Triangles in a Pentagon

There are $5$ isosceles triangles in the diagram given:

The first $3$ triangles can be shown to be isosceles by the symmetric properties of a regular pentagon. The last $2$ triangles can be shown to be isosceles by finding that the base angles are congruent.

Using the diagram above, $AB$ and $BC$ are sides of a regular pentagon, so they are congruent, which means $\triangle ABC$ is an isosceles triangle. Likewise, $BC$ and $CD$ are sides of a regular pentagon, so they are also congruent, which means $\triangle BCD$ is an isosceles triangle.

Since $\triangle ABC$ is an isosceles triangle, $\angle BAC = \angle BCA$ , and since the angles of a triangle add up to $180°$ and since the interior angle of a regular pentagon is $108°$ , $\angle BAC = \angle BCA = \frac{180° - 108°}{2} = 36°$ . Likewise, from isosceles $\triangle BCD$ , $\angle CBD = \angle CDB = 36°$ . Since $\angle EBC = \angle ECB = 36°$ , $\triangle BCE$ is an isoceles triangle.

Since the angles of $\triangle BCE$ add up to $180°$ , $\angle BEC = 180° - 36° - 36° = 108°$ , which means $\angle AEB = 72°$ and $\angle DEC = 72°$ . Since $\angle ABC = \angle ABE + \angle EBC$ , $\angle ABE = 108° - 36° = 72°$ . Likewise, $\angle DCE = 108° - 36° = 72°$ . Since $\angle ABE = \angle AEB$ , $\triangle ABE$ is an isosceles triangle, and since $\angle DCE = \angle CED$ , $\triangle DCE$ is an isosceles triangle.

Looking at the Pentagon the first 3 isoscelece triangles were easy to see then I visually discected the upper portion of the pentagon in half and saw two more isoscelece triangles 3+2=5

Use your eyes wisely

one at the bottom, two on the sides. But if you combine the bottom and the side isosceles triangles you get a fourth isosceles triangle, do the same with the triangle on the other side and you get five isosceles triangle.

Answer should not be 5 It should be 10

Let's call the points in the pentagon $A,B,C,D,E$ & the center $O$ .Then split the pentagon into 4 parts:quadrilateral $ABOE$ ,triangle $BOC$ ,triangle $COD$ & triangle $DOE$

There are 3 isosceles triangles:triangle $BOC$ ,triangle $COD$ & triangle $DOE$ .

There also have 2 more isosceles triangles:triangle $BCD$ & $CDE$

In total,the number of isosceles triangles in the diagram is $\boxed{\large{5}}$