Question

given: \\( \angle a b c = 90 ^ { \circ } \\)
which statements (if any) are true about the diagram? select all that apply:
\\( \square \\) a. \\( \angle 2 \\) and \\( \angle 3 \\) are complementary
\\( \square \\) b. points \\( a , b \\), and \\( d \\) are collinear
\\( \square \\) c. ray \\( a b \\) and ray \\( b c \\) are opposite rays
\\( \square \\) d. none of these

1. which of the following transformations will produce an image that is congruent to its pre-image? select all that apply:

\\( \square \\) a. a \\( 90 ^ { \circ } \\) rotation followed by a dilation of a scale factor of 3.
\\( \square \\) b. a dilation with a scale factor of \\( \frac { 1 } { 2 } \\) followed by a reflection over the line \\( y = x \\).
\\( \square \\) c. a translation of 4 units down, followed by a reflection over the \\( x \\)-axis.
\\( \square \\) d. a rotation of \\( 270 ^ { \circ } \\) followed by a reflection over the \\( y \\)-axis.

Explanation:

Step1: Analyze question 10

• For option A:

Since \(\angle ABC = 90^{\circ}\), and \(\angle ABC=\angle2+\angle3\), by the definition of complementary angles (two angles whose sum is \(90^{\circ}\)), \(\angle2\) and \(\angle3\) are complementary.

• For option B:

Collinear points lie on the same straight - line. Points \(A\), \(B\), and \(D\) do not lie on the same straight - line.

• For option C:

Opposite rays form a straight - line (a \(180^{\circ}\) angle). Ray \(AB\) and ray \(BC\) form a \(90^{\circ}\) angle, not a \(180^{\circ}\) angle.

Step2: Analyze question 11

• For option A:

A dilation with a scale factor \(k
eq1\) (here \(k = 3\)) changes the size of the figure. Congruent figures have the same shape and size. So, a \(90^{\circ}\) rotation followed by a dilation of a scale factor of \(3\) does not produce a congruent image.

• For option B:

A dilation with a scale factor \(k=\frac{1}{2}
eq1\) changes the size of the figure. So, a dilation with a scale factor of \(\frac{1}{2}\) followed by a reflection over the line \(y = x\) does not produce a congruent image.

• For option C:

A translation (which is a rigid transformation, it only moves the figure without changing its shape or size) of \(4\) units down and a reflection (a rigid transformation, it only flips the figure without changing its shape or size) over the \(x\) - axis. Rigid transformations preserve congruence.

• For option D:

A rotation (a rigid transformation, it only turns the figure without changing its shape or size) of \(270^{\circ}\) and a reflection (a rigid transformation) over the \(y\) - axis. Rigid transformations preserve congruence.

Answer:

1. A. \(\angle2\) and \(\angle3\) are complementary
2. C. A translation of 4 units down, followed by a reflection over the \(x\) - axis; D. A rotation of \(270^{\circ}\) followed by a reflection over the \(y\) - axis