Question

line m passes through points x and y. line n passes through points x and z. if m and n have equal slope, what can you conclude about points x, y, and z? explain
a. y and z are the same point. since a line is defined by a point and a slope, the fact that lines m and n have the same slope and both pass through x means that line m (which can be written as \overleftrightarrow{xy}) and line n (which can be written as \overleftrightarrow{xz}) must be the same line or \overleftrightarrow{xy} \cong \overleftrightarrow{xz}. by the segment addition postulate, x + y = x + z. therefore, by the addition property of equality, y = z.

b. y and z are on the opposite sides of x. since a line is defined by a point and a slope, the fact that lines m and n have the same slope and both pass through x means that line m (which can be written as \overleftrightarrow{yx}) and line n (which can be written as \overleftrightarrow{xz}) must be the same line. note that \overleftrightarrow{yx} is on \overleftrightarrow{yx} and \overleftrightarrow{xz} is on \overleftrightarrow{xz}. therefore, by the segment addition postulate, \overleftrightarrow{yx} + \overleftrightarrow{xz} = \overleftrightarrow{yxz}

c. points x, y, and z are the same point. since a line is defined by a point and a slope, the fact that lines m and n have the same slope and both pass through x means that lines m and n must be the same line or \overleftrightarrow{xy} \cong \overleftrightarrow{xz} \cong \overleftrightarrow{yz}. by using the multiplication property of equality on \overleftrightarrow{xy} \cong \overleftrightarrow{xz}, it can be shown that y = z. similar reasoning shows that x = y. by applying the transitive property of equality, x = y = z

d. points x, y, and z are collinear. since a line is defined by a point and a slope, the fact that lines m and n have the same slope and both pass through x means that lines m and n must be the same line. as a result, z lies on line m.

Explanation:

To solve this, we analyze the properties of lines and collinearity:

Step 1: Recall the definition of a line

A line is uniquely determined by its slope and a point it passes through (or by two points). If two lines have the same slope and share a common point, they are identical (the same line).

Step 2: Analyze the given lines

• Line \( m \) passes through \( X \) and \( Y \).
• Line \( n \) passes through \( X \) and \( Z \).
• Lines \( m \) and \( n \) have equal slopes.

Step 3: Conclude collinearity

Since \( m \) and \( n \) have the same slope and both pass through \( X \), they are the same line (a line is uniquely defined by its slope and a point). Thus, \( Y \) (on \( m \)) and \( Z \) (on \( n \)) must lie on this common line. Therefore, points \( X \), \( Y \), and \( Z \) are collinear (lie on the same line).

Now, evaluating the options (focusing on the correct logical flow):

• Option D states: "Points \( X \), \( Y \), and \( Z \) are collinear. Since a line is defined by a point and a slope, the fact that lines \( m \) and \( n \) have the same slope and both pass through \( X \) means that lines \( m \) and \( n \) must be the same line. As a result, \( Z \) lies on line \( m \)." This matches our reasoning.

Answer:

D. Points X, Y, and Z are collinear. Since a line is defined by a point and a slope, the fact that lines m and n have the same slope and both pass through X means that lines m and n must be the same line. As a result, Z lies on line m.