There are 4 criteria to determine that two triangles are congruent. In this article, we will learn to use the third criterion of congruence:
There are 4 criteria to determine that two triangles are congruent. In this article, we will learn to use the third criterion of congruence:
Definition: 2 triangles in which their three sides are of the same length are congruent triangles.
In this article, we will study this criterion and see examples of how to apply it.
Given the parallelogram ABCD
What can be said about triangles ACD and ABD?
Two triangles in which all three sides are of the same length are congruent triangles.
To prove that two triangles are congruent we can use one of the following postulates:
Given the triangles and such that
(edge)
(edge)
(edge)
Therefore, we can deduce that: and are congruent triangles according to the Side, Side, Side congruence criterion.
We will write it as follows:
according to the congruence criterion: Side, Side, Side (SSS)
From this we can also deduce that:
since these are corresponding angles and are equal in congruent triangles
Given the parallelogram ABCD
What can be said about triangles ACD and ABD?
Given the two triangles and such that is the common side.
We are also informed that:
Prove that the triangles and are congruent triangles.
Proof:
We will base our proof on the criterion we just learned.
Let's see
(side)
(side)
We realize that (side) is common to both triangles
From this, it follows that in both triangles and there are three pairs of equal sides.
Consequently, we can deduce that
according to the Side, Side, Side congruence criterion.
If you're interested in this article, you might also be interested in the following articles:
Congruence Criterion: Side, Angle, Side
Congruence Criterion: Angle, Side, Angle
Side, Side, and the Angle Opposite the Larger of the Two Sides
The Method of Writing a Formal Proof in Geometry
On the Tutorela blog, you'll find a variety of articles on mathematics.
Assignment
In the given figure:
By what theorem are the triangles congruent?
Solution
Since
Since
Common side
The triangles are congruent by
Answer
Congruent by
Given the parallelogram ABCD
What can be said about triangles ACD and ABD?
Assignment
In an isosceles triangle we draw the height .
According to which theorem of congruence do the triangles overlap?
Solution
Since triangle is isosceles
In an isosceles triangle, the height is also a median, and a median cuts the base into two equal parts.
Common side
The triangles overlap according to
Answer
Overlap according to
Assignment
The segments and intersect at point .
Which congruence theorem explains why the triangles are congruent?
Solution
and
Intersect at point
intersects
Vertically Opposite Angles
Overlapping triangles by
Answer
Overlapping by
Given the parallelogram ABCD
What can be said about triangles ACD and ABD?
Assignment
The triangles
In triangle we draw the median
and in triangle we draw the median .
We demonstrate:
Solution
Given that triangles and are congruent
In congruent triangles, the medians are necessarily equal
(coming from the same vertex to the same base)
The median bisects the base it reaches.
Congruent triangles by
Answer
Overlaid according to
Assignment
Given the isosceles trapezoid .
Inside it contains the square .
According to which theorem are the triangles congruent?
Solution
is an isosceles trapezoid (given)
Isosceles trapezoid
Since is a square
Since is a square and all sides in a square are equal
The base angles in an isosceles trapezoid are equal
In a square, all angles are right angles and measure degrees
if two angles are equal then the third is also equal
The triangles are congruent according to
Answer
Given the parallelogram ABCD
What can be said about triangles ACD and ABD?
What is the congruence criterion for two triangles?
There are four triangle congruence criteria, which allow us to determine if two triangles have the same dimensions in their sides and likewise the same length in their corresponding angles. In this way, we can say that the two triangles, even when they are in different positions or orientations, will have the same shape and size.
What is the SSS congruence criterion?
This criterion allows us to deduce if two triangles have the same shape and size. According to this criterion, two triangles are congruent when their three sides are equal.
What is the difference between the SSS congruence criterion and the SSS similarity criterion?
The SSS congruence criterion tells us that if two triangles have their three sides equal (congruent sides), then the two triangles are identical, meaning they have the same measurements in terms of sides and angles. Whereas the SSS similarity criterion tells us that if two triangles are similar, then their three sides are proportional, meaning they do not have the same measurement but they do have some proportion between them and they have the same shape, but with different measurements in terms of their sides.
Which pair of triangles are similar by the SSS criterion?
Two triangles will be similar when they have the same shape, regardless of orientation, that is, their corresponding angles are equal but their corresponding sides do not necessarily have the same length, instead, they must have a proportion between them.
What are the criteria for similarity and congruence of triangles?
Congruence criteria
The four triangle congruence criteria are:
Similarity criteria
Unlike the congruence criteria, there are only three triangle similarity criteria:
Given the parallelogram ABCD
What can be said about triangles ACD and ABD?