Please help me prove this 20 Use the Lemma A to prove Lemma

Please help me prove this!!

20-] Use the Lemma A to prove Lemma B. . [NonEuclidean Geometry emma A: If the angle of sum of a triangle AABC is less than 180 and M is the midpoint on BC, then the angle sums of AABM and MAMC are less than 180. Lemma : If angle Bthe of sum of a triangle AABC is less than 180 and D is any point on BCthen , the angle sums of AABD and MADC are less than 180. Proof B of Lemma : If B-D-C, the D must either coincide with the midpoint M of BC D lies or else on BM the is BM . Now, or MC. Assume for time being, D on let M, be the midpoint of BM, M, the midpoint of BM , M, the midpoint of BMFor some n, we must have M, on BD. For convenience, let N , , and so on. MnThen the sum of AABN is less than 180.(Question 1: Why?) angle Therefore, the angle sum of AABD is less than the angle sum of AABN (Question 2: Why?), concluding the angle sum of AABD being less than 180. Answers: Question 1 Question 2 NPD

Solution

Answer of question 1) : It is because of Lemma A that says that if the angle sum of triangle ABC is less than 180 degree and D is any point on line BC then surely the angle sum of triangle ABD will also be less than 180 degree and here it is so because point N is a point on line BC, so clearly its angle sum of triangle ABN will also be less than 180 degree.

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Answer of question 2) :It is because that angle ANB being an exterior angle for the triangle ADN will be equal to the sum of its two opposite interior angles, or in given figure also,

angle N= angle NAD+ angle ADN

or angle N > angle ADN or in other words angle ADN < angle ANB

so clearly in triangle ABD, its angle sum will be less than the angle sum of triangle ABN because angle D is less than angle N here even if two common angles A and B are of same measure.

It proves the result.