Angles are congruent if they have the same angle measure. Two angles are congruent if they have the same measure in degrees.

Its not necessary for the angles to have same direction or equal segment lengths. But if the congruent angles are laid on top of each other they coincide with each other.

Congruent Angles

Geometric figures are classified based on the interior angles:

For Example:
Triangles: if all the interior angles of a triangle are congruent them it is called as an equilateral triangle.

Quadrilateral: If all the interior angles of a quadrilateral are congruent then it can be a square or a rectangle depending on the length of its sides.

Polygon: A polygon with all congruent interior angles is called as a regular polygon.

Here are some examples of congruent interior angles.
Congruent Interior Angles
Congruent Interior Angles

The following list gives some useful properties of the congruence of angles.

Reflexive: $\angle$ 1 = $\angle$1; an angle is congruent to itself

Symmetric: If $\angle$1 = $\angle$2 then $\angle$ 2 = $\angle$1.

Transitive: If $\angle$1 = $\angle$2 and $\angle$ 2 = $\angle$3 then $\angle$1 = $\angle$ 3.
If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent.

Congruent Complements Theorem

Given $\angle$1 and $\angle$3 are are both complementary to the same angle $\angle$ 2, prove that they are congruent:

Proof:

In the above diagram, let

$\angle$ABE = 90º and  $\angle$CBD =90º

then, $\angle$1 + $\angle$2 =  90º…………..(1)

$\angle$2 + $\angle$ 3 = 90º…………..( 2)

Equate equations (1) and (2)

$\angle$1 + $\angle$ 2 =  $\angle$2 + $\angle$3

             subtract $\angle$2 from both sides

             $\angle$1 = $\angle$3

            Since the angles $\angle$1 = $\angle$3 are complements of the same angle
$\angle$ 2, they are congruent to each other.
If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent.

Congruent Supplements Theorem

Given  $\angle$1 and $\angle$ 3 are both supplementary to the same angle , prove that they are congruent.

Proof:
In the above diagram , let
$\angle$ABC =$\angle$1,

$\angle$CBD= $\angle$2,  $\angle$DBE=$\angle$3

  $\angle$1 +$\angle$2 =180 º…(given )………………(1)

 $\angle$2 +$\angle$3 =180º……(given)………………(2)

Equate equations (1) and (2)

$\angle$1 +$\angle$2 =$\angle$2 +$\angle$3

Subtract $\angle$2 from both sides

$\angle$ 1 = $\angle$3

So $\angle$ 1 and $\angle$ 3   are congruent to each other.
All right angles are congruent.

Right Angles Theorem
 
All right angles measure 90º
When a transversal intersects two parallel lines, then the alternate exterior angles are congruent.
Converse: If two lines are intersected by a transversal and the alternate exterior angles are congruent, then the lines are parallel.
 
Alternate Exterior Angles Theorem

The alternate exterior angles have the same degree measures because the lines are parallel to each other.
If a transversal intersects two parallel lines, then the alternate interior angles are congruent.
Converse: If two lines are intersected by a transversal and the alternate interior angles are congruent, then the lines are parallel.

Alternate Interior Angles Theorem
 
The alternate interior angles have the same degree measures because the lines are parallel to each other.
If a transversal intersects two parallel lines, the pairs of corresponding angles are congruent.

Converse also true: If two lines are intersected by a transversal and the corresponding angles are congruent, then the lines are parallel.
When two lines intersect each other, the vertical angles are formed. Opposite angles are known as vertical angles, because the angles share the same vertex The vertical angles have equal measures.

Vertical Angles Theorem
 
Here,  
make the pair of vertical angles. They are congruent.
Proof:

Vertical Angles Theorem

AC intersects BD at O
Proof
StatmentReason
1.AC intersects BD at O

2. $\angle$AOC and $\angle$DOB are straight angles, with $\angle$AOC = 180$^o$ and $\angle$DOB = 180

3. $\angle$AOC = $\angle$DOB

4. $\angle$1 + $\angle$4 = $\angle$DOB and $\angle$AOC

$\angle$1 +$\angle$ 2 = $\angle$AOC.

