Many angle relationships involve exactly a pair of angles. These relationships can be classified under the following categories:
1. Relationships based on the Measurement of Angles
2. Relationships based on the Position of Angles

Relationships based on the measurement of angles


On the basis of the measurement of the angles, the angles can be categorized into the following two categories:

Complementary Angles: Those angles for which the sum of the measures is $90^o$. That is, if $\angle$A + $\angle$B = $90^o$, each angle is the complement of the other.
Complementary Angles

Supplementary Angles: Those angles for which the sum of the measures is $180^o$. That is, if $\angle$A + $\angle$B = $180^o$, then each angle is the supplement of the other.Supplementary Angles

Congruent Angles: A pair of angles are said to be congruent angles if they have the same measure.Congruent Angles

Notation: If two angles $\angle$A and $\angle$B are congruent, we denote it by $\angle$A $\cong$ $\angle$B. It is customary to denote angles by small letters of the Greek Alphabet. That is, we use the Greek letters $\alpha$, $\beta$, $\gamma$ and so on to denote angles. Thus, if $\alpha$ and $\beta$ are congruent angles, we write $$\alpha \cong \beta$$

Relationships based on the Position of Angles


On the basis of the position of the angles, every pair of angles can be categorized into the following categories:

Adjacent Angles: Adjacent angles are a pair of angles which share a common vertex and side, but have no common interior points.Adjacent Angles.

Complementary angles and supplementary angles can be adjacent angles or non-adjacent angles.
Vertical Angles: Vertical angles are two non-adjacent angles formed by two intersecting lines.

In the figure, $\angle$1 and $\angle$2 are a pair of vertical angles. Another pair of vertical angles is $\angle$3 and $\angle$4.

Linear Pair of Angles: A linear pair is a pair of adjacent angles whose non-common sides are opposite rays.
Linear pair of Angles

In the figure,
$\angle$1 and $\angle$2 are vertical angles.
$\angle$3 and $\angle$4 are vertical angles.
$\angle$1 and $\angle$3 is a linear pair of angles.
$\angle$3 and $\angle$2 is a linear pair of angles.
$\angle$2 and $\angle$4 is a linear pair of angles.
$\angle$4 and $\angle$1 is a linear pair of angles.

Vertical angles are always congruent.
Thus, we have $\angle$1 $\cong$ $\angle$2 and $\angle$3 $\cong$ $\angle$4

Linear Pair Postulate: If two angles form a linear pair, then the angles are supplementary.

From the above diagram, it is clear that

m $\angle$1 + m $\angle$3 = $180^o$, m $\angle$2 + m$\angle$3 = $180^o$, m $\angle$2 + m $\angle$4 = $180^o$, and m $\angle$ 1 + m $\angle$4 = $180^o$

On the basis of the above discussion, we can find out the angles in a given diagram if the angles are expressed with unknowns.

Solved Examples

Question 1: Given that $\angle$A and $\angle$B are complementary angles, whose measures are given by

m $\angle$A = $(\frac{x}{2})^o$ and m $\angle$B = $(\frac{x}{3})^o$

Find the value of x and the given angles.
Solution:
 
Given that $\angle$A and $\angle$B are complementary angles.  Thus, we have
                 m $\angle$A + m $\angle$B = $90^o$

            $\rightarrow$ $(\frac{x}{2})^o$ + $(\frac{x}{3})^o$ = $90^o$

            $\rightarrow$ $\frac{x}{2}$ + $\frac{x}{3}$ = 90

            $\rightarrow$ $\frac{3x + 2x}{6}$ = 90

            $\rightarrow$ $\frac{5x}{6}$ = 90

            $\rightarrow$ x = 90 × $\frac{6}{5}$

            $\rightarrow$ x = 108
Thus,
    m $\angle$A = $(\frac{x}{2})^o$ = $(\frac{108}{2})^o$ = $54^o$ and m $\angle$B = $(\frac{x}{3})^o$ = $(\frac{108}{3})^o$ = $36^o$
 

Question 2: In the figure,
Solving Angle Relationships
        m $\angle$1 = (2x + 2y)$^o$
        m $\angle$2 = (4x - 2y)$^o$
        m $\angle$3 = (2x - y)$^o$

Find x and y.
Solution:
 
In the figure, $\angle$1 and $\angle$2 are vertical angles, and hence they are congruence.  Thus, we have
                 2x + 2y = 4x - 2y
            $\rightarrow$ 2x - 4y = 0
            $\rightarrow$ x - 2y = 0          (1)

Again, $\angle$1 and $\angle$3 are supplementary angles. Thus, we have
                  m $\angle$1 + m $\angle$3 = $180^o$
            $\rightarrow$(2x + 2y) + (2x - y) = 180
            $\rightarrow$4x + y = 180          (2)

From (1) and (2), we have
                 (x - 2y) + 2(4x + y) = 0 + 2(180)
            $\rightarrow$x - 2y + 8x + 2y = 360
            $\rightarrow$9x = 360
            $\rightarrow$x = $\frac{360}{9}$
            $\rightarrow$x = 40

From (2),
                 4x + y = 180
            $\rightarrow$4 (40) + y = 180
            $\rightarrow$160 + y = 180
            $\rightarrow$y = 180 - 160
            $\rightarrow$y = 20
 

Theorem 1 (Right Angles Congruence Theorem)

All right angles are congruent.
Given: $\angle$1 and $\angle$2 are right angles.
Proving Angle Relationships
To Prove: $\angle 1 \cong \angle 2$
Proof:
S. No.StatementReason
1∠1 and ∠2 are right angles.Given
2m ∠1 = 90oDefinition of Right Angle
3m ∠2 = 90oDefinition of Right Angle
490o = m ∠2(3) and by Symmetric Property of Equality
5m ∠1 = m ∠2(2), (4), and by Transitive Property of Equality
6∠1≅∠2Definition of Congruent Angles


Theorem 2 (Congruent Supplements Theorem)

If two angles are supplementary to the same angle or to congruent angles, the angles are congruent.

