Let M be a point of contact of two internally touching circles. Let line AMB be their common tangent.The chord CD of the bigger circle touches the smaller circle at point N and chord DM of the bigger circle touches the smaller circle at the points P and R respectively.
$∠ CMN ≅ ∠ DMN$

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Solution

Construction:- Join NR

$∠ CMA = ∠ CDM (angles in the alternate segment) Let ∠ CMA = ∠ CDM = x . . . . . . . . . . . . (1) ∠ NMA = ∠ NRM (angles in the alternate segment) Let ∠ NMA = ∠ NRM = y . . . . . . . . . . . . . . . (2) So, ∠ NMC = ∠ NMA - ∠ CMA = y - x . . . . . . . . . . . . . (3) ∠ NMR = ∠ DNR . . . . . . . (4) (angles in alternate segments) ∠ NRM is an exterior angle of △ NDR So, ∠ NRM = ∠ NDR + ∠ DNR or, y = ∠ CDM + ∠ DNR = x + ∠ DNR o r, ∠ DNR = y - x . . . . . . . . . . . . . . . . (5) \Rightarrow ∠ NMR = y - x [From (4) and (5)] \Rightarrow ∠ NMD = y - x . . . . . . . . (6) ∠ CMN = ∠ DMN [From (3) and (6)]$

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Let M be a point of contact of two internally touching circles. Let line AMB be their common tangent.The chord CD of the bigger circle touches the smaller circle at point N and chord DM of the bigger circle touches the smaller circle at the points P and R respectively. ∠CMN≅∠DMN

Let M be a point of contact of two internally touching circles. Let line AMB be their common tangent.The chord CD of the bigger circle touches the smaller circle at point N and chord DM of the bigger circle touches the smaller circle at the points P and R respectively.
$∠ CMN ≅ ∠ DMN$