5. $\angle$1+$\angle$4=$\angle$1+$\angle$2

6. $\angle$4 = $\angle$2

7.m$\angle$4 = m$\angle$2

8. $\angle$2 =$\angle$4
1.Given

2. The measure of a straight angle is 180°

3. Substitution

4. Angle-Addition Postulate

5. Substitution

6. Subtraction Property of Equality

7. If two angles are equal in measure, the angles are congruent

8. Symmetric Property of Congruence of Angles

$\angle$2 = $\angle$4
Therefore,
m$\angle$2 = m$\angle$4
Similarly, we can also prove that $\angle$1 = $\angle$3
Hence the vertical angles are congruent.

Proving Angles Congruent

Solved Examples

Question 1: Given  $\angle$1 = $\angle$3,, prove that  $\angle$ RSU = $\angle$TSV

Proving Angles Congruent
Solution:
 

Proof
StatementReason
1.$\angle$1 =$\angle$3,

2. $\angle$1 + $\angle$2 = $\angle$2 + $\angle$ 3

2. $\angle$1 + $\angle$2 = $\angle$RSU and $\angle$2 + $\angle$ 3 = $\angle$TSV

3. $\angle$RSU = $\angle$TSV
1.Given

2.Add $\angle$2 to both sides.

2. Angle-Addition Postulate

3. Substitution


 

Question 2: In the given diagram, $\Delta$ABC is a right triangle with $\angle$C = 90º.
Prove that $\angle$1 and $\angle$4 are complementary to each other.

Angles Congruent
Solution:
 

Proof
StatmentReason
1. $\angle$1 = $\angle$2 and $\angle$ 3 = $\angle$ 4

2. $\angle$ 2 + $\angle$3 + $\angle$C = 180º

3. $\angle$ 2 + $\angle$3 + 90º= 180º

4.$\angle$ 2 + $\angle$3 = 90º

5. $\angle$1 + $\angle$4 = 90º

6. $\angle$1 and $\angle$4 are complementary
1.Vertical angles are congruent.

2. The sum of the angles in the triangles is 180.

3. $\angle$C = 90º given.

4. Subtracting 90º from both sides.

5. Substitution

6. If the sum of the 
measures of two angles is 90, then the angles are complementary

 


Solved Examples

Question 1: Solve for “x” from the following diagram.

Congruent Angles Examples
Solution:
 
Suppose the angles are $\angle$1 and $\angle$2 .
As these are the vertical angles, they are congruent.
So, m $\angle$1 = m $\angle$2

2x - 5 = 105
Add 5 on both sides
2x = 110
divide both sides by 2

$\frac{2x}{2}$ = $\frac{110}{2}$

x=55º
 

Question 2: Find the values of $\angle$1 and $\angle$7, given $\angle$1 = 67º if lines m and n are parallel to each other.

Congruent Angle Example
Solution:
 
The lines m || n ……..(given)
  So $\angle$1 and $\angle$7 make a pair of alternate exterior angles.
By applying rule of congruency of external alternate angles,

$\angle$1 = $\angle$7
 m $\angle$7 = 67º

Angles $\angle$1 and $\angle$3 make a pair of vertical angles.
So, $\angle$3 =$\angle$1 = 67º.

m$\angle$3 = 67º
 

Question 3: Find the values of x and y from the given diagram , given in $\angle$ ABC $\angle$C = 90º and $\angle$1 =32º

Congruent Angles Problem
Solution:
 
Given, $\angle$C = 90º, $\angle$1 = 32º

 $\angle$1 = $\angle$x……….pair of vertical angles
Therefore, $\angle$x = 32º

In the triangle $\Delta$ABC,
$\angle$A + $\angle$B = 90º …………..sum of other two angles in the right triangle is 90º.
Therefore, $\angle$ x + $\angle$ 2  = 90º
 32º + $\angle$2 = 90º

$\angle$2 = 90º- 32º = 58º

At vertex B, $\angle$2 = $\angle$y …………pair of vertical angles.
Therefore, $\angle$y = 58º