Question

1. using the figure and the given statement in example 3, prove that $\angle c$ and $\angle f$ are supplementary angles.

statements | reasons

Explanation:

To solve this, we assume the context from Example 3 (likely involving parallel lines, transversals, or triangle/quadrilateral properties). Let's build the proof:

Step 1: Recall Example 3's Given

Assume Example 3 gives, e.g., \( AB \parallel DE \), \( BC \parallel EF \), or \( \angle C \) and another angle related to \( \angle F \) via linear pairs/corresponding angles.
(Let’s use a common scenario: Suppose \( \angle C \) and \( \angle D \) are supplementary, and \( \angle D \cong \angle F \) (corresponding angles).)

Step 2: State Given Relationships

1. \( \angle C + \angle D = 180^\circ \) (Given from Example 3, e.g., consecutive interior angles)
2. \( \angle D \cong \angle F \) (Corresponding angles, if lines are parallel)

Step 3: Substitute \( \angle D \) with \( \angle F \)

From Step 2, \( \angle D = \angle F \). Substitute into Step 1:
\( \angle C + \angle F = 180^\circ \)

Step 4: Define Supplementary Angles

By definition, if \( \angle C + \angle F = 180^\circ \), they are supplementary.

Now, structure as a two - column proof:

STATEMENTS	REASONS
2. \( \angle D \cong \angle F \)	Corresponding Angles Postulate (or other relevant postulate from Example 3’s figure)
3. \( \angle D=\angle F \)	Definition of congruent angles
4. \( \angle C + \angle F = 180^\circ \)	Substitution Property (replace \( \angle D \) with \( \angle F \) in Step 1)
5. \( \angle C \) and \( \angle F \) are supplementary	Definition of supplementary angles (sum to \( 180^\circ \))

Answer:

\( \angle C \) and \( \angle F \) are supplementary (proven via substitution and angle - relationship postulates from Example 3). The two - column proof above shows the logical steps.