Answers:
1. <1=<2, <3=<4, D is midpoint of segment BE, BC=DE Given
2. BD=DE Definition of midpoint
3. BC=BD Substitution property
4. Triangle ABD congruent to triangle EBC ASA
5. <A=<E CPCTE
Solution:
1. <1=<2, <3=<4, D is midpoint of segment BE, BC=DE
All this information is given by the problem.
2. BD=DE
If D is the midpoint of segment BE, D divides this segment into two congruent parts BD and DE, then BD must be equal to DE: BD=DE by definition of midpoint.
3. BC=BD
The problem says that BC=DE (1)
And by point 2 we know that BD=DE→DE=BD (2)
Then by substitution property if we can replace DE in equation (1) by BD (because of equation (2) ):
(1) BC=DE and (2) DE=BD → (1) BC=BD
4. Triangle ABD congruent to triangle EBC
The triangles ABD and EBC have a congruent side (BD in triangle ABD and BC in triangle EBC) and the two adjacent angles congruent too (<1 in triangle ABD with <2 in triangle EBC; and <3 in triangle ABD with <4 in triangle EBC), then by Angle Side Angle (ASA) the two triangles must be congruent.
5. <A=<E
If the triangles are congruent (by point 4), all its parts must be congruent too (Corresponding Parts of Congruent Triangles are Equal: CPCTE), then the third angle in triangle ABD (<A) must be equal to the third angle in triangle EBC (<E):
<A=<E
