Question 26

Let $$\triangle ABC$$ be a triangle with $$Ab = AC$$ and $$D$$ be a point on $$BC$$ such that $$\angle BAD = 30^\circ$$. If E is a point on $$AC$$ such that $$AD = AE$$, then $$\angle CDE$$ equals

IMG_0777 (1)

Based on the information provided, make the diagram as given above. We realise that the two angles on the unequal side in an isosceles triangle are equal. That both $$\triangle ABC$$ and $$\triangle ADE$$ are isosceles triangles.

Let $$\angle DAE$$ be $$2x$$, it follows that in $$\triangle ABC$$, $$\angle ABC + \angle ACB = 180-(30+2x)$$. Also, since $$\angle ABC = \angle ACB$$ we get $$\angle ABC = \angle ACB = 75-x$$

We also find in $$\triangle ADE$$, $$\angle ADE + \angle AED = 180-2x$$, and since $$\angle ADE = \angle AED$$ we get $$\angle AED = 90-x$$. 

Since $$\angle AED$$ and $$\angle DEC$$ are collinear, $$\angle DEC$$ becomes $$180-\angle AED = 180-(90-x) = 90+x$$

Finally, in $$\triangle CDE$$, we need to find the value of $$\angle CDE$$, which will be equal to $$180-(\angle DEC + \angle ACB) = 180- (75-x+90+x) = 180-165= 15^{\circ}$$

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