Proving Angle Relationships
Proof:
Given: $\angle$1 and $\angle$3 are supplementary angles to $\angle$2.
To Prove: $\angle$1 $\cong$ $\angle$3

Proof:
S. No.StatementReason
1∠1 and ∠2 are supplementary anglesGiven
2m ∠1 + m ∠2 = 180o(1), Definition of Supplementary Angels
3∠2 and ∠3 are supplementary anglesGiven
4m ∠3 + m ∠2=180o(3), Definition of Supplementary Angles
5180o = m ∠3 + m ∠2(4), Symmetric Property of Equality
6m ∠1 + m ∠2 = m∠3+m ∠2(2), (5), Transitive Property of Equality
7m ∠1 = m ∠3(6), Subtraction Property of Equality
8∠1≅∠3Definition of Congruent Angles


Theorem 3 (Congruent Complements Theorem)

If two angles are complement to the same angle or to congruent angles, the two angles are congruent.

Proving Angle Relationships

Given: $\angle$1 and $\angle$3 are complement to $\angle$2.
To Prove: $\angle$1 $\cong$ $\angle$3

Proof:
S. No. StatementReason
1∠1 and ∠2 are complementaryGiven
2m ∠1 + m ∠2 = 90oDefinition of complementary angles
3∠2 and ∠3 are complementaryGiven
4m ∠2 + m ∠3 = 90o Definition of complementary angles
590o = m ∠2 + m ∠3(4), Symmetric Property of Equality
6m ∠1 + m ∠2 = m ∠2 + m ∠3(2), (5), Transitive Property of Equality
7m ∠1 = m ∠3(6), Subtraction Property of Equality
8∠1≅∠3Definition of Congruent Angles


Theorem 4 (Vertical Angles Congruence Theorem)

Vertical Angles are equal in measure.

Construction: Let $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ be two intersecting lines in a plane, intersecting at the point P, as shown in the figure. Here, the angles $\angle$APD and $\angle$BPC are vertical angles.

Vertical Angles Congruence Theorem

To Prove:
$\angle$APD $\cong$ $\angle$BPC

S. No.StatementReason
1∠APD and ∠APC are a linear pair of anglesGiven
2m ∠APD+m ∠APC = 180o(1), Linear Pair Postulate
3∠BPC and ∠APC are a linear pair of anglesGiven
4m ∠BPC + m ∠APC = 180o(3), Linear Pair Postulate
5180o = m ∠BPC + m ∠APC(4), Symmetric Property of Equality
6m ∠APD + m ∠APC = m∠BPC + m ∠APC(2), (5), Transitive Property of Equality
7m ∠APC = m ∠BPC(6), Subtraction Property of Equality
8∠APC≅∠BPCDefinition of Congruent Angles

The interior and exterior angles in a triangle are related to one another. The following is the description of angle relationships in Triangles.

Theorem 5 (Triangle Sum Theorem)

The sum of the measures of the interior angles of a triangle is equal to 180o.

Given: Let $\Delta$ABC be a triangle.
Construction: Draw an auxiliary line $\overleftrightarrow{DE}$ which passes through B and parallel to $\overline{AC}$, as shown in the figure.

Triangle Sum Theorem

To Prove: m $\angle$1+ m $\angle$2 + m $\angle$3 = 180o

Proof:
S. No.StatementReason
1∠1≅∠4Alternate Interior Angles Theorem
2m ∠1= m ∠4(1), Definition of Congruent Angles
3∠3≅∠5Alternate Interior Angles Theorem
4m ∠3=m ∠5(3), Definition of Congruent Angles
5m ∠1+m ∠3=m ∠4+m ∠5(2), (4), Addition of Equalities
6m ∠4+m ∠2+m ∠5 = 180oAngle Addition Postulate and Definition of Straight Angle
7m ∠2+m ∠1+m ∠3=180o(5), (6), Substitution
8m ∠1+m ∠2+m ∠3=180o(7), Arrangement

Corollary 1: The acute angles of a right triangle are complementary.

Given: $\Delta$ABC be a right triangle, right-angled at B.

To Prove: $\angle$A and $\angle$C are complementary
Corollary
Proof
S. No.StatementReason
1∠B is a right angleGiven
2m ∠B= 90oDefinition of Right Angle
3m ∠A+m ∠B+m ∠C=180o Triangle Sum Theorem
4m ∠A+90o+m ∠C=180o(2), (3), Substitution
5m ∠A+m ∠C=90o(4), Substitution Property of Equality
6∠A and ∠C are complementaryDefinition of Complementary Angles

Similar to the previous corollary we can prove the following two statements:
1. The measure of each angle of an equiangular triangle is 60o.
2. The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles.

Vertex of Angle
Measure of Angle
Diagram
On the CircleMeasure of Angle = Half of the
measure of intercepted arc

m$\angle$ = $\frac{1}{2}$m$\angle$ACB
Angle Relationships in Circles