Question

statement reason

1. if ( mangle 2 = 47^{circ} ),

then, ( mangle 6=)

1. if ( mangle 4 = 104^{circ} ),

then, ( mangle 6=)

1. if ( mangle 5 = 66^{circ} ),

then, ( mangle 4=)

1. if ( mangle 7 = 71^{circ} ),

then, ( mangle 8=)

1. if ( mangle 1 = 127^{circ} ),

then, ( mangle 4=)
statement reason

1. if ( mangle 8 = 42^{circ} ),

then, ( mangle 3=)

1. if ( mangle 3 = 68^{circ} ),

then, ( mangle 7=)

1. if ( mangle 8 = 11^{circ} ),

then, ( mangle 1=)

1. if ( mangle 4 = 53^{circ} ),

then, ( mangle 2=)

1. if ( mangle 2 = 153^{circ} ),

then, ( mangle 1=)

Explanation:

Step1: Vertical angles and corresponding angles

For problem 5:

• $\angle2$ and $\angle6$ are corresponding angles.
• If two parallel lines are cut by a transversal, corresponding angles are equal.
• So, $m\angle6 = m\angle2$.
• Given $m\angle2 = 47^{\circ}$, then $m\angle6 = 47^{\circ}$.

Step2: Alternate interior angles and linear pairs

For problem 6:

• $\angle4$ and $\angle6$ are same - side interior angles.
• If two parallel lines are cut by a transversal, $m\angle4+m\angle6 = 180^{\circ}$ (supplementary).
• Given $m\angle4 = 104^{\circ}$, then $m\angle6=180^{\circ}-m\angle4$.
• $m\angle6 = 180^{\circ}-104^{\circ}=76^{\circ}$.

Step3: Vertical angles and alternate interior angles

For problem 7:

• $\angle5$ and $\angle3$ are alternate interior angles, so $m\angle5 = m\angle3 = 66^{\circ}$.
• $\angle3$ and $\angle4$ are linear - pair angles, so $m\angle3+m\angle4 = 180^{\circ}$.
• Then $m\angle4=180^{\circ}-m\angle3$.
• $m\angle4 = 180^{\circ}-66^{\circ}=114^{\circ}$.

Step4: Linear pair angles

For problem 8:

• $\angle7$ and $\angle8$ are linear - pair angles.
• So $m\angle7+m\angle8 = 180^{\circ}$.
• Given $m\angle7 = 71^{\circ}$, then $m\angle8=180^{\circ}-m\angle7$.
• $m\angle8 = 180^{\circ}-71^{\circ}=109^{\circ}$.

Step5: Vertical angles

For problem 9:

• $\angle1$ and $\angle4$ are vertical angles.
• Vertical angles are equal.
• Given $m\angle1 = 127^{\circ}$, then $m\angle4 = 127^{\circ}$.

Step6: Vertical angles and corresponding angles

For problem 10:

• $\angle8$ and $\angle5$ are vertical angles, so $m\angle8 = m\angle5 = 42^{\circ}$.
• $\angle5$ and $\angle3$ are corresponding angles.
• So $m\angle3 = m\angle5 = 42^{\circ}$.

Step7: Vertical angles and alternate interior angles

For problem 11:

• $\angle3$ and $\angle5$ are alternate interior angles, so $m\angle3 = m\angle5 = 68^{\circ}$.
• $\angle5$ and $\angle7$ are vertical angles.
• So $m\angle7 = m\angle5 = 68^{\circ}$.

Step8: Vertical angles and linear pair angles

For problem 12:

• $\angle8$ and $\angle5$ are vertical angles, so $m\angle8 = m\angle5 = 11^{\circ}$.
• $\angle5$ and $\angle1$ are linear - pair angles.
• So $m\angle1=180^{\circ}-m\angle5$.
• $m\angle1 = 180^{\circ}-11^{\circ}=169^{\circ}$.

Step9: Vertical angles

For problem 13:

• $\angle4$ and $\angle2$ are vertical angles.
• Vertical angles are equal.
• Given $m\angle4 = 53^{\circ}$, then $m\angle2 = 53^{\circ}$.

Step10: Linear pair angles

For problem 14:

• $\angle2$ and $\angle1$ are linear - pair angles.
• So $m\angle1 + m\angle2=180^{\circ}$.
• Given $m\angle2 = 153^{\circ}$, then $m\angle1=180^{\circ}-m\angle2$.
• $m\angle1 = 180^{\circ}-153^{\circ}=27^{\circ}$.

Answer:

1. $47^{\circ}$
2. $76^{\circ}$
3. $114^{\circ}$
4. $109^{\circ}$
5. $127^{\circ}$
6. $42^{\circ}$
7. $68^{\circ}$
8. $169^{\circ}$
9. $53^{\circ}$
10. $27^{\circ}$