satisfy the following:
∣∠CDO∣+∣∠EOC∣+∣∠OEC∣+∣∠ODC∣=180°.
Given that:
∣∠CDO∣=90° (since AD ⊥ BC)
∣∠EOC∣=90° (since BE ⊥ AC)
Thus, substituting these values gives us:
∣∠CDO∣+∣∠EOC∣=90°+90°=180°.
This shows both pairs of opposite angles sum to 180°. Therefore, DOEC is a cyclic quadrilateral.
Step 2
Prove that |∠DOC| = |∠DEC|.
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Answer
Since DOEC is a cyclic quadrilateral, the opposite angles are equal according to the properties of cyclic quadrilaterals.
Therefore, we conclude that:
∣∠DOC∣=∣∠DEC∣.
This completes the proof as required.
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