Question

15
given the figure below with sides as marked.
if ( mangle a = 44^circ ), determine ( mangle b ).

(figure of a triangle with marked sides and angles a, b, c, d, e)

if ( mangle b = 22^circ ), then go to question 2.
if ( mangle b = 68^circ ), then go to question 8.
if ( mangle b = 112^circ ), then go to question 12.
if ( mangle b = 44^circ ), then go to question 4.

Explanation:

Step1: Identify the triangle type

The triangle with sides marked as equal (two sides with ticks) is isosceles, so the base angles are equal. But first, find the vertical angle to \( \angle a \).
\( \angle a \) and its vertical angle (let's say \( \angle d \)) are equal, so \( m\angle d = 44^\circ \).

Step2: Find the vertex angle of the isosceles triangle

In the isosceles triangle, the two equal sides form base angles. Wait, actually, the triangle has a base angle equal to \( 44^\circ \) (from vertical angle). Then the vertex angle \( \angle c \) is calculated as \( 180^\circ - 2\times44^\circ = 180^\circ - 88^\circ = 92^\circ \)? No, wait, maybe I messed up. Wait, the line with the two ticks is a transversal? Wait, no, the triangle has two sides equal, so it's isosceles with base angles equal. Wait, actually, the angle adjacent to \( \angle b \) and the angle from the isosceles triangle: let's re-examine.

Wait, the triangle with sides marked (two sides) is isosceles, so the base angles are equal. The angle \( \angle a = 44^\circ \), its vertical angle is also \( 44^\circ \), which is a base angle of the isosceles triangle. Then the other base angle is also \( 44^\circ \), so the vertex angle is \( 180 - 44 - 44 = 92^\circ \)? No, that can't be. Wait, maybe the triangle is isosceles with the two equal sides forming angles at the base, and the angle \( \angle b \) is supplementary to the angle adjacent to it. Wait, no, let's use the exterior angle or linear pair.

Wait, the correct approach: The triangle is isosceles, so the two base angles are equal. The angle vertical to \( \angle a \) is \( 44^\circ \), so that's one base angle. Then the other base angle is also \( 44^\circ \), so the vertex angle is \( 180 - 44 - 44 = 92^\circ \)? No, that's not right. Wait, maybe the triangle has a base angle of \( 44^\circ \), so the other base angle is \( 44^\circ \), and the vertex angle is \( 92^\circ \), but then \( \angle b \) is supplementary to the angle adjacent to it. Wait, no, maybe the angle \( \angle b \) is equal to \( 180^\circ - 2\times44^\circ \)? No, that would be \( 92^\circ \), which is not an option. Wait, the options are \( 22^\circ, 68^\circ, 112^\circ, 44^\circ \). Wait, maybe I made a mistake.

Wait, the triangle is isosceles, so the two equal sides have base angles equal. The angle \( \angle a = 44^\circ \), its vertical angle is \( 44^\circ \), which is a base angle. Then the angle adjacent to \( \angle b \) is \( 180^\circ - 44^\circ - 44^\circ = 92^\circ \)? No, that's not. Wait, maybe the triangle is isosceles with the vertex angle equal to \( 44^\circ \), so the base angles are \( (180 - 44)/2 = 68^\circ \). Ah! That's it. So the vertex angle is \( 44^\circ \), so the base angles are \( (180 - 44)/2 = 68^\circ \). Then \( \angle b \) is equal to that base angle, so \( m\angle b = 68^\circ \).

Yes, that makes sense. So the vertex angle of the isosceles triangle is \( 44^\circ \) (vertical angle to \( \angle a \)), so the base angles are \( (180 - 44)/2 = 68^\circ \), and \( \angle b \) is equal to that base angle (since they are corresponding angles or vertical angles? Wait, no, \( \angle b \) is equal to the base angle of the isosceles triangle, so \( m\angle b = 68^\circ \).

Answer:

\( 68^\circ \)