Question

lab geometry congruence practice questions

1. describe the transformation for (x,y)→( - x, - y).
2. describe the transformation for (x,y)→(3x,y - 4).
3. describe the transformation for (x,y)→( - 2x, - 2y).
4. which sequence of transformations is equivalent to a reflection across the x - axis?

a. rotation of 180° about the origin
b. reflection across the y - axis, then reflection across the line y = x
c. translation left 3 units, then reflection across the y - axis
d. rotation of 90° clockwise, then reflection across the y - axis

1. maria states that the slope of a line segment does not change under a certain transformation. for each transformation, indicate whether it supports or does not support her claim.

	supports	does not support
reflection across the y - axis		√
rotation 180° about the origin	√
reflection across the line y = x

1. line p is parallel to line q, cut by the transversal z. decide whether each statement is always true, sometimes true, or never true:

	always true	sometimes true	never true
∠3≅∠6	√
line z ⊥ line p		√
∠4≅∠8
∠5+∠6 = 90°

Explanation:

Step1: Analyze transformation $(x,y)\to(-x,-y)$

This is a rotation of $180^{\circ}$ about the origin.

Step2: Analyze transformation $(x,y)\to(3x,y - 4)$

It is a horizontal stretch by a factor of 3 and a vertical translation of 4 units down.

Step3: Analyze transformation $(x,y)\to(-2x,-2y)$

It is a dilation with a scale factor of 2 and a rotation of $180^{\circ}$ about the origin.

Step4: Analyze equivalent transformation to reflection across x - axis

A reflection across the y - axis then across the line $y = x$ is equivalent to a reflection across the x - axis.

Step5: Analyze slope - change under transformations

Translation 3 units left and rotation $180^{\circ}$ about the origin do not change the slope of a line segment. Reflection across y - axis does not change slope, and reflection across $y = x$ changes slope.

Step6: Analyze angle and line relationships

$\angle1+\angle2 = 180^{\circ}$ as they are a linear pair. $\angle3\cong\angle6$ as they are alternate interior angles for parallel lines p and q cut by transversal z. Line z is not always perpendicular to line p. $\angle4\cong\angle8$ as they are corresponding angles. $\angle5+\angle6$ is not always $90^{\circ}$.

Answer:

1. Rotation of $180^{\circ}$ about the origin
2. Horizontal stretch by a factor of 3 and vertical translation 4 units down
3. Dilation with scale factor 2 and rotation of $180^{\circ}$ about the origin
4. b. Reflection across the y - axis, then reflection across the line $y = x$

5.
Translation 3 units left: Supports
Reflection across the y - axis: Supports
Rotation $180^{\circ}$ about the origin: Supports
Reflection across the line $y = x$: Does not support
6.
$\angle1+\angle2 = 180^{\circ}$: Always True
$\angle3\cong\angle6$: Always True
line z $\perp$ line p: Sometimes True
$\angle4\cong\angle8$: Always True
$\angle5+\angle6 = 90^{\circ}$: Sometimes True