PARALLELISM AND PERPENDICULARITY

Have you ever wondered how carpenters, architects and engineers design their work? What factors are being considered in making their designs? The use of parallelism and perpendicularity of lines in real life necessitates the establishment of these concepts deductively.

LEARNING GOALS AND TARGET

• The learner demonstrates understanding of the key concepts of parallel and perpendicular lines.
• The learner is able to communicate mathematical thinking with coherence and clarity in solving real-life problems involving parallelism and perpendicularity using appropriate and accurate representations.

PARALLELISM

Two lines are parallel if and only if they lie on the same plane (coplanar) and do not intersect (Figure 1).

Figure 1

Figure 2. m || n and t is a transversal line

From the figure above (figure 2), 

Transversal line - a line that intersects two or more lines at different points

Exterior Angles - angles made by a transversal that are outside the region between the two intersected lines

                    example : ∠1, ∠2, ∠7 and ∠8

Interior Angles - angles made by a transversal that lie inside the region between the two intersected lines

                    example : ∠3, ∠4, ∠5 and ∠6

Alternate-Exterior Angles - a pair of non-adjacent exterior angles on opposite sides of the transversal

                    example : ∠1 and ∠8, ∠2 and ∠7

Alternate-Interior Angles - a pair of non-adjacent interior angles on opposite sides of the transversal

                    example: ∠3 and ∠6, ∠4 and ∠5

Corresponding Angles - a pair of non-adjacent angles - one interior, one exterior - both on the same side of the transversal

                   example : ∠1 and ∠5, ∠3 and ∠7, ∠2 and ∠6, ∠4 and ∠8

Figure 3.

From the figure above (Figure 3)

1. ∠1 and ∠7 - alternate exterior angles
2. ∠3 and ∠5 - alternate interior angles
3. ∠4 and ∠8 - corresponding angles
4. ∠2 and ∠6 - corresponding angles
5. ∠4 and ∠6 - alternate interior angles
6. ∠2 and ∠8 - alternate exterior angles

If two lines are cut by a transversal, then the two lines are parallel if:

• corresponding angles are congruent.
• alternate-interior angles are congruent.
• alternate-exterior angles are congruent.
• interior angles on the same side of the transversal are supplementary.
• exterior angles on the same side of the transversal are supplementary.

You may visit the following sites to strengthen your knowledge regarding the different angles formed by parallel lines cut by a transversal line and how they are related with one another;

Properties of Parallel Lines (Part 1 of 2)

Properties of Parallel Lines (Part 2 of 2)

To enhance your knowledge, you may try to answer the question in the following link:

http://www.regentsprep.org/Regents/math/geometry/GP8/PracParallel.htm

 

Proving Lines are Parallel

The following theorems will lay down conditions which guarantee parallelism of two lines

• If 2 coplanar lines are both perpendicular to the same line, then they are parallel.
  
• CAP Theorem. If 2 lines have transversal and a pair of congruent Corresponding Angles, then the    lines are parallel.
  
• AIP Theorem. If 2 lines have a transversal and a pair of Alternate Interior Angles, then the lines are parallel.
  
• AEP Theorem. If 2 lines have a transversal and a pair of congruent Alternate Exterior angles, then the lines are parallel.
  
• If 2 lines have a transversal and interior angles on the same side of the transversal are supplementary, then the lines are parallel.
  
• If 2 lines are parallel to a third line, then they are parallel to each other. If a // b, and b //c, then a //c.
  

Criteria of parallelism of straight line and plane:

1)  If a straight line, lying out of plane, is parallel to some straight line, lying in the  

     plane, then it is parallel to this plane.

2)  If a straight line and a plane are perpendicular to the same straight line, then they  

     are parallel.

PERPENDICULARITY

Two lines that intersect to form right angles are said to be perpendicular. This is not  limited to lines only. Segments and rays can also be perpendicular. A perpendicular bisector of a segment is a line or a ray or another segment that is perpendicular to the segment and intersects the segment at its midpoint. The distance between two parallel lines is the perpendicular distance between one of the lines and any point on the other line.
Two lines are Perpendicular if they meet at a right angle (90°).

