Answer to Question #139276 in Trigonometry for riz

Question

Prove that, If 2 angles of a spherical triangle are equal, then the triangle is an isosceles spherical triangle.

Answer 1

Let the two equal angles be at vertex $A$ and $B$, and the third angle be at vertex $C$. Now, we draw a line segment from $C$ to the midpoint of $\overline{AB}$ which we call $M$.

Then we have two triangles $\triangle{AMC}$ and $\triangle{BMC}$, which are congruent by the SAS rule.

Since $\overline{AC}$ and $\overline{BC}$ are equal

$\therefore$ two sides of the spherical triangle $\triangle {ABC}$ are equal making it an isosceles spherical triangle