Ratio of sides in a triangle vs ratio of angles?

Question

Given a triangle with the ratio of sides being $X: Y : Z$, is it true that the ratio of angles is also $X: Y: Z$? Could I see a proof of this? Thanks

It is true if the triangle is equilateral. – Ross Millikan Jul 08 '12 at 00:23 — Ross Millikan, Jul 08 '12 at 00:23

score 9 · Answer 1 · 2013-03-28T02:29:09.547

9

This is false. The sides and angles are related by the law of sines: $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$

edited Mar 28 '13 at 02:29

answered Jul 08 '12 at 00:17

score 4 · Answer 2 · answered Jul 08 '12 at 00:17

4

No, it is not true. Consider a 45-45-90 right triangle:

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The sides are in the ratio $1:1:\sqrt{2}$, while the angles are in the ratio $1:1:2$.

answered Jul 08 '12 at 00:17

Zev Chonoles

2 Answers2