Question

given: line \overleftrightarrow{ab} intersects line \overleftrightarrow{dc} at point e; $\angle aeb$ and $\angle dec$ are straight angles. prove: $\angle aec \cong \angle deb$ statements reasons 1. $\underline{\quad\quad\quad\quad\quad\quad\quad}$ 1. $\underline{\quad\quad\quad\quad\quad\quad\quad}$ 2. $\angle aec$ is supplementary to $\angle ceb$. 2. $\underline{\quad\quad\quad\quad\quad\quad\quad}$ 3. $\angle deb$ is supplementary to $\angle ceb$. 3. def. of supplementary $\angle s$ 4. $\underline{\quad\quad\quad\quad\quad\quad\quad}$ 4. $\cong$ supplements theorem a. $\angle aeb$ and $\angle dec$ are straight angles. b. substitution c. angle addition postulate d. given e. addition property of equality f. linear pair theorem g. $\angle aec$ is congruent to $\angle deb$ h. def. supplementary angles

Explanation:

Step1: Determine Statement 1

The first statement should be the given information. The problem states "Line \( \overleftrightarrow{AB} \) intersects line \( \overleftrightarrow{DC} \) at point \( E \); \( \angle AEB \) and \( \angle DEC \) are straight angles." So the first statement is the given, which corresponds to option a (or d? Wait, the options: d is "Given", and a is the content. Wait, the first statement's reason should be "Given", and the statement is the given info. Wait, the options for statements and reasons:

For Statement 1: The given is "Line \( AB \) intersects line \( DC \) at point \( E \); \( \angle AEB \) and \( \angle DEC \) are straight angles." So the statement is option a: \( \angle AEB \) and \( \angle DEC \) are straight angles (part of the given), and the reason is d: Given.

Step2: Determine Reason 2

Statement 2 is \( \angle AEC \) is supplementary to \( \angle CEB \). The reason for this is the Linear Pair Theorem, because \( \angle AEC \) and \( \angle CEB \) form a linear pair (since \( \angle AEB \) is a straight angle, so they are adjacent and form a straight line), so the reason is f: Linear Pair Theorem.

Step3: Determine Statement 4

The reason for statement 4 is the \( \cong \) Supplements Theorem, which states that if two angles are supplementary to the same angle, then they are congruent. So \( \angle AEC \) and \( \angle DEB \) are both supplementary to \( \angle CEB \), so they are congruent. So statement 4 is g: \( \angle AEC \) is congruent to \( \angle DEB \).

Answer:

1. Statement: a. \( \angle AEB \) and \( \angle DEC \) are straight angles.

Reason: d. Given

1. Reason: f. Linear Pair Theorem
2. Statement: g. \( \angle AEC \) is congruent to \( \angle DEB \)

(For the blanks:

1. Statement: a; Reason: d
2. Reason: f
3. Statement: g)