Answer question with correct steps and answers please

Question

Use the figure shown to answer the question that follows.

What is the order of rotation of this figure?

A. 2

B.4

C.6

D.8

Two triangles are shown on the coordinate plane.

Name the congruent triangles and justify the reason for congruence.

 ΔABC ≅ ΔFDE by SAS 

ΔABC ≅ ΔFDE by HL 

ΔABC ≅ ΔFED by SSS 

ΔABC ≅ to ΔFED by ASA

If triangle ABC is translated 4 units to the left and 2 units downward, are the pre-image and image congruent? Why, or why not?

 Yes, distance is preserved and angle measure is not

 No, angle measure is preserved but distance is not 

Yes, distance and angle measure are preserved 

No, neither distance nor angle measure are preserved

Lines m and n are cut by a transversal, as shown in the figure.

Given line m is not parallel to line n, prove ∠4 is not congruent to ∠6 by contradiction. (2 points for the assumption statement, 4 points for the remainder of the proof)

Given the following conditional statement:

If m∠F = 30° and m∠G = 80°, then ∠F and ∠G are not complementary.

Part A: How is proof by contradiction different from a traditional proof? What assumption statement will begin proof by contradiction for the given conditional statement? (3 points)

Part B: Prove the given conditional statement is true by contradiction. (3 points)

Answer 1

What is the order of rotation of this figure?

Based on the image, the figure has rotational symmetry of order 4 (B). This means that the figure can be rotated 90 degrees 4 times to return to its original position.

Two triangles are shown on the coordinate plane.

1. Name the congruent triangles and justify the reason for congruence:
2. ΔABC ≅ ΔFDE by SAS
3. If ΔABC is translated 4 units to the left and 2 units downward, are the pre-image and image congruent? Why, or why not?
4. If ΔABC is translated 4 units to the left and 2 units downward, the pre-image and image will be congruent because the distance and angle measures are preserved.

Lines m and n are cut by a transversal, as shown in the figure. Given line m is not parallel to line n, prove ∠4 is not congruent to ∠6 by contradiction.

Assumption: We assume that ∠4 is congruent to ∠6

Proof:

1. Based on our assumption, we can conclude that ∠4 and ∠6 are corresponding angles as well.
2. Corresponding angles are congruent only if the lines they are on are parallel.
3. However, we are given that the lines m and n are not parallel.
4. By contradiction, our assumption that ∠4 is congruent to ∠6 must be false.
5. We conclude that ∠4 is not congruent to ∠6.

If m∠F = 30° and m∠G = 80°, then ∠F and ∠G are not complementary.

Part A: How is proof by contradiction different from a traditional proof? What assumption statement will begin proof by contradiction for the given conditional statement?

1. Proof by contradiction is a type of proof that begins by assuming the opposite of what we want to prove and then showing that this assumption leads to a contradiction. For the given conditional statement, our assumption would look something like this: ∠F and ∠G are complementary.

Part B: Prove the given conditional statement is true by contradiction.

Proof:

1. If our assumption is true, then ∠F and ∠G are complementary.
2. By definition, complementary angles add up to 90°.
3. However, (m∠F = 30°) + (m∠G = 80°) = 110°.
4. This is a contradiction, as 110° ≠ 90°.
5. We can conclude that if m∠F = 30° and m∠G = 80°, then ∠F and ∠G are not complementary.