Congruent Angles

A 101 Guide on Congruent Angles

Congruence means compatibility. But what does congruence mean in mathematics?In mathematics, we can say that two figures are congruent if they have the same size and shape, even though their orientation or position is different. The coincident parts of the congruent figures are called homologous parts.

An Introduction to Congruent Angles

In a nutshell, congruent angles are the angles that have equal measure. So, every angle that is equal in measurement will be named congruent angles. They are seen everywhere, for example, in an isosceles triangle, equilateral triangle, or when a transversal intersects two parallel lines. 

How to know if two numbers are congruent?

What are congruent angles?

Angles having same measurement are called congruent angle.

If two angles are congruent, it is denoted by the symbol"≅". So, if you want to represent

Let us look at more examples of congruent angles below:

In the above image, both the angles measure 60 degrees. Therefore, they can entirely overall each other. Thus, according to the definition of congruent angles, we can say that both the given angles are congruent

Congruent Angles Theorem

Vertical Angles

According to the vertical angle theorem, vertical angles are always congruent. Here is proof for you to check through the vertical angle theorem. 

Statement: Vertical angles are congruent.

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

So, in the above diagram:

• <1+<2=180 degrees as they are a linear pair.<1+<4=180 degrees as they are a linear pair as well.
• Therefore, <1+<2=180 degrees = <1+<4 by equating the above the two equations
• Therefore, <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)
• Therefore, <2=<4. If equals are subtracted from equals, the differences are equal. You can do this 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. Hence proven.

Alternate Interior Angle Theorem

The alternate interior angles theorem states that when a transversal cuts 2 parallel lines, the resultant angles are congruent.

In the following figure, if k||l, then <2 ≅ <8 and <3 ≅ <5.

And now we will prove that <2 ≅ <8.

Proof:

By the corresponding angles postulate, <1 ≅ <5 since k||l

Therefore, m<1=m<5 by the definition of congruent angles.

Now, m<1+m<2=180 degrees since <1 and <2 form a linear pair, they are supplementary.

Also, <5 and <8 are supplementary, therefore, m<5+m<8=180 degrees.

We were substituting m<1 for m<5, we get m<1+m<8=180 degrees.

On subtracting m<1 from both sides, we have m<8=180 degrees – m<1= m<2

Therefore, <2≅<8

This way, you can prove that <3≅<5 using the same process.

Congruent Supplement Theorem

Supplementary angles are those whose sum is 180 degrees. Thus, the theorem states that angles supplementing the same angle are congruent angles. 

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

• <1+<2=180 degrees (linear pair of angles)
• <2+<3=180 degrees (a linear couple of angles).
• From the above two equations, we get <1=<3.
• Therefore, supplementary angles to the same angle are congruent angles.
  
  

  Constructing Congruent Angles

  Follow these steps to construct a congruent angle:

  • Mark a point to designate the vertex of the first angle. You can name the angle "A".
  • Set a ruler that starts from point A and then extends to the right side.
  • Draw a straight line from A to any point on the ruler. You can mark the end of the line as "B".
  • Now, rotate the scale. Make sure the origin is at vertex A.
  • Then, draw a straight line from A on any inch mark of the ruler (you can make it of any length), and mark the chosen end as C.

  Now, let us see how to construct congruent angles:

  1. Mark a point to designate the vertex of the second angle and name the angle "D".
  2. Lay out the ruler and make sure it starts from point D and extends to the right.
  3. Draw a straight line from D to an inch on the ruler, and mark the chosen end as "E".
  4. Place the compass' origin on point A of the first angle. Set the compass so that the end of the pencil is somewhere between A and B. And mark this point as "L".
  5. Draw an arc from the point AB to the angle AC. You can then mark the end above AC as "M".
  6. Keep the aperture of the compass the same and in that way, place the compass at point D. Then place the end of the pencil at point E. You can designate this point as "H"/ 
  7. Next, draw an arc from H and make sure it looks like an arc from L to M.
  8. Retake the compass, set its origin to M and then set it so that the end is at L.
  9. Then, set the origin of the compass at H and rotate the pencil over the arc extending from H until it reaches the arc. Define this point as G.
  10. • Ultimately, position the ruler's edge to start at G and extend towards D. Draw a line from G to D.
        
        

        Congruent angles, tips and tricks

        Here are the tips to understand whether angles are congruent:

        • Vertical angles are congruent.
        • If two angles are complements of congruent angles, the two angles are congruent.
        • If a transversal cuts parallel lines, the corresponding angles are congruent.
        • If an angle is bisected, it divides it into two congruent angles.
        • If two angles in one triangle are congruent to two angles in another triangle, the third angles are congruent.

        If two lines are perpendicular, they form congruent adjacent angles.
        
        

        Congruent Angles Examples

        Question 1:

         

        Angles 1 and 2 are corresponding. For example, if the measure of angle 2 is 67 degrees, what is the measure of Angle 1?

        Answer: When a transversal cuts two parallel lines, the angles on the same side of the transversal and in matching corners will be congruent. Angles 1 and 2 are congruent, so both have an angle measure of 67 degrees. 

        Question 2:

         

        Line R is a transversal that crosses through the two parallel lines s and t. List all of the angles that are congruent to angle 6.

        • <8 is congruent to <6 because they are at vertical or opposite angles. 
        • <2 is congruent to <6 because they are corresponding angles (same side of the transversal and in matching corners).
        • <4 is congruent to <6 because they are alternate interior angles (alternate sides of the transversal and between the two parallel lines).

        Question 3:

         

        If the following irregular quadrilaterals are congruent, angle C must be congruent to what other angle?

        Corresponding angles are congruent for congruent polygons.

      • In conclusion,

      So, now you clearly understand what a congruent angle is, its theorems, the tips to understand when an angle is called a congruent angle, and their example. Congruent angles are frequently used in construction, art, architecture, and design. So, have an in-depth understanding of congruent angles and have a shining career in architecture. 

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  Most Popular Questions Searched By Students:

  Q1.What does Congruent Angles mean?

  Ans. Congruent angles are those angles having equal angles.
  

  Q2.What are the Conditions Required for Congruent Angles?

  Ans.Two angles can be called congruent angles when two pairs of corresponding sides and corresponding angles between them are equal.

  Q3.What are Congruent Angles in Parallel Lines?

  Ans.If two parallel lines are cut by a transversal, the alternate interior angles are congruent.