AM⊥BC and AN is the bisector of ∠A. If ∠ABC=70∘ and ∠ACB=20∘ find the value of ∠MAN

Question

Filo · Accepted Answer

We know that the sum of all the angles in triangle

A BC

is

18 0^{\circ}

.

∠ A + ∠ B + ∠ C = 18 0^{\circ}

By substituting the values

∠ A + 7 0^{\circ} + 2 0^{\circ} = 18 0^{\circ}

On further calculation

∠ A = 18 0^{\circ} - 7 0^{\circ} - 2 0^{\circ}

By subtraction

∠ A = 18 0^{\circ} - 9 0^{\circ}

∠ A = 9 0^{\circ}

We know that the sum of all the angles in triangle

A BM

is

18 0^{\circ}

.

∠ B A M + ∠ A BM + ∠ A MB = 18 0^{\circ}

By substituting the values

∠ B A M + 7 0^{\circ} + 9 0^{\circ} = 18 0^{\circ}

On further calculation

∠ B A M = 18 0^{\circ} - 7 0^{\circ} - 9 0^{\circ}

By subtraction

∠ B A M = 18 0^{\circ} - 16 0^{\circ}

∠ B A M = 2 0^{\circ}

It is given that

A N

is the bisector of

∠ A

So it can be written as

∠ B A N = \frac{1}{2} ∠ A

By substituting the values

∠ B A N = \frac{1}{2} (9 0^{\circ})

By division

∠ B A N = 4 5^{\circ}

From the figure we know that

∠ M A N + ∠ B A M = ∠ B A N

By substituting the values we get

∠ M A N + 2 0^{\circ} = 4 5^{\circ}

On further calculation

∠ M A N = 4 5^{\circ} - 2 0^{\circ}

By subtraction

∠ M A N = 2 5^{\circ}

Therefore,

∠ M A N = 2 5^{\circ}

.

Question Text	$A M ⊥ BC$ and $A N$ is the bisector of $∠ A$ . If $∠ A BC = 7 0^{\circ}$ and $∠ A CB = 2 0^{\circ}$ find the value of $∠ M A N$
Answer Type	Text solution:1
Upvotes	150

$A M ⊥ BC$ and $A N$ is the bisector of $∠ A$ . If $∠ A BC = 7 0^{\circ}$ and $∠ A CB = 2 0^{\circ}$ find the value of $∠ M A N$