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4.13: SSS

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Three sets of equal side lengths determine congruence.

Side-Side-Side Postulate

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

f-d_9906a5d9ed0e9d13de591f0716bdfaddc90289f1794b2cb40a98b4b8+IMAGE_TINY+IMAGE_TINY.png
Figure 4.13.1

¯BC¯YZ, ¯AB¯XY, and ¯AB¯XZ then ΔABCΔXYZ.

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.

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?

Example 4.13.1

Determine if the two triangles are congruent.

f-d_d3623fab1fc7e489f75f8c5af8e02fcabc340a447224677ebd4b816c+IMAGE_TINY+IMAGE_TINY.png
Figure 4.13.2

Solution

Start with ΔABC.

AB=(2(8))2+(2(6))2=(6)2+(4)2=36+16=52=213BC=(8(6))2+(6(9))2=(2)2+(3)2=4+9=13AC=(2(6))2+(2(9))2=(4)2+(7)2=16+49=65

Now find the sides of ΔDEF.

DE=(36)2+(94)2=(3)2+(5)2=9+25=34EF=(610)2+(47)2=(4)2+(3)2=16+9=25=5DF=(310)2+(97)2=(7)2+(2)2=49+4=53

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

Example 4.13.2

Fill in the blanks in the proof below.

Given:¯AB¯DC, ¯AC¯DB

Prove: ΔABCΔDCB

f-d_ed6b2ec8b08531a60a49c17b954b3a9b0ee7f95b3c5863d663018533+IMAGE_TINY+IMAGE_TINY.pngFigure 4.13.4

Solution

Statement Reason
1. 1.
2. 2. Reflexive PoC
3. ΔABCΔDCB 3.
Statement Reason
1. ¯AB¯DC,¯AC¯DB 1. Given
2. ¯BC¯CB 2. Reflexive PoC
3. ΔABCΔDCB 3. SSS Postulate

Example 4.13.3

Write a triangle congruence statement based on the picture below:

f-d_e14b58552fa520ee48e12fc24b1dd46c4fb23286171426dc1c111168+IMAGE_TINY+IMAGE_TINY.pngFigure 4.13.5

Solution

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

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

Example 4.13.4

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

f-d_bdcb9b47c6b83591762b3a6505065fc9f5f34d420aff42d3432d869a+IMAGE_TINY+IMAGE_TINY.pngFigure 4.13.6

Given: ¯AB¯DE

C is the midpoint of ¯AE and ¯DB.

Prove: ΔACBΔECD

Solution

Statement Reason

1. ¯AB¯DE

C is the midpoint of ¯AEand\(¯DB

1.Given
2. ¯AC¯CE,¯BC¯CD 2.Definition of a midpoint
3. ΔACBΔECD 3.SSS Postulate

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

Example 4.13.5

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

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

Solution

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

f-d_619cfa690f664fd1f35f3255360fcef2fe14beac50f338c99967b758+IMAGE_TINY+IMAGE_TINY.png
Figure 4.13.7

Begin with ΔABC and its sides.

AB=(6(2))2+(510)2=(4)2+(5)2=16+25=41BC=(2(3))2+(103)2=(1)2+(7)2=1+49=50=52AC=(6(3))2+(53)2=(3)2+(2)2=9+4=13

Now, find the lengths of all the sides in ΔDEF.

DE=(15)2+(32)2=(4)2+(5)2=16+25=41EF=(54)2+(2(5))2=(1)2+(7)2=1+49=50=52DF=(14)2+(3(5))2=(3)2+(2)2=9+4=13

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

Review

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

  1. f-d_2fa3c05ac23c205131da71478e45d02af3faee85a4eecd4ae49d5e67+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.13.8
  2. f-d_6819755197fcff08f9739463b9ae87616a1acc1b1f06ebdf91988498+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.13.9
  3. f-d_8ae601675e29b9710544a238dc2c41d3a7f4720d29c664024d462b8d+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.13.10
  4. f-d_ac2ca3ccd3de05abb2ab43bd98078d7069a7e8b0eb63622b9e8f6a28+IMAGE_TINY+IMAGE_TINY.pngFigure 4.13.11

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

  1. Use SSS
    f-d_241ceb6317d76cfd44148a92b63368fc3c0dee028f9b5fc2d6859287+IMAGE_TINY+IMAGE_TINY.pngFigure 4.13.12
  2. Use SSS
    f-d_fc90b95065daab8e8cb558cff608472a845b94d58d232372be4e0f39+IMAGE_TINY+IMAGE_TINY.pngFigure 4.13.13

Fill in the blanks in the proofs below.

  1. Given: B is the midpoint of ¯DC¯AD¯AC Prove: ΔABDΔABC
    f-d_150648577d2e3777ceaf7e05299f91a609ac26271341acdd63b63182+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.13.14
Statement Reason
1. 1.
2. 2. Definition of a Midpoint
3. 3. Reflexive PoC
4. ΔABDΔABC 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. f-d_532c993e268eb336db3a7e8fd2f0c5075ee5cfd59daaa921b6cbf04b+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.13.15
  2. f-d_de638de63222ff1e34b4c4aec86525121e28ef386a5a89167d666e82+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.13.16
  3. ΔABC:A(1,5),B(4,2),C(2,2) and ΔDEF:D(7,5),E(4,2),F(8,9)
  4. ΔABC:A(8,3),B(2,4),C(5,9) and ΔDEF:D(7,2),E(1,3),F(4,8)

Review (Answers)

To see the Review answers, open this PDF file and look for section 4.6.

Resources

Vocabulary

Term Definition
Congruent Congruent figures are identical in size, shape and measure.
Distance Formula The distance between two points (x1,y1) and (x2,y2) can be defined as d=(x2x1)2+(y2y1)2.
H-L (Hypotenuse-Leg) Congruence Theorem If the hypotenuse and leg in one right triangle are congruent to the hypotenuse and leg in another right triangle, then the two triangles are congruent.
Side Side Side Triangle A side side side triangle is a triangle where the lengths of all three sides are known quantities.
SSS SSS means side, side, side and refers to the fact that all three sides of a triangle are known in a problem.
Triangle Congruence Triangle congruence occurs if 3 sides in one triangle are congruent to 3 sides in another triangle.
Rigid Transformation A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.

Additional Resources

Interactive Element

Video: Introduction to Congruent Triangles

Activities: SSS Triangle Congruence Discussion Questions

Study Aids: Triangle Congruence Study Guide

Practice: SSS

Real World: SSS Triangle Congruence


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