Question

in triangle abc, ( mangle bac = 50^{circ} ).
if ( mangle acb = 30^{circ} ), then the triangle is triangle.
if ( mangle abc = 40^{circ} ), then the triangle is triangle.
if triangle abc is isosceles, and ( ab = 6 ) and ( bc = 4 ), then ( ac = ).

Explanation:

Step1: Find the third angle for the first case

In a triangle, the sum of angles is \(180^{\circ}\). Given \(m\angle BAC = 50^{\circ}\) and \(m\angle ACB=30^{\circ}\), then \(m\angle ABC=180-(50 + 30)=100^{\circ}\). Since one angle (\(100^{\circ}\)) is greater than \(90^{\circ}\), it is an obtuse - angled triangle.

Step2: Find the third angle for the second case

Given \(m\angle BAC = 50^{\circ}\) and \(m\angle ABC = 40^{\circ}\), then \(m\angle ACB=180-(50 + 40)=90^{\circ}\). Since one angle (\(90^{\circ}\)) is equal to \(90^{\circ}\), it is a right - angled triangle.

Step3: Find the length of \(AC\) for the isosceles triangle case

In an isosceles triangle, two sides are equal.
Case 1: If \(AB = AC\), then \(AC = 6\) (because \(AB = 6\))
Case 2: If \(BC=AC\), then \(AC = 4\). But we check the triangle inequality.
For \(AB = 6\), \(BC = 4\)
If \(AC = 4\): \(4 + 4>6\) (triangle inequality holds: \(a + b>c\) where \(a = 4\), \(b = 4\), \(c = 6\))
If \(AC = 6\): \(4+6>6\) (triangle inequality holds: \(a = 4\), \(b = 6\), \(c = 6\))

Answer:

1. Obtuse - angled
2. Right - angled
3. \(4\) or \(6\)