Question

1. in the diagram shown below, find the value of (x).

a. 25
b. 40
c. 65
d. 90

1. which of the following theorems or postulates you would use to prove that line (m) and (n) are parallel?

a. corresponding angles converse theorem
b. alternate interior angles converse theorem
c. congruent angles
d. vertical angles

1. to solve for (x), kaitlyn wrote (3x = 120). is kaitlyn correct? if not, write the correct statement and solve for (x).

1. in the diagram, select all the values that are true for (x) and (y).

a. (x = 100) and (y = 100)
b. (x = 95) and (y = 105)
c. (x = 85) and (y = 75)
d. (x = 63) and (y = 73)

1. in the diagram, find the values of (x) and (y).

Explanation:

Identify angle relationship in Question 11

Using the Consecutive Interior Angles knowledge point

$$
LATEXBLOCK0
$$

Identify converse theorem in Question 12

The marked angles are located between the two lines \(m\) and \(n\) on opposite sides of the transversal \(t\). These are alternate interior angles. To prove that lines \(m\) and \(n\) are parallel, we use the Alternate Interior Angles Converse Theorem, which states that if two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel.

Analyze Kaitlyn's work in Question 13

Using the Alternate Interior Angles knowledge point

$$
LATEXBLOCK1
$$

The two marked angles are alternate interior angles. Since the lines are parallel, these angles are congruent. Therefore, Kaitlyn's equation \(3x = 120\) is correct, and solving for \(x\) yields \(x = 40\).

Solve for variables in Question 14

The diagram shows two parallel lines \(a\) and \(b\) cut by a transversal \(t\). The angle \(y^\circ\) and the angle marked with the arc of \(180^\circ\) minus the other angle have a relationship. Specifically, the consecutive interior angles are supplementary:

$$
LATEXBLOCK2
$$

Since the question asks to "select all the values that are true", both B and D satisfy the relation \(y = x + 10\).

Solve for variables in Question 15

Using the Alternate Interior Angles knowledge point

$$
LATEXBLOCK3
$$

Answer:

Question 11

• (A) 25 (Correct answer)
• (B) 40
• (C) 65
• (D) 90

Question 12

• (A) Corresponding Angles Converse Theorem
• (B) Alternate Interior Angles Converse Theorem (Correct answer)
• (C) Congruent Angles
• (D) Vertical Angles

Question 13

Yes, Kaitlyn is correct because the two angles are alternate interior angles, which are congruent when lines are parallel.

$$3x = 120 \implies x = 40$$

Question 14

• (B) \(x = 95\) and \(y = 105\) (Correct answer)
• (D) \(x = 63\) and \(y = 73\) (Correct answer)

Question 15

$$x = 50,\quad y = 130$$