Answer:
(a, b) alternate interior angles at M and N, and at A and B are congruent
(c) the triangles are congruent by SAS and by ASA (and MX = NX)
(d) angles are no longer congruent, so the triangles are not congruent. The radii are given as congruent, but the chords cannot be shown congruent.
Explanation:
Given same-size circles A and B, externally tangent to each other at X, each with chords MX and NX, you want to know what can be concluded if AM║BN, and what is unprovable if those segments are not parallel.
Same-size circles
The circles being the same size means all the radii are congruent. This is shown by the single hash marks in the attached diagram.
(a) Angles
Alternate interior angles where a transversal crosses parallel lines are congruent. If AM║BN, this means the angles marked with a single arc are congruent, and the angles marked with a double arc are congruent. These are the alternate interior angles at transversal MN and at transversal AB.
(b) Corresponding parts
If AM║BN, in addition to the given congruences, we also know ...
- all radii are congruent — given in the problem statement
- angles M and N are congruent (see above)
- angles A and B are congruent (see above)
- the vertical angles at X are congruent to each other and to angles M and N (isosceles triangles) (AMBN is a parallelogram.)
(c) Congruent triangles
∆AMX ≅ ∆BNX by SAS or ASA (take your pick).
(d) Not parallel
If AM and BN are not parallel, MN is not a straight line through X, the angles at A and B are not congruent, and the angles at M and N are not congruent. (We assume segment AB still goes through X.)
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Additional comment
Triangles MAX and NBX are isosceles, so their base angles are congruent. If X lies on MN, then AM and BN must be parallel, since the vertical angles at X will be congruent along with the other base angles at M and N. If AM and BN are not parallel, point X cannot lie on segment AB.