Transformations and Angle Relationships

1. A translation 5 units left subtracts 5 from each x-coordinate. A translation 3 units down subtracts 3 from each y-coordinate.

For A(1, 2):

$A' = (1 - 5, 2 - 3) = (-4, -1)$

For B(4, 2):

$B' = (4 - 5, 2 - 3) = (-1, -1)$

For C(1, 6):

$C' = (1 - 5, 6 - 3) = (-4, 3)$

The translated coordinates are A'(-4, -1), B'(-1, -1), and C'(-4, 3).

2. a) Reflecting across the x-axis keeps x the same and changes the sign of y:

$(-3, 5) \to (-3, -5)$

b) Reflecting across the y-axis changes the sign of x and keeps y the same:

$(-3, 5) \to (3, 5)$

c) Rotating 180 degrees about the origin changes both signs:

$(-3, 5) \to (3, -5)$

3. a) Rotations and translations are rigid transformations, so they preserve side lengths. The image triangle has side lengths 5, 7, and 9.

b) Yes, the triangles are congruent. A rotation and a translation can move or turn a figure, but they do not change lengths or angle measures.

4. a) Vertical angles are congruent, so the vertical angle also measures 68 degrees.

b) Supplementary angles have a sum of 180 degrees:

$180 - 68 = 112$

The supplementary angle measures 112 degrees.

c) Corresponding angles formed by parallel lines are congruent, so the corresponding angle measures 68 degrees.

5. a) The angles in a triangle sum to 180 degrees:

$42 + 73 = 115$

$180 - 115 = 65$

The third angle measures 65 degrees.

b) Corresponding angles formed by parallel lines are congruent, so the corresponding angle measures 73 degrees.