Question

Im trying to find the measure of the numbered angle:)??

Answer 1

As Angle(2) and Angle(3) are Vertically Opposite Angles, they should be Equal.

Let us Take Angle(2) = Angle(3) = X

The Opposite Angle to 58° will also be 58° as they are Vertically Opposite Angles.

We can Notice that Angle(2), Angle(3) and Both the 58° Angles together make a Complete Angle (i.e. 360°)

⇒ Angle(2) + Angle(3) + 58° + 58° = 360°

⇒ Angle(2) + Angle(3) + 116° = 360°

⇒ Angle(2) + Angle(3) = 360° - 116°

⇒ Angle(2) + Angle(3) = 244°

⇒ X + X = 244°

⇒ 2X = 244°

⇒ X = 122°

We took Angle(2) = Angle(3) = X

⇒ Angle(2) = Angle(3) = 122°

Now let us go for the Upper Triangle :

As Sum of Angles in a Triangle = 180°

Angle (1) + Angle(2) + 22° = 180°

Angle (1) + 122° + 22° = 180°

Angle (1) + 144° = 180°

Angle (1) = 180° - 144°

Angle (1) = 36°

Now let us go for the Lower Triangle :

As Sum of Angles in a Triangle = 180°

Angle (3) + Angle(4) + 20° = 180°

Angle (4) + 122° + 20° = 180°

Angle (4) + 142° = 180°

Angle (4) = 180° - 142°

Angle (4) = 38°

Answer 2

3 and 2 would be 122 since 3 and the 58° are a linear pair making them add up to 180. And 3 and 2 are vertical so they would be congruent.
4 would then have to be 38 since the three angles would have to measure up to 180.
And the 1 would have to be 36 with the same reasoning as before.
Disclamier (I might have done some math wrong since I did it in my head.)