In triangle ABC , AM is perpendicular to BC and AN is the bisector of angle A Find the - Maths - Lines and Angles

Question

In triangle ABC , AM is perpendicular to BC and AN is the
bisector of angle A Find the measure of angle MAN, when angle B =
65 degree and angle C = 30 degree

Shridhar Kaushik · Accepted Answer

Since âˆ A + âˆ B + âˆ C = 180Â° â‡’ âˆ A = 85Â° Also, AN is angle bisector of âˆ BAC âˆ´ âˆ BAN = âˆ NAC = $42 \frac{1}{2} °$ As âˆ ANC is exterior angle of Î” ABN $∴ ∠ N = 65 ° + 42 \frac{1}{2} ° = 107 \frac{1}{2} °$ Since BC is a straight line â‡’ âˆ ANB + âˆ ANC = 180Â° $\Rightarrow ∠ ANB = 180 ° - 107 \frac{1}{2} ° = 72 \frac{1}{2} °$ In right Î”AMN, âˆ MAN + âˆ ANM + âˆ AMN = 180Â° â‡’ âˆ MAN + âˆ ANB + âˆ AMN = 180Â° $\Rightarrow ∠ MAN + 72 \frac{1}{2} ° + 90 ° = 180 ° \Rightarrow ∠ MAN = 90 ° - 72 \frac{1}{2} ° \Rightarrow ∠ MAN = 17 \frac{1}{2} °$

In triangle ABC , AM is perpendicular to BC and AN is the bisector of angle A. Find the measure of angle MAN, when angle B = 65 degree and angle C = 30 degree.