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.

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 Angles Theorem

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.

## Congruent Complements Theorem

If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent.

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.

## Congruent Supplements Theorem

If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent.

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.

## Right Angles Theorem

All right angles are congruent.

All right angles measure 90º

## Alternate Exterior Angles Theorem

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.

The alternate exterior angles have the same degree measures because the lines are parallel to each other.

## Alternate Interior Angles Theorem

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.

The alternate interior angles have the same degree measures because the lines are parallel to each other.

## Corresponding Angles Postulate

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.

## Vertical Angles Theorem

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.

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

AC intersects BD at O
 Proof Statment Reason 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.

### Solved Examples

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

Solution:

 Proof Statement Reason 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.

Solution:

 Proof Statment Reason 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

## Congruent Angles Examples

### Solved Examples

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

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
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.

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

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