Question 10

$$ABC$$ is a triangle. Point $$D$$ lies on $$AC$$ such that $$AB=BD=CD$$. All the angles in the diagram are positive whole numbers of degrees. The largest possible size, in degrees of $$\angle ABC$$ is

Solution

Let $$\angle BAD=\angle ADB=x$$ because $$AB=BD$$, and let $$\angle DBC=\angle BCD=y$$ because $$BD=CD$$. The straight line at $$D$$ gives $$x=2y$$. Hence $$\angle ABC=(180^\circ-2x)+y=180^\circ-3y$$, which is largest for the smallest positive integer $$y=1$$, giving $$177^\circ$$.

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$$ABC$$ is a triangle. Point $$D$$ lies on $$AC$$ such that $$AB=BD=CD$$. All the angles in the diagram are positive whole numbers of degrees. The largest possible size, in degrees of $$\angle ABC$$ is

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