Standardized Test Practice: Transparency 3-6 5-Minute Check on Lesson 3-5 Given the following information, determine which segments, if any, are parallel. State the postulate or theorem that justifies your answer. 1. 9  13 2. 2  5 3. m2 + m4 = 180 4. 5  14 5. Refer to the figure above. Find x so that AB || CD if m1 = 4x + 6 and m5 = 7x – 27. 6. If ℓ, m and p are coplanar lines such that ℓ  p and p  m, which statement is valid? Standardized Test Practice: A m || ℓ B m  ℓ C p || ℓ D ℓ and m intersect

Standardized Test Practice: Transparency 3-6 5-Minute Check on Lesson 3-5 Given the following information, determine which segments, if any, are parallel. State the postulate or theorem that justifies your answer. 1. 9  13 2. 2  5 FG || HI; if corresp. none s are , lines are || 3. m2 + m4 = 180 AB || CD; if consecutive interior s are supplementary, lines are ||. 4. 5  14 CD || HI; if alternate interior s are , lines are ||. 5. Refer to the figure above. Find x so that AB || CD if m1 = 4x + 6 and m14 = 7x – 27. x = 11 6. If ℓ, m and p are coplanar lines such that ℓ  p and p  m, which statement is valid? Standardized Test Practice: A m || ℓ B m  ℓ C p || ℓ D ℓ and m intersect

Objectives Find the distance between a point and a line Find the distance between parallel lines

Vocabulary Equidistant – has the same distance (parallel lines are equidistant everywhere) Locus – the set of all points that satisfy a given condition (parallel lines can be described as the locus of points in a plane equidistant from a given line)

Key Concepts The (shortest) distance between a line and a point not on the line is the length of the segment perpendicular to the line from the point The distance between two parallel lines is the distance between one of the lines and any point on the other line Theorem 3.9: In a plane, if two lines are equidistant from a third line, then the two lines are parallel to each other.

Draw the segment that represents the distance from Answer: Since the distance from a line to a point not on the line is the length of the segment perpendicular to the line from the point,

Draw the segment that represents the distance from Answer:

Construct a line perpendicular to line s through V(1, 5) not on s Construct a line perpendicular to line s through V(1, 5) not on s. Then find the distance from V to s.

Graph line s and point V. Place the compass point at point V Graph line s and point V. Place the compass point at point V. Make the setting wide enough so that when an arc is drawn, it intersects s in two places. Label these points of intersection A and B.

Put the compass at point A and draw an arc below line s Put the compass at point A and draw an arc below line s. (Hint: Any compass setting greater than will work.)

Using the same compass setting, put the compass at point B and draw an arc to intersect the one drawn in step 2. Label the point of intersection Q.

Draw . and s. Use the slopes of and s to verify that the lines are perpendicular.

The segment constructed from point V(1, 5) perpendicular to the line s, appears to intersect line s at R(–2, 2). Use the Distance Formula to find the distance between point V and line s. Answer: The distance between V and s is about 4.24 units.

Construct a line perpendicular to line m through Q(–4, –1) not on m Construct a line perpendicular to line m through Q(–4, –1) not on m. Then find the distance from Q to m. Answer:

Find the distance between the parallel lines a and b whose equations are y = 2x + 3 and y = 2x – 3 respectively. You will need to solve a system of equations to find the endpoints of a segment that is perpendicular to both a and b. The slope of lines a and b is 2.

First, write an equation of a line p perpendicular to a and b First, write an equation of a line p perpendicular to a and b. The slope of p is the opposite reciprocal of 2, Use the y-intercept of line a, (0, 3), as one of the endpoints of the perpendicular segment. Point-slope form Simplify. Add 3 to each side.

Next, use a system of equations to determine the point of intersection of line b and p. Substitute 2x–3 for y in the second equation.

Group like terms on each side. Simplify on each side. Substitute 2.4 for x in the equation for p. The point of intersection is (2.4, 1.8).

Then, use the Distance Formula to determine the distance between (0, 3) and (2.4, 1.8). Answer: The distance between the lines is or about 2.7 units.

Find the distance between the parallel lines a and b whose equations are and respectively. Answer:

Summary & Homework Summary: Homework: Distance between a point and a line is measured by the perpendicular segment from the point to the line. Homework: pg 163: 21, 25, 27 Chapter 3 Review: Practice Test on page 171