What does congruence mean in mathematics?
In maths, two figures are congruent if they have the same shape and size, even though their position or orientation is different. The coincident parts of the congruent figures are called homologous or corresponding.
How to know if two numbers are congruent?
Two integers a and b are said to be congruent with respect to a natural number (positive integer) called congruent modulo “m” if the divisions a/m and b/m give the same remainder.
It’s written like this: a ≡ b mod (m).
What are congruent angles?
Two angles are said to be congruent if their corresponding sides and angles are of equal measure. Two angles are also congruent if they coincide when superimposed. That is, if by turning it and/or moving it, they coincide with each other. The diagonals of a parallelogram also set up congruent vertex angles, or even pairs of angles.
Congruence criteria for triangles
In many cases, it’s enough to establish the equality between three corresponding parts and use one of the criteria to deduce the congruence of two triangles. The congruence criteria correspond to the postulates and theorems that state which are the minimum conditions that two or more triangles must meet to be congruent. These are:
- Congruence of their sides
- Congruence of their angles
For two triangles to be congruent, it’s enough that only some sides and/or angles are equal.
SSS criterion
If you have two triangles and the three sides of one are respectively congruent with those of the other, then the triangles are congruent. SSS stands for: side, side, side.
SAS criterion
Two triangles are congruent if two of their sides and the angle between them are respectively equal. SAS stands for: side, angle, side.
ASA criterion
If two angles and the side between them are respectively congruent with the same of another triangle, then the triangles are congruent. ASA stands for: angle, side, angle.
SSA criterion
Two triangles are congruent if they have two sides respectively congruent and the angles opposite the greater of the sides are also congruent. SSA stands for: side, side, angle.
What are adjacent angles?
An angle is a geometric figure that is formed with two rays that share the same vertex as an origin. Adjacent is an adjective that describes what is located next to something.
Adjacent angles share a side and the vertex, while the other two sides are opposite rays. This definition allows you to infer that adjacent angles are also contiguous or consecutive angles (because they have a common side and the same vertex) and supplementary angles (the sum of both results in 180°, which is known as a straight angle) .
Other properties of adjacent angles is that their cosines have the same value, although they have inverse signs. For example, if you take two adjacent angles, one of 120° and the other of 60°, the cosine of the first is equal to that of the second multiplied by -1. The sine of these angles, on the other hand, is the same.
What type of angles are congruent?
Vertical angles
When two lines intersect, 4 angles are formed. Those that are opposite each other are vertical angles, these are always congruent.
Alternate angles
A line that intersects two parallel lines forms congruent angles. In the following figure, you will see that the following pairs are congruent (interior angles to exterior angles):
Isosceles triangles
In this type of triangle, two of its internal angles are congruent.
Equilateral triangles
In this type of triangle, its three internal angles are congruent and always measure 60⁰.
Squares and rectangles
The four internal angles are congruent and measure 90⁰.
How to find congruent angles?
You can determine the congruence of triangles by using their angles together with the congruence of their sides. Two triangles will be equal if any of the following conditions are true:
- If the three sides of two triangles A and B are equal, then they are congruent.
- If in two triangles A and B two of their sides and the angle formed by them are equal, then A and B are congruent.
- If in two triangles A and B two of their angles are congruent and the side between them are equal, then A and B are congruent.
How to construct a congruent angle?
It can be very useful to know how to construct congruent angles, whether you’re drawing architectural diagrams or doing some trigonometry homework. To do this, you must first draw an angle with a ruler and then use a compass to copy the angle.
The important thing is to make sure they both maintain the same angle, as this is the definition of congruence. Follow these steps:
- Mark a point with your pencil to designate the vertex of the first angle. You can name it “A”.
- Set a straight edge ruler so that it starts at A and extends to the right.
- Draw a straight line from A to one of the inch marks on the ruler, depending on how long you want to make it. You can mark the end of the line as “B”.
- Now, rotate the rule. Make sure the origin remains at vertex “A”.
- Next, draw a straight line from “A” to one of the inch marks on the ruler, depending on how long you want it. You can mark the chosen end as “C”.
Now, let’s see how to construct the congruent angles:
- Mark a point with your pencil to designate the vertex of the second angle. You can mark it as “D”.
- Lay out a ruler so that it starts at “D” and extends to the right.
- Draw a straight line from “D” to one of the marks on the ruler, depending on the length you want. Mark the chosen end as “E”.
- Place the origin of your compass on the vertex (“A”) of the original angle (the first). Set the compass so that the end of the pencil is somewhere between “A” and “B.” Mark this point as “L.”
- Draw an arc from the point on “AB” to the point on the angle “AC”. You can mark the point above “AC” as “M”.
- Take the compass without changing the aperture, place it with its origin at point “D” and the end of the pencil at the point of the angle “ED” it reaches. You can designate this point as “H”.
- Draw an arc from “H” that looks like an arc that goes from “L” to “M”.
- Take the compass again, set its origin to “M” and set it so that the end is at “L”.
- Now, take the compass, set its origin at “H”, and rotate the pencil over the arc extending from “H” until it touches the arc. Define this point as “G”.
- Finally, position the edge of the ruler so that it starts at “G” and extends toward “D.” Draw a line from “G” to “D”.
Need help with congruence of angles?
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Frequently asked questions
How do I get the measure of angles?
In general, the best tool for measuring an angle is a protactor.
What is an angle bisector?
In geometry, the bisector of an angle—also called the bisector of an angle—is a ray that divides an angle into two equal parts.
Is there another way to draw congruent angles?
Another way of drawing congruent angles is to draw a right angle or a right-angled triangle. Two 45° congruent angles (acute angles) are obtained by cutting the right angle (90-degree angle) with an angle bisector.
What is a congruent geometric figure?
Two geometric figures are said to be congruent when they have the same shape and size, or even the same angle measure in the case of angles. For example, there are congruent triangles.
Are right angles congruent?
Yes, all right angles are congruent. Two angles are congruent if they are complements of the same angle.
Are congruent angles always 45 degrees?
Congruent angles are not always 45 degrees, but they can be if a 45-degree angle is congruent to another angle of the same measure.
Are congruent angles 180 degrees?
Congruent angles are not supplementary angles. However two angles can be congruent and supplementary if they are right angles.
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