Question

7.

1. ( mangle cda cong mangle f )

9.

1. ( angle c ) and ( angle f ) are supplementary angles.

1. definition of congruent angles

8.

1. substitution property of equality

10.

given
parallelogram opposite angles theorem
( mangle c + mangle cda = 180^circ )
parallelogram consecutive angles theorem
( angle cda cong angle edg )
( mangle cda = mangle edg )
definition of congruent angles
( mangle c + mangle f = 180^circ )
definition of supplementary angles

Explanation:

Step1: Analyze Question 7

We need to find the reason for a statement related to congruent angles. The "Definition of congruent angles" is a reason, but we need to match the statement. Wait, the first blank (7) is for a statement with reason "Definition of congruent angles". The option "Definition of congruent angles" is a reason, but maybe the statement is \( m\angle CDA = m\angle EDG \) with reason "Definition of congruent angles" (since \( \angle CDA\cong\angle EDG \) implies \( m\angle CDA = m\angle EDG \) by definition of congruent angles). Wait, no, the reason for 7 is "Definition of congruent angles", so the statement should be \( m\angle CDA = m\angle EDG \) (since \( \angle CDA\cong\angle EDG \) gives this by definition). So for 7, the statement is \( m\angle CDA = m\angle EDG \) with reason "Definition of congruent angles".

Step2: Analyze Question 8

The statement is \( m\angle CDA\cong m\angle F \)? Wait, no, \( m\angle CDA\cong m\angle F \) is not correct, should be \( \angle CDA\cong\angle F \) (but the statement is \( m\angle CDA\cong m\angle F \)? Wait, the given options: "Given" might be the reason for \( m\angle CDA\cong m\angle F \)? Wait, no, let's list the options:

Options:
- Given
- Parallelogram Opposite Angles Theorem
- \( m\angle C + m\angle CDA = 180^\circ \)
- Parallelogram Consecutive Angles Theorem
- \( \angle CDA\cong\angle EDG \)
- \( m\angle CDA = m\angle EDG \)
- Definition of congruent angles
- \( m\angle C + m\angle F = 180^\circ \)
- Definition of supplementary angles
- Transitive Property of Equality

Wait, for question 8: statement is \( m\angle CDA\cong m\angle F \)? No, the statement is \( m\angle CDA\cong m\angle F \)? Wait, the reason for 8: let's see, the statement is \( m\angle CDA\cong m\angle F \) (but actually, angle measures are equal, so \( m\angle CDA = m\angle F \) by... Wait, maybe "Given" is the reason for \( m\angle CDA\cong m\angle F \) (but the notation is off, maybe it's \( \angle CDA\cong\angle F \) with \( m\angle CDA = m\angle F \) by given? Wait, the option "Given" is a reason, so for 8, the reason is "Given" for the statement \( m\angle CDA = m\angle F \) (or \( \angle CDA\cong\angle F \)).

Step3: Analyze Question 9

The reason is "Substitution Property of Equality". We have \( m\angle C + m\angle CDA = 180^\circ \) (from Parallelogram Consecutive Angles Theorem) and \( m\angle CDA = m\angle F \) (from 8, given), so substituting \( m\angle CDA \) with \( m\angle F \) gives \( m\angle C + m\angle F = 180^\circ \). So the statement for 9 is \( m\angle C + m\angle F = 180^\circ \) with reason "Substitution Property of Equality".

Step4: Analyze Question 10

The statement is \( \angle C \) and \( \angle F \) are supplementary angles. The reason is "Definition of supplementary angles" (since \( m\angle C + m\angle F = 180^\circ \) implies they are supplementary by definition).

Now, let's match each blank:

7. Statement: \( m\angle CDA = m\angle EDG \) (since \( \angle CDA\cong\angle EDG \) and by definition of congruent angles, their measures are equal), Reason: Definition of congruent angles.

8. Statement: \( m\angle CDA = m\angle F \) (or \( \angle CDA\cong\angle F \)), Reason: Given (from the options, "Given" is a reason, so this statement is given).

9. Statement: \( m\angle C + m\angle F = 180^\circ \), Reason: Substitution Property of Equality (substituting \( m\angle CDA \) with \( m\angle F \) in \( m\angle C + m\angle CDA = 180^\circ \)).

