Question

given that △abc is equilateral, and \\(\overline{ad}\\) bisects \\(\angle a\\), show that the side lengths of \\(\triangle adb\\) are in a ratio of \\(1 : \sqrt{3} : 2\\).

1. equilateral triangles are also equiangular, so \\(m\angle a = m\angle b = m\angle c = 60^\circ\\).
2. by definition of an angle bisector, \\(m\angle bad = m\angle cad = 30^\circ\\)
3. equilateral triangles are isosceles triangles.
4. because the bisector of the vertex angle of an isosceles triangle is the \\(\perp\\) bisector of the base, \\(m\angle bda = m\angle cda = \\) dropdown with 30°, 60°, 90°.

Explanation:

Step1: Recall properties of isosceles triangles

In an isosceles triangle, the angle bisector of the vertex angle is also the perpendicular bisector of the base. This means it forms a right angle with the base.

Step2: Determine the measure of ∠BDA and ∠CDA

Since \( \triangle ABC \) is equilateral (hence isosceles with \( AB = AC \)) and \( AD \) bisects \( \angle A \), by the property of isosceles triangles, \( AD \perp BC \). So \( m\angle BDA = m\angle CDA = 90^\circ \).

Answer:

\( 90^\circ \)