Question

naming triangles based on side lengths and angle measures

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 two cases

• For the first case:
• We know that the sum of angles in a triangle is \(180^{\circ}\). Given \(m\angle BAC = 50^{\circ}\) and \(m\angle ACB=30^{\circ}\), then \(m\angle ABC=180-(50 + 30)=100^{\circ}\).
• For the second case:
• Given \(m\angle BAC = 50^{\circ}\) and \(m\angle ABC = 40^{\circ}\), then \(m\angle ACB=180-(50 + 40)=90^{\circ}\).

Step2: Classify the triangles based on angles

• For the first case:
• Since one angle (\(m\angle ABC = 100^{\circ}\)) is greater than \(90^{\circ}\), the triangle is an obtuse - angled triangle.
• For the second case:
• Since one angle (\(m\angle ACB = 90^{\circ}\)) is equal to \(90^{\circ}\), the triangle is a right - angled triangle.

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

• In an isosceles triangle, two sides are equal.
• If \(AB = AC\), then \(AC = 6\) (because \(AB = 6\)).
• We check the triangle inequality. For a triangle with sides \(a,b,c\), \(a + b>c\), \(a + c>b\) and \(b + c>a\).
• If \(AC = 6\), with \(AB = 6\) and \(BC = 4\), \(6+6>4\), \(6 + 4>6\) and \(4+6>6\).
• If \(AC=4\), then \(4 + 4>6\), \(4+6>4\) and \(6 + 4>4\) also holds. But since \(AB = 6\) and \(BC = 4\), if the triangle is isosceles with \(AB\) as one of the equal sides, \(AC = 6\) (because if \(AC = 4\), then \(AB

eq AC
eq BC\) which is not an isosceles triangle in the sense of two equal sides when \(AB = 6\) and \(BC = 4\)).

Answer:

• Obtuse - angled
• Right - angled
• \(6\)