3. in the figure shown, lines f and g are parallel. select the angle that is congruent to angle 1.
a. angle 2
b. angle 6
c. angle 7
d. angle 8
(from unit 1, lesson 20.)
4. angle bde is congruent to angle bac. name another pair of congruent angles. explain how you know.
(from unit 1, lesson 20.)
5. a. describe a transformation that could be used to show that corresponding angles are congruent.
b. describe a transformation that could be used to show that alternate interior angles are congruent.

Question

Accepted Answer

### Problem 3
# Explanation:
## Step1: Recall parallel line angle relationships
When two parallel lines $f$ and $g$ are cut by a transversal, corresponding angles, alternate - interior angles, and alternate - exterior angles are congruent.
## Step2: Analyze each option
- **Option A**: Angle 1 and angle 2: There is no direct parallel - line - angle - relationship (like corresponding, alternate - interior, etc.) that makes them congruent.
- **Option B**: Angle 1 and angle 6: There is no direct parallel - line - angle - relationship (like corresponding, alternate - interior, etc.) that makes them congruent.
- **Option C**: Angle 1 and angle 7: Since $f\parallel g$ and they are alternate - exterior angles. By the Alternate - Exterior - Angles Theorem, if two parallel lines are cut by a transversal, then alternate - exterior angles are congruent.
- **Option D**: Angle 1 and angle 8: There is no direct parallel - line - angle - relationship (like corresponding, alternate - interior, etc.) that makes them congruent.

# Answer:
C. Angle 7

### Problem 4
# Explanation:
Since $m$ is a line and $\angle BDE$ and $\angle BDA$ form a linear pair ($\angle BDE+\angle BDA = 180^{\circ}$), and $\angle BAC$ and $\angle BAD$ form a linear pair ($\angle BAC+\angle BAD=180^{\circ}$). Given $\angle BDE\cong\angle BAC$, by the Congruent - Supplements Theorem (if two angles are congruent, then their supplements are congruent), $\angle BDA\cong\angle BAD$. Also, since $m\parallel l$ (assuming $m$ and $l$ are related as in a parallel - line - transversal setup from the figure), $\angle BED\cong\angle BCA$ (corresponding angles, if $BE$ is a transversal for parallel lines $m$ and $l$).

# Answer:
$\angle BDA\cong\angle BAD$ (by the Congruent - Supplements Theorem: if $\angle BDE\cong\angle BAC$ and $\angle BDE+\angle BDA = 180^{\circ}$, $\angle BAC+\angle BAD = 180^{\circ}$, then $\angle BDA\cong\angle BAD$) or $\angle BED\cong\angle BCA$ (if $m\parallel l$ and $BE$ is a transversal, by the Corresponding - Angles Theorem)

### Problem 5
## Part (a)
# Explanation:
A translation can be used to show that corresponding angles are congruent. When we translate a figure (a set of parallel lines and a transversal), the position of the figure changes, but the measure of the angles remains the same. A translation is a rigid transformation (a transformation that preserves shape and size). If we have two parallel lines $l_1$ and $l_2$ cut by a transversal $t$, and we translate the figure such that one of the parallel lines maps onto the other parallel line, the corresponding angles (e.g., $\angle1$ and $\angle2$) will overlap, showing their congruence.

# Answer:
A translation. A translation is a rigid transformation that slides a figure. When we translate the figure with parallel lines and a transversal,  corresponding angles (formed by the parallel lines and the transversal) will coincide, proving their congruence.

## Part (b)
# Explanation:
A rotation of $180^{\circ}$ about the mid - point of the segment joining a pair of points (one on each parallel line and on the transversal) can be used. Let $O$ be the mid - point of the segment $AB$ (where $A$ is on one parallel line and $B$ is on the other parallel line and both are on the transversal). A $180^{\circ}$ rotation about $O$ is a rigid transformation. When we rotate the figure $180^{\circ}$ about $O$, alternate - interior angles (e.g., $\angle3$ and $\angle4$) will map onto each other.

# Answer:
A $180^{\circ}$ rotation. A $180^{\circ}$ rotation about the mid - point of the segment joining a pair of points (one on each parallel line and on the transversal) is a rigid transformation. Alternate - interior angles will map onto each other under this rotation, proving their congruence.

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QUESTION IMAGE

Question

Explanation:

Problem 3

Step1: Recall parallel line angle relationships

Step2: Analyze each option

Answer:

Problem 4