Question

1. select all the pairs of angles that are alternate interior angles.

□ a. ∠1 and ∠3
□ b. ∠5 and ∠6
□ c. ∠8 and ∠9
□ d. ∠8 and ∠17
□ e. ∠16 and ∠17

1. point b has coordinates (-8, 1). what are the coordinates of the point when reflected across the y-axis and then rotated 180° about the origin?

ⓐ (8, 1)
ⓑ (-8, -1)
ⓒ (-8, 1)
ⓓ (8, -1)

1. select all the equations that represent a line that is perpendicular to the line with equation y = 2x - 8.

□ a. $y = \frac{1}{2}x + 1$
□ b. $y = -\frac{1}{2}x + 1$
□ c. $x + 2y = 5$
□ d. $-x + 2y = -3$
□ e. $-x - 2y = 9$

1. quadrilateral abcd has coordinates a(-2, 0), b(0, 4), c(4, 6), and d(2, 2). what are the coordinates of the image of abcd after applying a reflection across the y-axis, and then applying a reflection across the x-axis?

a( , ),
b( , ),
c( , ),
d( , )

1. what rule maps abcd to abcd?

ⓐ $r_{y\text{-axis}} \circ t_{\langle -1, 0 \
angle}$
ⓑ $r_{x\text{-axis}} \circ t_{\langle 1, 0 \
angle}$
ⓒ $t_{\langle 0, -1 \
angle} \circ r_{y\text{-axis}}$
ⓓ $t_{\langle 0, 1 \
angle} \circ r_{x\text{-axis}}$

1. △abc has vertices a(1, 3), b(2, 5), and c(5, 3). what are the coordinates of b after the transformation (x, y) → (-y, x), then translation 1 unit right and 4 units up?

( , )

Explanation:

Question 5

Step1: Recall alternate interior angles

Alternate interior angles are formed when a transversal crosses two parallel lines, lying between the lines and on opposite sides of the transversal.

Step2: Analyze each option

• Option A: ∠1 and ∠3 are vertical angles, not alternate interior.
• Option B: ∠5 and ∠6 are adjacent angles, not alternate interior.
• Option C: ∠8 and ∠9 are not between the lines (exterior or not in the right position).
• Option D: ∠8 and ∠17 – Check the lines: the transversal and parallel lines. ∠8 and ∠17 lie between the two parallel lines (the horizontal ones) and on opposite sides of the transversal (the slant line), so they are alternate interior.
• Option E: ∠16 and ∠17 – These are adjacent angles, not alternate interior.

Step1: Reflect across y - axis

The rule for reflecting a point \((x,y)\) across the \(y\) - axis is \((x,y)\to(-x,y)\). For point \(B(-8,1)\), after reflection across \(y\) - axis, it becomes \((8,1)\).

Step2: Rotate 180° about the origin

The rule for rotating a point \((x,y)\) 180° about the origin is \((x,y)\to(-x,-y)\). Now take the point \((8,1)\) from step 1. Applying the 180° rotation: \((8,1)\to(-8,-1)\)? Wait, no, wait. Wait, 180° rotation: \((x,y)\to(-x,-y)\). Wait, initial reflection: \((-8,1)\) across \(y\) - axis is \((8,1)\). Then 180° rotation: \((8,1)\to(-8,-1)\)? But wait, let's re - check. Wait, maybe I made a mistake. Wait, the options: Let's re - do.

Wait, reflection across \(y\) - axis: \((x,y)\to(-x,y)\). So \(B(-8,1)\) becomes \((8,1)\). Then rotation 180° about origin: \((x,y)\to(-x,-y)\). So \((8,1)\to(-8,-1)\)? But option B is \((-8,-1)\), option D is \((8,-1)\). Wait, no, wait, maybe I messed up the rotation. Wait, 180° rotation: the formula is \((x,y)\to(-x,-y)\). So after reflecting \((-8,1)\) over \(y\) - axis to \((8,1)\), then rotating 180°: \((8,1)\) becomes \((-8,-1)\)? But let's check the options. Option B is \((-8,-1)\), option D is \((8,-1)\). Wait, maybe I made a mistake in the order. Wait, the problem says: reflect across \(y\) - axis and then rotate 180° about origin.

Wait, another way: The composition of reflection over \(y\) - axis (\(r_y\)) and rotation 180° (\(R_{180}\)): \(R_{180}(r_y(x,y))=R_{180}(-x,y)= (x,-y)\). Wait, that's different. Wait, \(R_{180}(a,b)=(-a,-b)\). So \(r_y(-8,1)=(8,1)\), then \(R_{180}(8,1)=(-8,-1)\)? No, \(R_{180}(8,1)=(-8,-1)\). But let's check the options. Option B is \((-8,-1)\), option D is \((8,-1)\). Wait, maybe my initial reflection was wrong. Wait, no: reflection across \(y\) - axis: \((x,y)\to(-x,y)\). So \((-8,1)\) becomes \((8,1)\). Then 180° rotation: \((8,1)\) becomes \((-8,-1)\), which is option B? But wait, let's check the options again. Wait, the options are:

A. \((8,1)\)

B. \((-8,-1)\)

C. \((-8,1)\)

D. \((8,-1)\)

Wait, maybe I made a mistake in the rotation. Wait, 180° rotation: the rule is \((x,y)\to(-x,-y)\). So \((8,1)\) rotated 180° is \((-8,-1)\), which is option B? But let's check with the options. Wait, but let's think again. Wait, maybe the order is different. Wait, maybe the reflection is first, then rotation. Let's confirm with the formula for 180° rotation: yes, \((x,y)\to(-x,-y)\). So after reflection: \((8,1)\), then rotation: \((-8,-1)\), which is option B? But wait, the answer given in the options, let's check the options again. Wait, the options: A is \((8,1)\), B is \((-8,-1)\), C is \((-8,1)\), D is \((8,-1)\). Wait, maybe I made a mistake in the reflection. Wait, no: reflection across \(y\) - axis: \((x,y)\to(-x,y)\). So \((-8,1)\) becomes \((8,1)\). Then 180° rotation: \((8,1)\to(-8,-1)\), which is option B? But let's check the options. Wait, the correct answer is B? Wait, but let's re - check. Wait, maybe the problem is that I mixed up the rotation. Wait, 180° rotation: the coordinates \((x,y)\) become \((-x,-y)\). So yes, \((8,1)\) becomes \((-8,-1)\), so option B.

Step1: Find the slope of the given line

The given line is \(y = 2x-8\), its slope \(m_1 = 2\). For a line to be perpendicular to this line, the slope \(m_2\) of the perpendicular line should satisfy \(m_1\times m_2=-1\). So \(m_2=-\frac{1}{2}\).

Step2: Analyze each option

• Option A: \(y=\frac{1}{2}x + 1\), slope \(m=\frac{1}{2}\), not \(-\frac{1}{2}\), so not perpendicular.
• Option B: \(y =-\frac{1}{2}x+1\), slope \(m =-\frac{1}{2}\), so perpendicular.
• Option C: Rewrite \(x + 2y=5\) as \(y=-\frac{1}{2}x+\frac{5}{2}\), slope \(m =-\frac{1}{2}\), so perpendicular.
• Option D: Rewrite \(-x + 2y=-3\) as \(y=\frac{1}{2}x-\frac{3}{2}\), slope \(m=\frac{1}{2}\), not perpendicular.
• Option E: Rewrite \(-x-2y = 9\) as \(y=-\frac{1}{2}x-\frac{9}{2}\), slope \(m =-\frac{1}{2}\), so perpendicular.

Answer:

D. ∠8 and ∠17

Question 6