The document discusses parallel and perpendicular lines, determining if pairs of lines are parallel or perpendicular based on their slopes, and working through geometry problems involving lines and triangles. It provides examples of finding slopes of lines from their equations in slope-intercept form and using slopes to determine if lines are parallel or if their slopes are negative reciprocals, indicating perpendicular lines. Several practice problems are included involving classifying lines, identifying types of quadrilaterals, writing equations of lines, and solving geometry problems about triangles and parallelograms.

1. Mathematics III: Geometry Pineda, KWP
PERPENDICULAR AND PARALLEL LINES
Parallel Lines
If two nonvertical lines are parallel,
then their slopes are equal.
m1 = m2
Perpendicular Lines
If two nonvertical lines are
perpendicular, then their slopes are
negative reciprocals.
m1 x m2 = -1
EXERCISE: Determine which pairs of lines are parallel and perpendicular.
L1 : 5x – 2y = 10
L2 : 2x + 5y = 7
L3 : 10x – 4y = 5
Solution:
Change each equation into slope-intercept form to identify the slopes of each equation.
L1 : 5x – 2y = 10
-2y = -5x + 10 (Transpose -5x to the right side)
- (Divide each term by -2)
m1 =
4th Quarter – Plane Coordinate Geometry Copyright 2013

2. Mathematics III: Geometry Pineda, KWP
L2 : 2x + 5y = 7
5y = -2x + 7 (Transpose 2x to the right side)
-
(Divide each term by 5)
m2 =
L3 : 10x – 4y = 5
-4y = -10x + 5 (Transpose 10x to the right side)
(Divide each term by 4)
Since m1 and m3 are equal, therefore L1 and L3 are parallel lines.
Since m1 and m2, m1 and m3 are negative reciprocals, therefore L1 and L2 , and L1 and L3 are perpendicular lines.
4th Quarter – Plane Coordinate Geometry Copyright 2013

3. Mathematics III: Geometry Pineda, KWP
TRY THE FOLLOWING:
A. Determine if the following pair of lines are parallel, perpendicular, or neither.
__________________________ 1. y = 5x + 2
y = 5x + 7
__________________________ 2. -
-
__________________________ 3. y = 3x – 4
-
__________________________ 4. y = 3x + 1
3y = - x – 7
__________________________ 5. 2x – y = 7
x + 2y = 3
__________________________ 6. 3x – y = 6
x – 3y = 4
__________________________ 7. y–7=x
x+y=y
__________________________ 8. x+y=7
2x + 2y =1
B. Determine which of the following quadrilaterals with the given vertices are parallelograms. Write parallelogram on the
blank, otherwise, identify what kind of quadrilateral is formed.
__________________________ 1. A(0 , 0), B(6 , 4), C(11 , 3) and D(5 , -1)
__________________________ 2. E(-4 , -1), F(-3 , 3), G(5 , 7) and H(4 , 2)
__________________________ 3. J(3 , 4), K(5 , -5), L(-4 , -14) and M(-6 , -5)
__________________________ 4. P(-4 , 5), Q(4 , 6), R(8 , 1) and S(-2, -4)
4th Quarter – Plane Coordinate Geometry Copyright 2013

4. Mathematics III: Geometry Pineda, KWP
C. Write an equation of a line in standard form of the line passing through the given point and perpendicular to the given
line:
1. (-2 , 1) ; y = 4x + 1
2. (-1 , -2) ; 3x – 2y = 6
3. (0 , 3) ; y = -2x – 3
4. (4 , 3) ; 5x – 3y = -15
D. Write an equation of a line in standard form of the line passing through the given point and parallel to the given line:
1. (2 , -1) ; y = 4x + 1
2. (-1 , -2) ; 3x – 2y = 6
3. (-3 , 0) ; y = -2x – 3
4. (3 , -4) ; 5x – 3y = -15
E. Answer each problem below and box your final answer.
1. Show that the triangle with vertices P(2 , 2),
Q(1 , 3) and R(7 , 7) is a right triangle.
2. Show that the figure with vertices A(2 , 6), B(1 ,
3), C(2 , 1), and D(3 , 4) is a parallelogram.
3. Find the equation of a line containing the
altitude to KJ of ∆KJH with K(-2 , 1), J(-4, -5)
and H(-3 , 4).
4. The line containing points (0 , 2) and (-3 , 0) is
parallel to the line through (4 , C) and (0 , -1).
What is the value of c?
BIBLIOGRAPHY:
Jose Dilao, S. & Bernabe, J. (2009). Geometry: Textbook for Third Year. SD Publications Inc., Araneta Avenue,
Quezon City.
Mercado, et. al. (2003). Next Century Mathematics Third Year High School: Geometry. Phoenix Publishing House Inc.,
Quezon Avenue, Quezon City.
Oronce, O. & Mendoza, M. (2003). Exploring Mathematics III: Geometry. Sampaloc, Manila.
4th Quarter – Plane Coordinate Geometry Copyright 2013