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Question 1 (Essay Worth 10 points)

(01.05 MC)

The following conditional statement is given: "If it is Friday, then there is no school tomorrow."

Part A: Explain how to change a statement into the converse and write the converse of the given conditional statement. (5 points)

Part B: Is the converse of the conditional statement true or false? Explain your answer. If the converse is false, give a counterexample. (5 points)

Question 2 (Essay Worth 10 points)
(01.02 MC)

Jaymie wants to construct congruent angles with a compass and straightedge, while Annie wants to construct congruent segments with a compass and straightedge. Describe the similarities and differences between their construction steps.

Question 3 (Essay Worth 10 points)
(01.06 MC)

The figure shows adjacent angles BAC and CAD.

Adjacent angles BAC and CAD sharing common ray AC

Given:
m∠BAD = 134°
m∠BAC = (2x + 3)°
m∠CAD = (4x −1)°

Part A: Using the angle addition postulate, write and solve an equation for x. Show all your work. (6 points)

Part B: Find the m∠CAD. Show all your work. (4 points)

Question 4 (Essay Worth 10 points)
(01.07 HC)

Tonya and Pearl each completed a separate proof to show that alternate interior angles AKL and FLK are congruent. Who completed the proof incorrectly? Explain.

Line AB is parallel to EF, transversal GJ crosses line AB at K and crosses line EF at L.

Tonya's Proof

Statement Justification
1. line AB ∥ line EF with transversal segment GJ 1. Given
2. angle AKL is congruent to angle GKB 2. Vertical Angles Theorem
3. angle GKB is congruent to angle FLK 3. Corresponding Angles Theorem
4. angle AKL is congruent to angle FLK 4. Transitive Property

Pearl's Proof

Statement Justification
1. line AB ∥ line EF with transversal segment GJ 1. Given
2. angle AKL is congruent to angle GKB 2. Definition of Adjacent Angles
3. angle GKB is congruent to angle FLK 3. Corresponding Angles Theorem
4. angle AKL is congruent to angle FLK 4. Transitive Property

1 answer

Question 1

Part A: The converse of a conditional statement reverses the hypothesis and the conclusion. For the given statement "If it is Friday, then there is no school tomorrow," the hypothesis is "it is Friday" and the conclusion is "there is no school tomorrow." Therefore, the converse is: "If there is no school tomorrow, then it is Friday."

Part B: The converse of the conditional statement is not necessarily true. In fact, it can be false because there are other instances in which there is no school tomorrow, such as holidays or snow days, which are not Fridays. A counterexample could be a Monday after a snowstorm, where there is no school, but it is not Friday. This shows that the condition of having no school tomorrow does not exclusively occur on Fridays.

Question 2: Both Jaymie and Annie are focused on creating congruent geometric elements using a compass and straightedge; however, their goals differ.

Similarities:

  1. Both constructions use the compass for measuring and marking distances and the straightedge for drawing lines.
  2. Each method relies on fundamental geometric principles, such as congruence, to ensure the resulting constructions are equal to each other.

Differences:

  1. Jaymie's construction steps to create congruent angles typically involve drawing a segment, constructing a circle with the compass to create an arc that intersects the angle, and then replicating that arc to form a new angle congruent to the original.
  2. Annie's construction steps for congruent segments would involve measuring a segment with the compass, marking that length on a new location, and then connecting the endpoints with a straightedge.

In summary, while both tasks require the same tools and general idea of replication, they differ primarily in their focus (angles for Jaymie versus segments for Annie) and the specific steps used in the construction process.

Question 3

Part A: Using the angle addition postulate, the equation can be set up as follows: m∠BAD = m∠BAC + m∠CAD.

Substituting the given values: 134° = (2x + 3) + (4x - 1).

Combine like terms: 134° = 2x + 3 + 4x - 1, 134° = 6x + 2.

Now, subtract 2 from both sides: 132° = 6x.

Next, divide both sides by 6: x = 22.

Part B: Now that we have x, we can find m∠CAD using the equation: m∠CAD = 4x - 1.

Substituting x = 22: m∠CAD = 4(22) - 1, m∠CAD = 88 - 1, m∠CAD = 87°.

Thus, m∠CAD is 87°.

Question 4: To evaluate who completed the proof incorrectly, let's examine each proof’s justifications.

Tonya's Proof:

  1. The first step is valid and given.
  2. Her second step claims that angle AKL is congruent to angle GKB using the Vertical Angles Theorem. However, this is incorrect because AKL and GKB are not vertical angles; therefore, this statement is not justified correctly.
  3. The third step, with the use of the Corresponding Angles Theorem, is correct because GKB and FLK are indeed corresponding angles.
  4. The transitive property in her last step is valid as it relies on the congruence established in the previous step.

Conclusion: Tonya's proof is incorrect due to a misapplication of the Vertical Angles Theorem in statement 2.

Pearl's Proof:

  1. The first statement in Pearl's proof is valid and also given.
  2. Her second step, claiming that angle AKL is congruent to angle GKB using the Definition of Adjacent Angles, is incorrect since AKL and GKB are not adjacent. They share a common vertex but are not adjacent angles in the sense of sharing a common side.
  3. The third step referring to the corresponding angles is still valid.
  4. The last step using the transitive property is also valid since it is based on a true congruence claim.

Conclusion: Pearl's proof is also incorrect because of the erroneous justification for step 2 related to adjacent angles.

Final Conclusion: Both Tonya and Pearl made mistakes, but Tonya's primary fault lies in her misunderstanding of vertical angles, while Pearl's misunderstanding is related to adjacent angles.