Explanation:
10.
the various sides and line segments have been marked as equal already.
so, SSS proves that both triangles are congruent.
but to reprove things with the midpoint theorem :
remember
the line segment in a triangle connecting the midpoints of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.
so,
D, F are the midpoints of BA and BC.
so, AC is parallel to DF and twice as long as DF.
D, E are the midpoints of BA and AC.
so, BC is parallel to DE and twice as long as DE.
E, F are the midpoints of AC and BC.
so, BA is parallel to EF and twice as long as EF.
so, DF = AE = EC
DE = BF = FC
EF = BD = DA
so, the EF of DEF corresponds to DA of ADE.
DF corresponds to AE.
and DE is shared between both triangles.
so, again, all 3 sides correspond 1:1 to sides in the other triangle. and SSS proves they are congruent.
11.
equilateral means all 3 sides are equal :
PS = PT = ST
TSR can only be isoceles (both legs are equal) when
ST = SR
because PM = MQ, M is the midpoint of PQ.
and because of MS || QR, S is the midpoint of PR.
that means PS = SR.
and because PS = ST, ST = SR
therefore, TSR is isoceles.