a) Segment BD in triangle ABC is a special segment as it serves as both a median and altitude, perpendicular bisecting side AC.
b) The measure of angle EBC is 25 degrees, obtained by solving for x in the equation 2x + 17 = 35.
c) Triangle BDE is congruent to triangle BDC by ASA. Corresponding congruent parts include angles DBE and BDC, angle BDE and angle C, and side BD.
a) Special Segment BD:
Segment BD is the perpendicular bisector of side AC. This makes BD a median and an altitude of triangle ABC. It is also the segment joining the vertex of the right angle (B) to the midpoint of the hypotenuse (AC) in a right-angled triangle.
b) Measure of Angle EBC:
Since BD is the perpendicular bisector, it divides triangle ABC into two congruent right-angled triangles, ABD and CBD. Therefore, angle C = angle DBE. Given that angle C = 35 degrees, we can write:
2x + 17 = 35
Solving for x:
2x = 18
x = 9
So, the measure of angle EBC is 3(9) - 2 = 25 degrees.
c) Congruent Triangle to △BDE:
Triangle BDE is congruent to triangle BDC. They are congruent by ASA (Angle-Side-Angle) as both triangles share angle DBE (equal to angle C) and angle BDE (equal to angle BDC), and the included side BD is common.
In summary, triangle BDE is congruent to triangle BDC, and the corresponding congruent parts are:
- Angle DBE ≅ Angle BDC
- Angle BDE ≅ Angle C
- Side BD ≅ Side BD (common side)