Summarize the rotations that turn a regular triangle onto itself. Enter your answers in ascending order.(2 points)

Question

The rotations that turn a regular triangle onto itself are
°,
°, and
°.

Answer 1

The rotations that turn a regular triangle onto itself are 0°, 120°, and 240°.

Answer 2

How will the following transformation rule move a preimage?

(x,y)→(−x,−y)

(1 point)
Responses

It will be dilated to become smaller.
It will be dilated to become smaller.

It will be translated up and to the right.
It will be translated up and to the right.

It will be rotated 180 degrees.
It will be rotated 180 degrees.

It will be translated down and to the left.

Answer 3

The transformation rule \((x,y) \to (-x,-y)\) will move a preimage rotated 180 degrees.

Answer 4

Identify which two lines are parallel.(1 point)
Responses

y=12x+3 and y=12x−5
y equals Start Fraction 1 over 2 End Fraction x plus 3 and y equals Start Fraction 1 over 2 End Fraction x minus 5

y=13x+4 and y=13x+4
y equals Start Fraction 1 over 3 End Fraction x plus 4 and y equals Start Fraction 1 over 3 End Fraction x plus 4

y=23x−6 and y=−32x−6
y equals two-thirds x minus 6 and y equals negative three-halves x minus 6

y=−12x+6 and

Answer 5

To determine which lines are parallel, we need to identify the slopes of each pair of lines. Parallel lines have the same slope.

1. y = 12x + 3 and y = 12x - 5

   • Both lines have a slope of 12.

2. y = \(\frac{1}{2}\)x + 3 and y = \(\frac{1}{2}\)x - 5

   • Both lines have a slope of \(\frac{1}{2}\).

3. y = 13x + 4 and y = 13x + 4

   • Both lines have a slope of 13 (they are identical).

4. y = \(\frac{2}{3}\)x - 6 and y = -\(\frac{3}{2}\)x - 6

   • The slopes are \(\frac{2}{3}\) and -\(\frac{3}{2}\), which are not the same.

5. y = -12x + 6 and the last line is not completely provided, so we cannot check its slope.

The pairs of parallel lines are:

• \(y = 12x + 3\) and \(y = 12x - 5\)
• \(y = \frac{1}{2}x + 3\) and \(y = \frac{1}{2}x - 5\)
• \(y = 13x + 4\) and \(y = 13x + 4\) (these lines are identical)

If asked to identify just one set, the most straightforward answer would be the first set: y = 12x + 3 and y = 12x - 5.

Answer 6

What type of angles are used to help prove that the sum of the measures of the interior angles of a triangle is always 180° ? (1 point)
Responses

right angles
right angles

vertical angles
vertical angles

alternate interior angles
alternate interior angles

obtuse angles

Answer 7

The type of angles used to help prove that the sum of the measures of the interior angles of a triangle is always 180° are alternate interior angles.

Answer 8

PROOF: Given isosceles △ABC with AB¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯, I can construct BD←→, the angle bisector of ∠B. Then, by the definition of an angle bisector, I know that ∠ABD≅∠CBD. I also know that line segments are congruent to themselves, so BD¯¯¯¯¯¯¯¯≅BD¯¯¯¯¯¯¯¯ by the reflexive property of congruence. I now have two pairs of sides and an included angle that are congruent, so I know that △ABD≅△CBD by the SAS Congruence Theorem.

What statement should be placed at the end of the proof?

(1 point)
Responses

 Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so ∠A≅∠B.
 Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so angle upper A congruent to angle upper B .

Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so AB¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯.
Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so Modifying above upper A upper B with bar congruent to Modifying above upper B upper C with bar .

Finally, ∠A≅∠C by the Base Angles Theorem.
Finally, angle upper A congruent to angle upper C by the Base Angles Theorem.

Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so ∠A≅∠C.

Answer 9

The appropriate statement to place at the end of the proof is:

Finally, corresponding parts of congruent triangles are congruent by the CPCTC Theorem, so ∠A≅∠C by the Base Angles Theorem.

Answer 10

Lulu marks the points (9,2) and (3,6) on the coordinate plane. What is the distance between these two points, rounded to the nearest tenth of a unit?(1 point)
Responses

7.2 units
7.2 units

7.8 units
7.8 units

10.0 units
10.0 units

6.0 units

Answer 11

To calculate the distance \( d \) between the points \((9, 2)\) and \((3, 6)\), we use the distance formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting in the coordinates \((x_1, y_1) = (9, 2)\) and \((x_2, y_2) = (3, 6)\):

\[ d = \sqrt{(3 - 9)^2 + (6 - 2)^2} \] \[ = \sqrt{(-6)^2 + (4)^2} \] \[ = \sqrt{36 + 16} \] \[ = \sqrt{52} \] \[ = \sqrt{4 \times 13} \] \[ = 2\sqrt{13} \] Calculating \( \sqrt{13} \):

\[ \sqrt{13} \approx 3.60555 \]

Thus,

\[ 2\sqrt{13} \approx 2 \times 3.60555 \approx 7.2111 \]

Rounding \( 7.2111 \) to the nearest tenth gives:

7.2 units

So the correct answer is 7.2 units.