Question

find each measure.

1. \\( \angle l y f \\)
2. \\( \angle n j z \\)
3. \\( \angle m y w \\)

Explanation:

Step1: Use the property of angle bisector

In the first figure, since \( LY \) is the angle bisector of \( \angle JYF \), and \( \angle JYL=\angle LYF \). Given \( \angle JYL = 58^{\circ} \), so \( \angle LYF=58^{\circ} \)

Step2: Use the property of right - angled triangle and angle sum

In the second figure, in right - angled triangle \( JIN \), \( \angle I = 90^{\circ} \), \( \angle IJN=38^{\circ} \), then \( \angle JNI = 90^{\circ}-38^{\circ}=52^{\circ} \). Since \( JN = JZ \), \( \angle NJZ = 180^{\circ}- 2\times52^{\circ}=76^{\circ} \)

Step3: Use the property of angle bisector and equation solving

In the third figure, since \( YM \) and \( YW \) are angle bisectors, \( 4x - 1=2x + 5 \). Solve the equation \( 4x-2x=5 + 1 \), \( 2x=6 \), \( x = 3 \). Then \( \angle MTW=(4x - 1)+(2x + 5)=6x+4 \). Substitute \( x = 3 \) into it, \( \angle MTW=6\times3 + 4=22^{\circ} \)

Answer:

1. \( 58^{\circ} \)
2. \( 76^{\circ} \)
3. \( 22^{\circ} \)