Congruence In Lines And Angles

KAPDEC® | Elite STEM Learning Platform | https://kapdec.com

---

Unit: Congruency

Chapter: Congruence in Lines and Angles

Reference: - Basic Angle Relationships, Parallel Lines and Transversals, Perpendicular Lines and Angle Congruence, Triangle Angle Properties, Angle Congruence Theorems & Proofs, Applications & Problem-Solving

After studying this chapter, you should be able to understand:

• Geometry: - Lines & Angles
• Basic Angle Relationships & Parallel Lines and Transversals
• Perpendicular Lines and Angle Congruence
• Triangle Angle Properties & Angle Congruence Theorems & Proofs

Geometry: Lines and Angles

Theorems:

Axiom 1- If a ray stands on a line, then the sum of two adjacent angles so formed is 180°.

                      

As we can see that PR is a straight line, thus angle formed on a straight line is 180°.

Therefore, ∠A + ∠B = 180°, thus the sum of adjacent angles is equal to 180°.

Also, when the sum of two adjacent angles is 180°, then they are called a linear pair of angles.

Axiom 2- If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line.

• The two axioms above together are called the Linear Pair Axiom.

Theorems related to Lines and Angles

Theorem 1- If two lines intersect each other, then the vertically opposite angles are equal.

                      

Proof- ∠A & ∠C and ∠B & ∠D are vertically opposite angles.

As we can see, MN is a straight line and ∠A & ∠B are adjacent angles on it,

∠A + ∠B = 180°     .... (a) (Axiom 1)

Similarly, PQ is also a straight line and ∠A & ∠D are adjacent angles on it, so

∠A + ∠D = 180°     .... (b) (Axiom 1)

Equating (a) & (b), we can say that ∠B = ∠D.

Similarly, we can proof this theorem for ∠A & ∠C.

Transversal Line

A line which intersects two or more lines at distinct points is called a transversal line.

                     

Line R intersects lines P and Q at points X and Y respectively. Therefore, line R is a transversal for lines P and Q.

 

 

Here, we can observe that four angles are formed at each of the points X and Y.

Let us name them, ∠1, ∠2, ∠3 ... ∠7 & ∠8.

Nomenclature of angles related to transversal line.

                     

• ∠ 1, ∠ 2, ∠ 7 and ∠ 8 are called exterior angles and ∠ 3, ∠ 4, ∠ 5 and ∠ 6 are called interior angles.
• Corresponding angles-
  • ∠ 1 and ∠ 5
  • ∠ 2 and ∠ 6
  • ∠ 4 and ∠ 8
  • ∠ 3 and ∠ 7
• Alternate interior angles-
  • ∠ 4 and ∠ 5
  • ∠ 3 and ∠ 6
• Alternate exterior angles-
  • ∠ 1 and ∠ 8
  • ∠ 2 and ∠ 7
• Consecutive interior angles-
  • ∠ 4 and ∠ 6
  • ∠ 3 and ∠ 5

Relationship between angles

Axiom 3- If a transversal line intersects two parallel lines, then each pair of corresponding angles is equal.

Therefore, 

• • ∠ 1 = ∠ 5
  • ∠ 2 = ∠ 6
  • ∠ 4 = ∠ 8
  • ∠ 3 = ∠ 7

Also,

Axiom 4- If a transversal line intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other.

Theorem 2- If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.

Proof- As we know ∠QXY = ∠AXP....... (Vertically Opposite angles)

Also,            ∠AXP = ∠XYR..... (Axiom 3, Corresponding angles)

Equating both, we can conclude that,

∠QXY = ∠XYR

               

Similarly, we can prove this for,

∠PXY = ∠XYS

Converse, of Theorem 2 is also true.

Therefore,

Theorem 3- If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines is parallel.

Theorem 4- If a transversal intersects two parallel lines, then the consecutive interior angles are supplementary.

So, ∠QXY + ∠XYS = 180°     and ∠PXY + ∠XYR = 180°

Converse, of Theorem 4 is also true, so,

 Theorem 5- If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines is parallel.                             

Lines Parallel to Same Lines

        

Given, Line A∥

B and A∥ C, where line P is the transversal line.

 

So as, A∥ B, ∠1= ∠2..... (Corresponding angles)

Also, A∥ C, ∠1= ∠3..... (Corresponding angles)

Thus, we can say that ∠2 = ∠3

Now as ∠2 = ∠3, we can say that line B ∥  C.

Theorem 6- Lines which are parallel to the same line are parallel to each other.
 

• Congruent Angles – Two angles are congruent if they have the same measure.
• Angle Addition Postulate – If a point lies inside an angle, the sum of the two smaller angles equals the measure of the larger angle.
• Complementary Angles – Two angles are complementary if the sum of their measures is 90°.
• Supplementary Angles – Two angles are supplementary if the sum of their measures is 180°.
• Vertical Angles – When two lines intersect, the opposite (vertical) angles formed are always congruent.
• Parallel Lines and Transversals – When a transversal intersects two parallel lines, special angle relationships are created, including corresponding, alternate interior, alternate exterior, and consecutive interior angles.
• Perpendicular Lines – Two lines are perpendicular if they intersect at a 90° angle.
• Perpendicular Bisector Theorem – A point on the perpendicular bisector of a segment is equidistant from the segment’s endpoints.
• Right Angles Congruence Theorem – All right angles are congruent, meaning they have the same measure of 90°.
• Triangle Sum Theorem – The sum of the interior angles of any triangle is always 180°.
• Exterior Angle Theorem – The measure of an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.
• Congruent Supplements Theorem – If two angles are supplementary to the same angle, then they are congruent.
• Congruent Complements Theorem – If two angles are complementary to the same angle, then they are congruent.
• Geometric Proofs – A logical sequence of statements and reasons used to justify geometric relationships and congruence.