Question

in triangle abc, $m\angle bac = 50^\circ$.

if $m\angle acb = 30^\circ$, then the triangle is \\(\boxed{}\\) triangle.

if $m\angle abc = 40^\circ$, then the triangle is \\(\boxed{}\\) triangle.

if triangle abc is isosceles, and $ab = 6$ and $bc = 4$, then $ac = \boxed{}$.

Explanation:

Step1: Calculate third angle for first case

Sum of angles in triangle: $180^\circ - 50^\circ - 30^\circ = 100^\circ$ (obtuse angle exists)

Step2: Calculate third angle for second case

Sum of angles: $180^\circ - 50^\circ - 40^\circ = 90^\circ$ (right angle exists)

Step3: Analyze isosceles triangle sides

Isosceles triangle has two equal sides:

• If $AC=AB$, then $AC=6$ (check triangle inequality: $6+6>4$, $6+4>6$ ✔️)
• If $AC=BC$, then $AC=4$ (check triangle inequality: $6+4>4$, $4+4>6$ ✔️)

Answer:

obtuse
right
4 or 6