Two triangles are said to be congruent if they have same shape and same size. When triangles are congruent corresponding sides (sides in same position) and corresponding angles (angles in same position) are congruent (equal).
There are two theorems and three postulates that are used to identify congruent triangles.
Angle-Angle-Side Theorem (AAS theorem)
As per this theorem the two triangles are congruent if two angles and a side not between these two angles of one triangle are congruent to two corresponding angles and the corresponding side not between the angles of the other triangle.
Example:
Hypotenuse-Leg Theorem (HL theorem)
If the hypotenuse and one of the legs (sides) of a right triangle are congruent to hypotenuse and corresponding leg of the other right triangle, the two triangles are said to be congruent.
Side-Side-Side Postulate (SSS postulate)
If all three sides of a triangle are congruent to corresponding three sides of other triangle then the two triangles are congruent.
Angle-Side-Angle Postulate (ASA postulate)
According to this postulate the two triangles are said to be congruent if two angles and the side between these two angles of one triangle are congruent to corresponding angles and the included side (side between two angles) of the other triangle.
Side-Angle-Side Postulate (SAS postulate)
If two sides and the included angle (angle between these two sides) of one triangle are congruent to the corresponding two sides and the included angle of a second triangle, then the two triangles are congruent.
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In today’s geometry lesson, we’re going to learn two more triangle congruency postulates.
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
The Angle-Side-Angle and Angle-Angle-Side postulates.
These postulates (sometimes referred to as theorems) are know as ASA and AAS respectively.
Here we go!
Triangle Congruence Postulates
Proving two triangles are congruent means we must show three corresponding parts to be equal.
From our previous lesson, we learned how to prove triangle congruence using the postulates Side-Angle-Side (SAS) and Side-Side-Side (SSS). Now it’s time to look at triangles that have greater angle congruence.
Angle-Side-Angle
The Angle-Side-Angle Postulate (ASA) states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
And as seen in the figure to the right, we prove that triangle ABC is congruent to triangle DEF by the Angle-Side-Angle Postulate.
ASA Postulate Example
Angle-Angle-Side
Whereas the Angle-Angle-Side Postulate (AAS) tells us that if two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the two triangles are congruent.
And as seen in the accompanying image, we show that triangle ABD is congruent to triangle CBD by the Angle-Angle-Side Postulate.
AAS Postulate Example
As you will quickly see, these postulates are easy enough to identify and use, and most importantly there is a pattern to all of our congruency postulates.
Can you can spot the similarity?
Yep, you guessed it. Every single congruency postulate has at least one side length known!
And this means that AAA is not a congruency postulate for triangles. Likewise, SSA, which spells a “bad word,” is also not an acceptable congruency postulate.
We will explore both of these ideas within the video below, but it’s helpful to point out the common theme.
You must have at least one corresponding side, and you can’t spell anything offensive!
Knowing these four postulates, as Wyzant nicely states, and being able to apply them in the correct situations will help us tremendously throughout our study of geometry, especially with writing proofs.
So together we will determine whether two triangles are congruent and begin to write two-column proofs using the ever famous CPCTC: Corresponding Parts of Congruent Triangles are Congruent.
Triangle Congruency – Lesson & Examples (Video)
38 min
- Introduction ASA and AAS postulates
- 00:00:24 – What are Angle-Side-Angle and Angle-Angle-Side postulates?
- 00:13:17 – If possible, write a congruency statement using ASA, AAS, SSS, or SAS (Examples #1-6)
- Exclusive Content for Member’s Only
- 00:28:41 – If possible, write a congruency statement using AAS, ASA, SAS, or SSS (Examples #7-10)
- 00:40:18 – Complete the two column proof (Examples #11-13)
- Practice Problems with Step-by-Step Solutions
- Chapter Tests with Video Solutions
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