10. Statement: \( \angle C \) and \( \angle F \) are supplementary angles, Reason: Definition of supplementary angles (since \( m\angle C + m\angle F = 180^\circ \) fits the definition of supplementary angles).

Now, let's pick the correct options:

7. The statement with reason "Definition of congruent angles" is \( m\angle CDA = m\angle EDG \) (because \( \angle CDA\cong\angle EDG \) implies \( m\angle CDA = m\angle EDG \) by definition of congruent angles). So for 7, the answer is \( m\angle CDA = m\angle EDG \) (option: \( \boldsymbol{m\angle CDA = m\angle EDG} \)) with reason "Definition of congruent angles".

8. The statement \( m\angle CDA\cong m\angle F \) (wait, no, \( m\angle CDA = m\angle F \)) is given, so reason is "Given" (option: \( \boldsymbol{Given} \)).

9. The statement is \( m\angle C + m\angle F = 180^\circ \) (option: \( \boldsymbol{m\angle C + m\angle F = 180^\circ} \)) with reason "Substitution Property of Equality".

10. The reason for \( \angle C \) and \( \angle F \) being supplementary is "Definition of supplementary angles" (option: \( \boldsymbol{Definition\ of\ supplementary\ angles} \)).

Wait, let's recheck:

- For 7: Reason is "Definition of congruent angles", so the statement must be a measure equality from a congruent angle. The option \( m\angle CDA = m\angle EDG \) comes from \( \angle CDA\cong\angle EDG \) (option \( \angle CDA\cong\angle EDG \)) by definition of congruent angles. So 7: statement is \( m\angle CDA = m\angle EDG \), reason: Definition of congruent angles.

- For 8: Statement is \( m\angle CDA\cong m\angle F \)? Wait, no, the statement is \( m\angle CDA\cong m\angle F \) (but actually, angle measures are equal, so \( m\angle CDA = m\angle F \)) and the reason is "Given" (since it's a given fact). So 8: reason is "Given".

- For 9: Reason is "Substitution Property of Equality", so we substitute \( m\angle CDA \) with \( m\angle F \) in \( m\angle C + m\angle CDA = 180^\circ \) (from Parallelogram Consecutive Angles Theorem) to get \( m\angle C + m\angle F = 180^\circ \). So 9: statement is \( m\angle C + m\angle F = 180^\circ \).

- For 10: Statement is \( \angle C \) and \( \angle F \) are supplementary angles, reason is "Definition of supplementary angles" (since \( m\angle C + m\angle F = 180^\circ \) fits the definition).

Now, let's list the answers:

7. \( m\angle CDA = m\angle EDG \) (option: \( m\angle CDA = m\angle EDG \))

8. Given (option: Given)

9. \( m\angle C + m\angle F = 180^\circ \) (option: \( m\angle C + m\angle F = 180^\circ \))

10. Definition of supplementary angles (option: Definition of supplementary angles)

Wait, but let's check the options again:

Options:

- Given

- Parallelogram Opposite Angles Theorem

- \( m\angle C + m\angle CDA = 180^\circ \)

- Parallelogram Consecutive Angles Theorem

- \( \angle CDA\cong\angle EDG \)

- \( m\angle CDA = m\angle EDG \)

- Definition of congruent angles

- \( m\angle C + m\angle F = 180^\circ \)

- Definition of supplementary angles

- Transitive Property of Equality

So:

7. Reason: Definition of congruent angles, so statement is \( m\angle CDA = m\angle EDG \) (because \( \angle CDA\cong\angle EDG \) implies this by definition). So 7: \( m\angle CDA = m\angle EDG \)

8. Statement: \( m\angle CDA\cong m\angle F \) (wait, no, \( m\angle CDA = m\angle F \)) with reason "Given". So 8: Given

9. Reason: Substitution Property of Equality, so substitute \( m\angle CDA \) with \( m\angle F \) in \( m\angle C + m\angle CDA = 180^\circ \) (from Parallelogram Consecutive Angles Theorem) to get \( m\angle C + m\angle F = 180^\circ \). So 9: \( m\angle C + m\angle F = 180^\circ \)

10. Reason: Definition of supplementary angles (since \( m\angle C + m\angle F = 180^\circ \) means they are supplementary). So 10: Definition of supplementary angles

Answer:

1. \( m\angle CDA = m\angle EDG \)
2. Given
3. \( m\angle C + m\angle F = 180^\circ \)
4. Definition of supplementary angles