Congruent Angles - Definition, Theorem, Examples, Construction

Congruent Angles

Congruent angles are the angles that have equal measure. So all the angles that have equal measure will be called congruent angles. They are seen everywhere, for example, in equilateral triangles, isosceles triangles, or when a transversal intersects two parallel lines. Let us learn more about the congruence of angles along with their construction in this article.

1.	What are Congruent Angles?
2.	Congruent Angles Theorem
3.	Constructing Congruent Angles
4.	FAQs on Congruent Angles

What are Congruent Angles?

In mathematics, the definition of congruent angles is "angles that are equal in the measure are known as congruent angles". In other words, equal angles are congruent angles. It is denoted by the symbol "≅", so if we want to represent ∠A is congruent to ∠X, we will write it as ∠A ≅ ∠X. Look at a congruent angles example given below.

In the above image, both the angles are equal in measurement (60∘ each). They can completely overlap each other. So, as per the definition, we can say that both the given angles are congruent angles.

Congruent Angles Theorem

There are many theorems based on congruent angles. Using the congruent angles theorem we can easily find out whether two angles are congruent or not. Those theorems are listed below:

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

Let's understand each of the theorems in detail along with its proof.

Vertical Angles Theorem

According to the vertical angles theorem, vertical angles are always congruent. Let us check the proof of it.

Statement: Vertical angles are congruent.

Proof: The proof is simple and is based on straight angles. We already know that angles on a straight line add up to 180°.

So in the above figure:

Statement	Reason
∠1+∠2 = 180°	Linear Pair
∠1+∠4 = 180°	Linear Pair
∴ ∠1+∠2 = 180∘ = ∠1+∠4	By equating the above two equations
∴ ∠1+∠2 =∠1+∠4	Quantities equal to the same quantity are equal to each other. (Transitive: if a=b and b=c that implies a=c)
∴ ∠2 =∠4	If equals are subtracted from equals, the differences are equal. (By eliminating ∠1 on both sides)
Also, ∠1=∠3	Similarly, we can prove for ∠1 and ∠3

Conclusion: Vertically opposite angles are always congruent angles.

Corresponding Angles Theorem

The corresponding angles definition tells us that when two parallel lines are intersected by a third one, the angles that occupy the same relative position at each intersection are known to be corresponding angles to each other.

When a transversal intersects two parallel lines, corresponding angles are always congruent to each other. In this figure, ∠1 = ∠2. It's a postulate so we do not need to prove this. It is always stated as true without proof.

Alternate Angles Theorem

When a transversal intersects two parallel lines, each pair of alternate angles are congruent.

Refer to the figure above. We have:

∠1 = ∠5 (corresponding angles)

∠3 = ∠5 (vertically opposite angles)

Thus, ∠1 = ∠3

Similarly, we can prove the other three pairs of alternate congruent angles too.

Congruent Supplements Theorem

Supplementary angles are those whose sum is 180°. This theorem states that angles supplement to the same angle are congruent angles, whether they are adjacent angles or not.

We can prove this theorem by using the linear pair property of angles, as,

∠1+∠2 = 180° (Linear pair of angles)

∠2+∠3 = 180° (Linear pair of angles)

From the above two equations, we get ∠1 = ∠3.

∴ Angles supplement to the same angle are congruent angles.

Congruent Complements Theorem

Complementary angles are those whose sum is 90°. This theorem states that angles that complement the same angle are congruent angles, whether they are adjacent angles or not. Let us understand it with the help of the image given below.

We can easily prove this theorem as both the angles formed are right angles.

∠a+∠b = 90° (∵∠a and ∠b form 90° angle)

∠a+∠c = 90° (∵∠a and ∠c form 90° angle)

So, from the above two equations, we get, ∠b ≅ ∠c.

∴ Two angles complementary to the same angle are congruent angles.

Constructing Congruent Angles

In this section, we will learn how to construct two congruent angles in geometry. There are two cases that come up while learning about the construction of congruent angles, and they are:

• Construction of two congruent angles with any measurement.
• Construction of a congruent angle to the given angle.

Construction of Two Congruent Angles

Let's learn the construction of two congruent angles step-wise.

Step 1- Draw two horizontal lines of any suitable length with the help of a pencil and a ruler or a straightedge.

Step 2- Take any arc on your compass, less than the length of the lines drawn in the first step, and keep the compass tip at the endpoint of the line. Draw the arc keeping the lines AB and PQ as the base without changing the width of the compass.

Step 3 - Keep the compass tip on point D and expand the legs of the compass to draw an arc of any suitable length. Draw that arc and repeat the same process with the same arc by keeping the compass tip on point S.

Step 4- Draw lines that will join AC and PR.

This is how we get two congruent angles in geometry, ∠CAB, and ∠RPQ.

Construction of a Congruent Angle to the Given Angle

By now, you have learned about how to construct two congruent angles in geometry with any measurement. But what if any one angle is given and we have to construct an angle congruent to that? Let's learn it step-wise.

Suppose an angle ∠ABC is given to us and we have to create a congruent angle to ∠ABC.

Step 1 - Draw a horizontal line of any suitable measurement and name it YZ.

Step 2 - Keep compass tip at point B in the given angle and draw an arc by keeping BC as the base and name that point D.

Step 3 - With the same width, draw an arc by keeping the compass tip at point Y and name the point at line YZ as O.

Step 4 - Keep compass tip at point D and measure the arc from point D to the point of intersection of the arc at segment AB.

Step 5 - With the same arc, keep your compass tip at point O and mark a cut at the arc drawn in step 3, and name that point as X.

Step 6 - Draw a line and join points X and Y.

Here, we get ∠ABC ≅ ∠XYZ, which satisfies the definition of the congruent angle. This is how we can construct an angle congruent to the given angle.

Congruent Angles Tips and Tricks:

• Congruent angles are just another name for equal angles.
• All vertically opposite angles are congruent angles.
• All alternate angles and corresponding angles formed by the intersection of two parallel lines and a transversal are congruent angles.
• According to the definition of congruent angles "For any two angles to be congruent, they need to be of the same measurement."

► Related Topics

Check these interesting articles related to congruent angles definition.

• Transitive Property of Congruence
• Consecutive Angles
• Consecutive Interior Angles
• Congruence in Triangles