Question

1. find $m\angle 1$

$m\angle 1 = \square ^\circ$

Explanation:

Step1: Identify equal side triangles

The tick marks show:

1. The larger triangle has two equal sides, so it is isosceles with the base angle $40^\circ$.
2. The smaller triangle formed by the segment also has two equal sides, so it is isosceles.

Step2: Find vertex angle of large triangle

Sum of angles in a triangle is $180^\circ$. Let the vertex angle be $\alpha$:
$$\alpha = 180^\circ - 2\times40^\circ = 100^\circ$$

Step3: Split vertex angle for small triangle

The segment bisects the vertex angle, so the base angles of the small isosceles triangle are each $\frac{100^\circ}{2}=50^\circ$.

Step4: Calculate $\angle 1$

Sum of angles in the small triangle is $180^\circ$:
$$m\angle 1 = 180^\circ - 2\times50^\circ = 80^\circ$$

Answer:

$80$