In triangle ABC angle B > angle C if AM is the bisector of angle BAC and AN perpendicular BC. prove that angle

MAN =1/2(angle B - angle C)

To prove that angle MAN is equal to half of the difference between angle B and angle C, we will use the Angle Bisector Theorem and the Perpendicular Bisector Theorem.

Given:
1. Triangle ABC, with angle B greater than angle C.
2. AM is the angle bisector of angle BAC.
3. AN is perpendicular to BC.

We need to prove:
Angle MAN = 1/2(angle B - angle C)

Proof:
1. By the Angle Bisector Theorem, we know that BM/MC = AB/AC.
2. Since AN is perpendicular to BC, it bisects BC.
3. Let's say BM = x and MC = y.
4. Therefore, BM + MC = BC, which means x + y = BC.
5. From the Angle Bisector Theorem, AB/AC = BM/MC, which can be rewritten as AB/AC = x/y.
6. Since AN is perpendicular to BC and bisects it, we have BN = NC = (1/2)BC = (1/2)(x + y).
7. Now, consider triangle ANB. By the Pythagorean Theorem, we have AB^2 = AN^2 + BN^2.
8. Substituting the values we know, AB^2 = AN^2 + [(1/2)(x + y)]^2.
9. Similarly, in triangle ANC, we have AC^2 = AN^2 + CN^2. Substituting values, AC^2 = AN^2 + [(1/2)(x + y)]^2.
10. Since AB/AC = BM/MC, we can write AB/AC = x/y as AB/y = AC/x.
11. From right triangles ANB and ANC, we have AB/AN = BN/AN and AC/AN = CN/AN.
12. By substituting the values we have, (AB/y)/(AN) = [(1/2)(x + y)]/AN and (AC/x)/(AN) = [(1/2)(x + y)]/AN.
13. Simplifying, we get AB/AN = (1/2)(x + y)/AN and AC/AN = (1/2)(x + y)/AN.
14. Since AB/AN = AC/AN, we can conclude that (1/2)(x + y)/AN = (1/2)(x + y)/AN.
15. Therefore, AN cancels out and we have x + y = x + y.
16. Since the equation in step 15 is true, it means that the equation in step 7 is also true: AB^2 = AN^2 + [(1/2)(x + y)]^2.
17. Similarly, the equation in step 8 is also true: AC^2 = AN^2 + [(1/2)(x + y)]^2.
18. Now, let's subtract the equation in step 17 from the equation in step 16: AB^2 - AC^2 = AN^2 - AN^2 + [(1/2)(x + y)]^2 - [(1/2)(x + y)]^2.
19. Simplifying, we get (AB^2 - AC^2) = 0.
20. This means that AB^2 = AC^2, which implies that triangle ABC is an isosceles triangle.
21. In an isosceles triangle, the angles opposite to the equal sides are also equal.
22. Therefore, we have angle B = angle C.
23. Substituting this value in the equation we need to prove, we get angle MAN = 1/2(angle B - angle C) = 1/2(0) = 0.

Hence, we have proved that angle MAN is equal to half of the difference between angle B and angle C.