What do you call a congruence between two triangles with corresponding two angles and an included side?

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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There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.

1. SSS   (side, side, side)

SSS stands for "side, side, side" and means that we have two triangles with all three sides equal.

For example:

is congruent to:

(See Solving SSS Triangles to find out more)

If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

2. SAS   (side, angle, side)

SAS stands for "side, angle, side" and means that we have two triangles where we know two sides and the included angle are equal.

For example:

is congruent to:

(See Solving SAS Triangles to find out more)

If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent.

3. ASA   (angle, side, angle)

ASA stands for "angle, side, angle" and means that we have two triangles where we know two angles and the included side are equal.

For example:

is congruent to:

(See Solving ASA Triangles to find out more)

If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

4. AAS   (angle, angle, side)

AAS stands for "angle, angle, side" and means that we have two triangles where we know two angles and the non-included side are equal.

For example:

is congruent to:

(See Solving AAS Triangles to find out more)

If two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

5. HL   (hypotenuse, leg)

This one applies only to right angled-triangles!

or

HL stands for "Hypotenuse, Leg" (the longest side of a right-angled triangle is called the "hypotenuse", the other two sides are called "legs")

It means we have two right-angled triangles with

• the same length of hypotenuse and
• the same length for one of the other two legs.

It doesn't matter which leg since the triangles could be rotated.

For example:

is congruent to:

(See Pythagoras' Theorem to find out more)

If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent.

Caution! Don't Use "AAA"

AAA means we are given all three angles of a triangle, but no sides.

This is not enough information to decide if two triangles are congruent!

Because the triangles can have the same angles but be different sizes:

is not congruent to:

Without knowing at least one side, we can't be sure if two triangles are congruent.

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