To prove that triangle ABC is isosceles triangle, we need to show all the three given conditions. Thus, answer is E. I,II and III.
What is isosceles triangle?
An isosceles triangle therefore has both two equal sides and two equal angles.
When, the slopes of AB and CD are negative reciprocals, ∠ADC =∠CDB = 90°.
Also CD = AD implies that ∠CAD = ∠DCA and CD = DB implies that ∠DCB = ∠DBC.
In triangle ACD, ∠ACD + ∠ADC + ∠ CAD = 180°.
= ∠ACD + 90 + ∠ CAD = 180
= ∠ACD + ∠ CAD = 90
(∠ACD = ∠ CAD)
∠ACD = ∠ CAD = 45°
Similarly, ∠DCB = ∠DBC =45°
This implies, ∠CAB = ∠CBA = 45°
Implying thereby, triangle ABC is isosceles triangle.
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