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SSS Triangle Congruence

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SSS Triangle Congruence

What if you were given two triangles and provided with information only about their side lengths? How could you determine if the two triangles were congruent? After completing this concept, you'll be able to use the Side-Side-Side (SSS) shortcut to prove triangle congruency.

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CK-12

Watch the portions of the following two videos that deal with SSS triangle congruence.

James Sousa: Introduction to Congruent Triangles

James Sousa: Determining If Two Triangles are Congruent

Guidance

If 3 sides in one triangle are congruent to 3 sides in another triangle, then the triangles are congruent.

\overline{BC} \cong \overline{YZ}, \ \overline{AB} \cong \overline{XY} , and \overline{AC} \cong \overline{XZ} then \triangle ABC \cong \triangle XYZ .

This is called the Side-Side-Side (SSS) Postulate and it is a shortcut for proving that two triangles are congruent. Before, you had to show 3 sides and 3 angles in one triangle were congruent to 3 sides and 3 angles in another triangle. Now you only have to show 3 sides in one triangle are congruent to 3 sides in another.

Example A

Write a triangle congruence statement based on the picture below:

From the tic marks, we know \overline{AB} \cong \overline{LM}, \ \overline{AC} \cong \overline{LK}, \ \overline{BC} \cong \overline{MK} . From the SSS Postulate, the triangles are congruent. Lining up the corresponding sides, we have \triangle ABC \cong \triangle LMK .

Don’t forget ORDER MATTERS when writing congruence statements. Line up the sides with the same number of tic marks.

Example B

Write a two-column proof to show that the two triangles are congruent.

Given : \overline{AB} \cong \overline{DE}

C is the midpoint of \overline{AE} and \overline{DB} .

Prove : \triangle ACB \cong \triangle ECD

Statement Reason

1. \overline{AB} \cong \overline{DE}

C is the midpoint of \overline{AE} and \overline{DB}

1.Given
2. \overline{AC} \cong \overline{CE}, \ \overline{BC} \cong \overline{CD} 2.Definition of a midpoint
3. \triangle ACB \cong \triangle ECD 3.SSS Postulate

Note that you must clearly state the three sets of sides are congruent BEFORE stating the triangles are congruent.

Example C

The only way we will show two triangles are congruent in an x-y plane is using SSS.

Find the lengths of all the line segments from both triangles to see if the two triangles are congruent.

To do this, you need to use the distance formula.

Begin with \triangle ABC and its sides.

AB & = \sqrt{(-6-(-2))^2+(5-10)^2}\\& = \sqrt{(-4)^2+(-5)^2}\\& = \sqrt{16+25}\\& = \sqrt{41}

BC & = \sqrt{(-2-(-3))^2+(10-3)^2}\\& = \sqrt{(1)^2+(7)^2}\\& = \sqrt{1+49}\\& = \sqrt{50} = 5 \sqrt{2}

AC & = \sqrt{(-6-(-3))^2 + (5-3)^2}\\& = \sqrt{(-3)^2+(2)^2}\\& = \sqrt{9+4}\\& = \sqrt{13}

Now, find the lengths of all the sides in \triangle DEF .

DE & = \sqrt{(1-5)^2 + (-3-2)^2}\\& = \sqrt{(-4)^2 + (-5)^2}\\& = \sqrt{16+25}\\& = \sqrt{41}

EF & = \sqrt{(5-4)^2+(2-(-5))^2}\\& = \sqrt{(1)^2+(7)^2}\\& = \sqrt{1+49}\\& = \sqrt{50} = 5\sqrt{2}

DF & = \sqrt{(1-4)^2+(-3-(-5))^2}\\& = \sqrt{(-3)^2+(2)^2}\\& = \sqrt{9+4}\\& = \sqrt{13}

AB = DE, \ BC = EF , and AC = DF , so the two triangles are congruent by SSS.

CK-12

Guided Practice

1. Determine if the two triangles are congruent.

2. Fill in the blanks in the proof below.

Given : \overline{AB} \cong \overline{DC}, \ \overline{AC} \cong \overline{DB}

Prove : \triangle ABC \cong \triangle DCB

Statement Reason
1. 1.
2. 2. Reflexive PoC
3. \triangle ABC \cong \triangle DCB 3.

3. Is the pair of triangles congruent? If so, write the congruence statement and why.

Answers:

1. Start with \triangle ABC .

AB & = \sqrt{(-2-(-8))^2+(-2-(-6))^2}\\& = \sqrt{(6)^2+(4)^2}\\& = \sqrt{36+16}\\& = \sqrt{52} = 2\sqrt{13}

BC & = \sqrt{(-8-(-6))^2+(-6-(-9))^2}\\& = \sqrt{(-2)^2+(3)^2}\\& = \sqrt{4+9}\\& = \sqrt{13}

AC & = \sqrt{(-2-(-6))^2+(-2-(-9))^2}\\& = \sqrt{(4)^2+(7)^2}\\& = \sqrt{16+49}\\& = \sqrt{65}

Now find the sides of \triangle DEF .

DE & = \sqrt{(3-6)^2 + (9-4)^2}\\& = \sqrt{(-3)^2 + (5)^2}\\& = \sqrt{9+25}\\& = \sqrt{34}

EF & = \sqrt{(6-10)^2+(4-7)^2}\\& = \sqrt{(-4)^2+(-3)^2}\\& = \sqrt{16+9}\\& = \sqrt{25} = 5

DF & = \sqrt{(3-10)^2+(9-7)^2}\\& = \sqrt{(-7)^2+(2)^2}\\& = \sqrt{49+4}\\& = \sqrt{53}

No sides have equal measures, so the triangles are not congruent.

2.

Statement Reason
1. \overline{AB} \cong \overline{DC}, \ \overline{AC} \cong \overline{DB} 1. Given
2. \overline{BC} \cong \overline{CB} 2. Reflexive PoC
3. \triangle ABC \cong \triangle DCB 3. SSS Postulate

3. The triangles are congruent because they have three pairs of sides congruent.  \triangle DEF \cong \triangle IGH .

Practice

Are the pairs of triangles congruent? If so, write the congruence statement and why.

State the additional piece of information needed to show that each pair of triangles is congruent.

  1. Use SSS
  2. Use SSS

Fill in the blanks in the proofs below.

  1. Given : B is the midpoint of \overline{DC} \overline{AD} \cong \overline{AC} Prove : \triangle ABD \cong \triangle ABC
Statement Reason
1. 1.
2. 2. Definition of a Midpoint
3. 3. Reflexive PoC
4. \triangle ABD \cong \triangle ABC 4.

Find the lengths of the sides of each triangle to see if the two triangles are congruent. Leave your answers under the radical.

  1. \triangle ABC: \ A(-1, 5), \ B(-4, 2), \ C(2, -2) and \triangle DEF: \ D(7, -5), \ E(4, 2), \ F(8, -9)
  2. \triangle ABC: \ A(-8, -3), \ B(-2, -4), \ C(-5, -9) and \triangle DEF: \ D(-7, 2), \ E(-1, 3), \ F(-4, 8)

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