Question

statements reasons

1. ( mangle acb = 48^{circ}, mangle chi = 132^{circ} ) 1. given
2. ( mangle dch = 48^{circ} ) 2. ? a. what property justifies ( mangle dch = 48^{circ} )?
3. ( 48^{circ}+132^{circ}=180^{circ} ) 3. addition
4. ( mangle dch + mangle chi = 180^{circ} ) 4. substitution property of equality
5. ( angle dch ) and ( angle chi ) are supplementary. 5. definition of supplementary angles
6. ( pparallel q ) 6. ? b. what property justifies ( pparallel q )?
7. ( mangle efy = 44^{circ}, mangle fyz = 44^{circ} ) 7. given
8. ( angle efycongangle fyz ) 8. definition of congruent angles
9. ( pparallel r ) 9. ? c. what property justifies ( pparallel r )?
10. ( qparallel r ) 10. ? d. what allows you to write ( qparallel r )?

Explanation:

• A: Since \(m\angle ACB = 48^{\circ}\) and \(m\angle DCH = 48^{\circ}\), by the definition of vertical angles (vertical angles are equal), this property justifies \(m\angle DCH=48^{\circ}\).
• B: When two lines are cut by a transversal and the consecutive - interior angles are supplementary (\(m\angle DCH + m\angle CHI=180^{\circ}\)), the lines are parallel. So, the property is "If consecutive - interior angles are supplementary, then the lines are parallel".
• C: When two lines are cut by a transversal and the alternate - interior angles are congruent (\(\angle EFY\cong\angle FYZ\)), the lines are parallel. So, the property is "If alternate - interior angles are congruent, then the lines are parallel".
• D: If \(p\parallel q\) and \(p\parallel r\), then by the transitive property of parallel lines (\(a\parallel b\) and \(a\parallel c\) implies \(b\parallel c\)), we can say \(q\parallel r\).

Answer:

A. Vertical Angles Theorem.
B. If consecutive - interior angles are supplementary, then the lines are parallel.
C. If alternate - interior angles are congruent, then the lines are parallel.
D. Transitive Property of Parallel Lines.