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Congruent Triangles - SSS and SAS

SSS Congruence

When two triangles are congruent, all corresponding sides and angles are congruent.

It is obvious that these two figures are the same size when all the information is given.

2 Triangles. Every angle and side in the first triangle is marked corresponding and congruent to the other triangle

There are six corresponding congruent statements because each triangle has six parts.

Three Angles:

  • ∠A ≅ ∠X
  • ∠B ≅ ∠Y
  • ∠C ≅ ∠Z

Three Sides:

  • segment BC ≅ segment YZ
  • segment AC ≅ segment XZ
  • segment AB ≅ segment XY

2 Triangles ABC and XYZ. Segments AB and XY have two congruence marks. Segments XZ and AC have one congruence mark. Segments YZ and BC have three congruence marks. Angles A and X have one congreunce arc. Angles Z and C have two congruence arcs. Angles B and Y have three congruence arcs.

This lesson will focus on proving that two triangles are congruent by using shortcut methods.

Sometimes you may not know all the information about two different triangles but they may look congruent. Look at the triangles here.

Just because two of the sides are equal does not necessarily mean the triangles are congruent.

Two triangles with two sides lengths of 6 and 10 units.

SSS Congruence

There are methods you will learn that may show two triangles are congruent even though all the information about the angles and sides may not be given.

The first method is called the SSS Congruence Postulate. The postulate states: If three sides of one triangle are congruent to three sides of another, then the triangles are congruent.

SSS Example

Let's look at an example using the SSS postulate.

Example: List the corresponding congruent parts of the triangles. Is this enough information to prove the triangles are congruent? If so, write the congruence statement and the method used to prove they are congruent.

Solution: First we will list all given corresponding congruent parts.

segment AB ≅ segment FE

segment AC ≅ segment FD

segment BC ≅ segment ED

Triangles ABC and DEF. Segment AB is congruent to segment EF. Segment AC is congruent to segment DF. Segment DE is congruent to segment BC.

Now we have enough information to state the triangles are congruent. When writing the following congruence statement, the corresponding vertices must be in the same location:

Yes, △ABC ≅ △FED by SSS postulate.

Proof Using Prior Knowledge

Sometimes information is not given with congruent symbols. You must use prior knowledge to determine that sides and angles are congruent.

The Math Warehouse has an example of this concept.

Scroll past Example 1 to A Side by Side Proof.

Click the Next icon to reveal the information in the two column proof.

Example #1

Open Prove That Two Triangles Are Congruent Using SSS Congruence Postulate in a new tab

Application

Rigid Motions

Recall the transformations used in prior lessons.

Reflection is a flipA triangle point to the left is opposite a similar triangle facing to the right. A vertical reflection line between them mirrors the 2 triangles.

Rotation is a turnA triangle with the right angle at the bottom left is rotated to where the triangle's right angle is now at the top left

Translation is a slideA right triangle is simply moved to the right.

Dilation is a scaleA right triangle is shrunk

Which transformation does not hold congruence?

  • Reflection
  • Rotation
  • Translation
  • Dilation

Answer: Dilation; See the change in size? These figures are not congruent. It would not be the others because they are the same size, so they do not hold congruence.

Example #2

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SAS Congruence

Included Angle

Great, you have just learned one method to prove two triangles are congruent.

Before we move on to the next method, let's go over a concept that you will need to learn to better understand the SAS postulate.

An included angle is the angle that is formed by two sides. In the diagram, the included angle formed by the sides AB and BC is ∠B because that is where the two sides intersect.

Triangle A B CTriangle A B C with length A B and length B C highlighted with arrows

Example #3

Let's see if you can identify included angles.

What is the included angle formed by the sides segment EG and segment FG? ∠___

Answer: ∠G

Triangle E F G

Example #4

What is the included angle formed by the sides segment XY and segment XZ?

Triangle X Y Z

Answer: ∠X

SAS Postulate

Terrific, now that you can identify an included angle, you can use the next method to prove triangles are congruent.

The SAS Postulate states that if two sides and an included angle of one triangle are congruent to two sides and an included angle of a second triangle, then the triangles are congruent.

2 triangles. 2 sides and the angle between the two sides of one triangle are congruent to two sides and the angle between the tw sides of the other triangle.

Examples and Practice

For examples and interactive practice using the SAS Postulate, go to Math Warehouse.

If you would like to see a video of someone explaining these concepts along with solving problems go to Brightstorm: SSS and SAS.

Example #5

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