Congruent Angles: Theorem, Rules & Symbol

Congruent angles are two or more angles that are similar to each other. Congruent means one figure that is identical to another in terms of shape and size, meaning if one puts one figure on the other, it will be superimposed exactly.

• These figures could be either polygon, angle, line segment, or 3D shape. 
• However, the angles measured are equal to each other. for example, isosceles triangles and equilateral triangles. 
• The type of angles does not make any difference in the congruence of angles, which means they can be interior, exterior, acute, obtuse, or angles.

Key Terms: Congruent, Angles, Vertical, Corresponding, Alternate, Supplement, Complements

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What are Congruent Angles?

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In mathematics terms, the congruent angle is an angle that is equal in the measure known as congruent angles. Or one can say that equal angles are congruent angles and the "≅"symbol denotes it, so if one wants to represent ∠P as congruent to ∠X, one will write it as ∠P ≅ ∠X. 

Look at the congruent angles example given below.

Congruent Angles

In the figure given above, ∠A ≅ ∠B is read as “The angle A is congruent to the angle B”.

• In the above diagram, both the angles are equal in measurement i.e. 45∘ each. 
• They can completely overlap each other. 
• However, as per the definition, both the given angles are congruent angles.

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Detailed Solved Examples

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Example 1. Find the measurement of angle f.

Solution: Here, ∠AOC and ∠DOE are congruent angles. 

So, ∠AOC = ∠DOE.

113∘ = 90º+f

f = 113°-90°

f = 23°

Therefore, the f is 23 degrees.

Example 2: In the given below example, two lines are parallel to each other and are intersected by a transversal. What will be the measure of ∠y and ∠x?

Solution: In the above diagram lines are parallel and according to the congruent alternate angles theorem, the given angle of measure is 85∘ and ∠x are alternate congruent angles. 

So, 85∘ = ∠x.Similarly, 

95∘ and y are congruent alternate angles. 

So, 95∘ = ∠y.

Therefore, the angle of ∠x is 85°, and ∠y is 95°.

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Congruent Angles Symbol

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The symbol of congruence in geometry term is represented by the “≅“. In the above example, ∠A and ∠B have the same measure, then they are said to be equal or congruent. Therefore ∠A is congruent to ∠B and ∠A = ∠B or ∠X ≅ ∠Y.

Read More: Difference Between Square and Rhombus

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Congruent Angles Theorem

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There are many theorems that are based on congruent angles and through which we can easily find out whether two angles are congruent to each other or not. These theorems are shown in below bullets points:

• Vertical angles theorem
• Corresponding angles theorem
• Alternate angles theorem
• Congruent supplements theorem
• Congruent complements theorem

Vertical Angles Theorem

As per the theorem of vertical angles, vertical angles are always congruent. Look at the proof of it.

Statement: Vertical angles are congruent.

Proof: The proof is based on straight angles. The angle on a straight line is 180°.

Vertical Angles

So in the above figure:

Statement Reason

∠1+∠2 = 180° Linear Pair

∠1 +∠4 = 180° Linear Pair

∴ ∠3+∠4 = 180°ˆ˜ = ∠3+∠2

By equating the above two equations

∴ ∠2+∠4 = ∠3+∠1

Conclusion: Hence, according to that vertically opposite angles are always congruent angles.

Corresponding Anglеs Thеorеm

In thе contеxt of parallеl linеs intеrsеctеd by a transvеrsal, corresponding anglеs arе еqual in mеasurе. Corresponding angles arе located on thе samе sidе of thе transvеrsal, with onе anglе insidе thе parallеl linеs and thе оthеr outsidе, sharing thе samе relative position.

Altеrnatе Anglеs Thеorеm

Whеn parallel linеs arе intersected by a transvеrsal, altеrnatе intеrior anglеs arе congruеnt. Thеsе anglеs arе situatеd on opposite sidеs of thе transvеrsal and arе locatеd bеtwееn thе two parallel linеs, sharing thе samе rеlativе position.

Altеrnatе Anglеs Thеorеm

Here,

∠4= ∠6 (alt. angles)

∠5= ∠3 (alt. angles)

Congruent Supplеmеnts Theorem

If two anglеs arе supplеmеnts of thе samе anglе (or congruеnt anglеs), thеy havе thе samе measure and add up to 180 dеgrееs. This theorem is applicablе when two anglеs together form a straight linе.

Congruent Complеmеnts Theorem

Whеn two angles arе complements of thе samе anglе (or congruеnt anglеs), thеy havе equal measures and add up to 90 degrees. This theorem is rеlеvant whеn two anglеs togеthеr form a right anglе. 

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Congruent Angles Rules

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The Below points highlight the rules of congruence angles:

• The measures of angle measures are the same.
• The length and the direction of the two triangles making up these congruent angles are irrelevant.

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Congruent Angles Construction

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Make an angle congruent to a given angle with the help of a ruler and a compass; however, this is just to copy the angle construction.

• However, the easiest way to construct congruent angles is to draw two parallel lines cut by a transversal
• The corresponding angles are congruent in this case.

Congruent Angles

In the above figure, l is the transversal made by two parallel lines a and b. Below are the pairs of corresponding angles that are congruent. Thus, the pair of congruent angles are:

∠1 and ∠7

∠4 and ∠6

∠2 and ∠8

∠3 and ∠5

Read More: Faces, Edges, and Vertices

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Congruent Angles Examples

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Below are some examples of congruent angles.

The above figure highlights congruent angles of measure of 60 degrees even though they have unequal arms.

In the above example, the direction of terminals by which the angles made are different, yet they are said to be congruent.

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How to find Congruent Angles?

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Look into the below figure to understand how to find congruent angles.

Here, ∠OCA and ∠ODB have an equal measures of angles, that's why they are called congruent.

Read More:

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Things to Remember

• Congruent angles are another name for equal angles.
• All vertically opposite angles are known as congruent angles.
• All corresponding angles and alternate angles formed by the intersection of two parallel lines and a transversal are also called congruent angles.
• As per the definition of congruent angles "For any two angles to be congruent, they need to be of the same measurement."
• The length and the direction of the two triangles making up these congruent angles are irrelevant.
• If two anglеs arе supplеmеnts of thе samе anglе (or congruеnt anglеs), thеy havе thе samе measure and add up to 180 dеgrееs. Whеn two angles arе complements of thе samе anglе (or congruеnt anglеs), they have equal measures and add up to 90 degrees.

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