        

                                              

Line PY is a perpendicular bisector and XZ ⊥ PY

The distance between the two right angles is the perpendicular distance between the two parallel lines

To prove that two lines are perpendicular, one of the following theorems must be true:

• If two lines are perpendicular to each other, then they form four right angles.
  

If m ⊥ n, then we can conclude that ∠1, ∠2, ∠3

and ∠4 are right angles.

• If the angles in a linear pair are congruent, then the lines containing their sides are perpendicular.
  

If ∠1 and ∠2 form a linear pair and ∠1 ≅ ∠2, then l1 ⊥ l2.

• If two angles are adjacent and complementary, the non-common sides are perpendicular.
  

If ∠1 and ∠2 form a linear pair and ∠1 ≅ ∠2, then l1 ⊥ l2.

∠CAR and ∠EAR are complementary and adjacent, then AC ⊥ AE.

Proving Theorems on Perpendicular Lines

1. If two lines are perpendicular, then they form four right angles.
2. If the angles in a linear pair are congruent, then the lines containing their sides are perpendicular.
3. In a plane, through a point on a given line there is one and only one line perpendicular to the given line.
4. In a plane, a segment has a unique perpendicular bisector.
5. If two angles are adjacent and complementary, the non-common sides are perpendicular.

 

The following video will explain how to construct a perpendicular line to a point and a perpendicular line through a point not on a line.

Constructing Perpendicular Lines

Criteria of perpendicularity of straight line and plane:

1)  If a straight line is perpendicular to two interesting straight lines, lying in a plane,  

     then it is perpendicular to this plane.

2)  If a plane is perpendicular to one of parallel straight lines, then it is perpendicular  

     to the other.

Theorem about three perpendiculars. A straight line, that lies in a plane and is perpendicular to a projection of straight line inclined to this plane, is perpendicular to this straight line inclined to plane.

Theorem about common perpendicular to two crossing straight lines.  Any two crossing straight lines have the only common perpendicular.

 KINDS OF QUADRILATERALS

Quadrilateral is a polygon with four sides. Quadrilaterals are classified as follows:

1. Trapezium – a quadrilateral with no pair of parallel sides.
   

   

   Trapezium

   
   

   
   
2. Trapezoid – a quadrilateral with exactly one pair of parallel sides. If the non-parallel sides are congruent, the trapezoid is said to be isosceles.
   

   

   Trapezoid

   
   

   
   
3. Parallelogram – a quadrilateral with two pairs of parallel and congruent sides. There are two special kinds of parallelogram: the rectangle which has four right angles and the rhombus which has four congruent sides. A square which has four congruent angles and four congruent sides can be a rectangle or a rhombus because it satisfies the definition for a rectangle and a rhombus.
   

   

   Parallelogram

 

Conditions to Prove that a Quadrilateral is a Parallelogram

Given a parallelogram, related definition and theorems are stated as follows:

1. A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
2. If a quadrilateral is a parallelogram, then 2 pairs of opposite sides are congruent.
3. If a quadrilateral is a parallelogram, then 2 pairs of opposite angles are congruent.
4. If a quadrilateral is a parallelogram, then the consecutive angles are supplementary.
5. If a quadrilateral is a parallelogram, then the diagonals bisect each other.
6. If a quadrilateral is a parallelogram, then the diagonals form two congruent triangles.

 To prove a parallelogram, related definition and theorems are stated as follows:

(Many of these theorems are converses of the previous theorems.)

1. A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
2. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
3. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
4. If one angle is supplementary to both consecutive angles in a quadrilateral, then the quadrilateral is a parallelogram.
5. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a  parallelogram.
6. If one pair of opposite sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a parallelogram.

To have more understanding of the properties of a parallelogram, you may visit the following link:

https://www.youtube.com/watch?feature=player_detailpage&v=0rNjGNI1Uzo