Grade 7 → Geometry → Congruence of Triangles ↓
Criteria for Congruence
In geometry, it is important to understand the concept of congruence, especially when dealing with triangles. Congruent triangles are triangles that are the same in shape and size, meaning that all corresponding sides and angles are equal. To determine if two triangles are congruent, we can use specific criteria also known as congruence rules.
What is congruence?
Congruence in geometry means that two figures or objects are the same in shape and size. When it comes to triangles, congruence means that all the sides and angles of one triangle are exactly equal to the corresponding sides and angles of another triangle.
Importance of congruent triangles
Similar triangles have important applications in various fields including engineering, architecture, and even art. They are used to ensure uniformity and stability. For example, when building structures, similar triangles can help create stable and balanced designs.
Criteria for congruence in triangles
There are several criteria to establish the congruence of triangles. These criteria are based on the equality of sides, angles, or a combination of both. Here are the main criteria:
1. Side-side-side (SSS) criterion
According to the SSS criterion if three sides of a triangle are respectively equal to three sides of another triangle, then those triangles are congruent.
Example:
If in triangle ABC and triangle DEF:
AB = DE, BC = EF, AC = DF
Then, according to the SSS criterion, triangle ABC is equivalent to triangle DEF.
2. Side-angle-side (SAS) criterion
According to the SAS criterion, if two sides and the angle between these sides of a triangle are respectively equal to two sides and the angle between these sides of another triangle, then the triangles are congruent.
Example:
Consider triangle PQR and triangle XYZ:
PQ = XY, ∠PQR = ∠XYZ, QR = YZ
Using the SAS criterion, triangle PQR is equilateral to triangle XYZ.
3. Angle-side-angle (ASA) criterion
The ASA criterion stipulates that if two angles of a triangle and the side between these angles are equal to two angles and the corresponding sides of another triangle, then those triangles are congruent.
Example:
For triangle ABC and triangle DEF, if:
∠BAC = ∠EDF, AC = DF, ∠ACB = ∠DFE
So according to the ASA criterion, triangle ABC is equivalent to triangle DEF.
4. Angle-angle-side (AAS) criterion
The AAS criterion tells us that if two angles and a non-conjunct side of a triangle are equal to the corresponding two angles and a side of another triangle, then the two triangles are congruent.
Example:
Consider triangle LMN and triangle OPQ where:
∠LMN = ∠OPQ, ∠MNL = ∠PQR, MN = PQ
According to the AAS criterion, triangle LMN is equivalent to triangle OPQ.
5. Right angle-hypotenuse-side (RHS) criterion
The RHS criterion is specific to right triangles. It states that if the hypotenuse and one side of a right triangle are equal to the hypotenuse and one side of another right triangle, then the triangles are congruent.
Example:
For right-angled triangle ABC and right-angled triangle DEF:
AC (hypotenuse) = DF, AB = DE,
The triangles are congruent by RHS criterion.
Understanding compatible parts
When we talk about corresponding parts of similar triangles, we mean parts (sides or angles) that match in position and measure in both triangles. The phrase "corresponding parts of similar triangles are equal" is often abbreviated as CPCTE.
Example:
In congruent triangles ABC and DEF:
AB = DE, BC = EF, Ca = Fd, ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
In both triangles the sides and angles in the same position are equal.
How to prove that triangles are congruent?
To prove that two triangles are congruent, you must show that all corresponding aspects satisfy one of the criteria mentioned above. Based on the given information, you can use SSS, SAS, ASA, AAS, or RHS to work out the congruence.
Common mistakes
It is important to ensure full alignment with a criterion. A common mistake is to mix criteria or assume conformity through incomplete data.
Incorrect example:
If you know two angles and a side from different locations in two triangles, you cannot imagine the congruence unless it satisfies one of the specific criteria like ASA or AAS.
Practice problems
Now, let's try some practice scenarios to determine whether triangles are congruent:
- Triangles GHI and JKL are given:
GH = JK, HI = KL, ig = lj,
What criterion can be used to prove that they are identical? - Consider the triangles MON and PQR, is there enough information to claim that they are congruent if:
MO = PQ, ∠MON = ∠PQR, ∠OMN = ∠QRP
- If in triangles ABC and XYZ:
AC = XZ, AB = XY, ∠CAB = ∠ZXY
Are these triangles congruent?
Conclusion
Understanding the criteria for congruence of triangles is foundational not just in geometry but also in a wide range of mathematical concepts and applications. Using the SSS, SAS, ASA, AAS, and RHS criteria, you can confidently determine whether two triangles are similar in shape and size, opening up avenues for deeper mathematical understanding and real-world applications.