Triangle BDE is congruent to any equilateral triangle. The corresponding parts of the congruent triangles are BD, DE, and BE. This congruence arises from the properties of a median in triangle ABC, leading to equal side lengths and angles in triangle BDE, aligning with the characteristics of an equilateral triangle.
a) Segment BD is a median of triangle ABC. A median is a line segment that joins a vertex of a triangle to the midpoint of the opposite side. In this case, BD joins vertex B to the midpoint of side AC.
b) To find the measure of angle BDE, we can use the fact that the angles of a triangle add up to 180 degrees. We are given that angle DAE is 35 degrees, and we know that angles BAD and EAC are congruent (since BD bisects angle BAC). Therefore, we can find the measure of angles BAD and EAC as follows:
(angle BAC) / 2 = angle BAD = angle EAC
Substituting the given value for angle BAC:
(90 degrees) / 2 = 45 degrees
Therefore, angle BDE = 180 degrees - (angle BAD + angle DAE + angle EAC) = 180 degrees - (45 degrees + 35 degrees + 45 degrees) = 60 degrees.
c) A triangle that could be congruent to triangle BDE is an equilateral triangle. An equilateral triangle is a triangle where all three sides are equal in length, and all three angles are equal in measure (60 degrees).
In triangle BDE, we are given that BD = DE (since BD is a median), and we can infer from the symmetry of the diagram that BE = BD. Therefore, all three sides of triangle BDE are equal.
The corresponding parts of the congruent triangles are:
Triangle BDE: BD, DE, BE
Equilateral triangle: any side (since all sides are congruent), any side, any side