Question

given abcd and gdef are parallelograms. prove ∠c and ∠f are supplementary angles. statements 1. abcd and gdef are parallelograms. 2. ∠c and ∠cda are supplementary angles. 3. 4. ∠edg ≅ ∠f 5. ∠edg = ∠f 6. reasons 1. 2. 3. definition of supplementary angles 4. 5. 6. vertical angles congruence theorem

Explanation:

Step1: Identify Given Information

The problem states \(ABCD\) and \(GDEF\) are parallelograms. This is the given data, so the reason for statement 1 is "Given".

Step2: Use Parallelogram Property (Consecutive Angles)

In a parallelogram, consecutive angles are supplementary. For parallelogram \(ABCD\), \(\angle C\) and \(\angle CDA\) are consecutive angles, so the reason for statement 2 is "Consecutive angles of a parallelogram are supplementary".

Step3: Write Supplementary Angles Definition

By the definition of supplementary angles, if two angles are supplementary, their sum is \(180^\circ\). So statement 3 should be "\(\angle C + \angle CDA = 180^\circ\)".

Step4: Use Parallelogram Property (Opposite Angles)

In parallelogram \(GDEF\), opposite angles are congruent. So \(\angle EDG\) and \(\angle F\) are opposite angles, so the reason for statement 4 is "Opposite angles of a parallelogram are congruent".

Step5: Define Congruent Angles

If two angles are congruent, their measures are equal. So the reason for statement 5 is "Definition of congruent angles".

Step6: Identify Vertical Angles

\(\angle CDA\) and \(\angle EDG\) are vertical angles, so by the Vertical Angles Congruence Theorem, \(\angle CDA \cong \angle EDG\). So statement 6 should be "\(\angle CDA \cong \angle EDG\)".

Step7: Substitute and Prove

From step 3: \(\angle C + \angle CDA = 180^\circ\). From step 5 and 6: \(\angle CDA = \angle F\) (since \(\angle CDA \cong \angle EDG\) and \(\angle EDG = \angle F\)). Substituting \(\angle CDA\) with \(\angle F\), we get \(\angle C + \angle F = 180^\circ\), so \(\angle C\) and \(\angle F\) are supplementary.

Answer:

1. Reason for Statement 1: Given
2. Reason for Statement 2: Consecutive angles of a parallelogram are supplementary
3. Statement 3: \(\boldsymbol{\angle C + \angle CDA = 180^\circ}\)
4. Reason for Statement 4: Opposite angles of a parallelogram are congruent
5. Reason for Statement 5: Definition of congruent angles
6. Statement 6: \(\boldsymbol{\angle CDA \cong \angle EDG}\)

(And finally, by substitution \(\angle C + \angle F = 180^\circ\), proving \(\angle C\) and \(\angle F\) are supplementary.)