In these lessons, we will learn

- congruent triangles
- how to tell if triangles are congruent using SSS, SAS, ASA and AAS rules
- why AAA and SSA does not work as congruence shortcuts
- the Hypotenuse Leg Rule for right triangles
- how to use CPCTC (corresponding parts of congruent triangles are congruent)

Related Topics: More Geometry Lessons

Congruent triangles are triangles that have the same **size** and **shape**. This means that the
corresponding sides are equal and the corresponding angles are equal.

In the above diagrams, the corresponding sides are *a* and *d*; *b* and *e* ; *c* and *f*.

The corresponding angles are *x*
and *s*; *y*
and *t*; *z*
and *u*.

We can tell whether two triangles are congruent without testing all the sides and all the angles of the two triangles. There are four rules to check for congruent triangles. They are called the SSS rule, SAS rule, ASA rule and AAS rule. There is also another rule for right triangles called the Hypotenuse Leg rule. As long as one of the rules is true, it is sufficient to prove that the two triangles are congruent.

The Side-Side-Side (SSS) rule states that

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

The Side-Angle-Side (SAS) rule states that

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

An included angle is the angle formed by the two given sides.

Included Angle Non-included angle

The Angle-Side-Angle (ASA) Rule states that

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

An included side is the side between the two given angles.

The Angle-Angle-Side (AAS) Rule states that

If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, then the triangles are congruent.(This rule may sometimes be referred to as SAA).

For the ASA rule the given side must be included and for AAS rule the side given must not be included. We must use the same rule for both the triangles that we are comparing.

Compare AAS with AAS

Compare ASA with ASA

Compare AAS with ASA

If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent (Side-Side-Side or SSS).

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent (Side-Angle-Side or SAS).

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent (Angle-Side-Angle or ASA).

**Example:**

Which of the following conditions would be sufficient for the above triangles to be congruent?

a)

a = e, x = u, c = fb)

a = e, y = s, z = tc)

x = u, y = t, z = sd)

a = f, y = t, z = s

**Solution for a):**

Step 1:a = egives the S

x = ugives the A

c = fgives the S

Step 2:Beware!xanduare not the included angles. This is not SAS but ASS which isnotone of the rules. Note that you cannot compare donkeys with triangles!

**Answer:**
*a = e, x = u, c = f *is **not** sufficient for the above triangles to be congruent.

**Solution for b):**

Step 1:

a = egives the S

y = sgives the A

z = tgives the AStep 2:

aandeare non-included sides. Follows the AAS rule.

**Answer**:
*a = e, y = s, z = t *is sufficient show that the above are
congruent triangles.

**Solution for c):**

Step 1:x = ugives the A

y = tgives the A

z = sgives the A

Step 2:AAA isnotone of the rules.

**Answer:**
*x = u, y = t, z = s* is **not** sufficient for the above triangles to be congruent.

**Solution for d):**

Step 1:

a, y, zfollows AAS (non-included side)

f ,t, sfollows the ASA (included side)Step 2: Comparing AAS with ASA is not allowed

Triangles with all three corresponding angles equal may not be congruent. These triangles will have the same shape but not necessarily the same size. They are called similar triangles.

SSA Does not WorkTriangles with two corresponding sides and one non-included angle equal may not be congruent.

SSA Can't Be Used to Prove Triangles are CongruentThis video explains why there isn't an SSA Triangle Congruence Postulate or Theorem.

Now that we know 4 ways to prove two triangles are congruent, lets look at two non-congruence theorems. Students often use these to prove triangles are congruent which is incorrect.

The Hypotenuse-Leg (HL) Rule states that

If the hypotenuse and one leg of a right triangle are equal to the hypotenuse and one leg of another right triangle, then the two right triangles are congruent.

In the right triangles Δ*ABC* and Δ*PQR* , if *AB* = *PR*, *AC = QR *then Δ*ABC* ≡ Δ*RPQ* .

How to use the Hypotenuse - Leg Congruence Theorem

CPCTC stands for * Corresponding Parts of Congruent Triangles are Congruent*.

CPCTC states that

If two or more triangles are proven congruent by: ASA, AAS, SSS, HL, or SAS, then all of their corresponding parts are congruent as well. This can be used to prove various geometrical problems and theorems.

CPCTC Triangle Congruence

The following video shows how to use the principle that corresponding parts of congruent triangles are congruent, or CPCTC.

How do we use congruent triangles to prove line segments or angles congruent?

Proving triangles are congruent using their corresponding parts.

You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

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