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4.14: SAS

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Two sets of corresponding sides and included angles prove congruent triangles.

Side-Angle-Side Postulate

If two sides and the included angle in one triangle are congruent to two sides and the included angle in another triangle, then the two triangles are congruent. (When an angle is between two given sides of a polygon it is called an included angle.)

f-d_77ca1d4d5a96259729f68cffe461702e9c92b5abb6018335683fa888+IMAGE_TINY+IMAGE_TINY.png
Figure 4.14.1

¯AC¯XZ, ¯BCYZ, and CZ, then ΔABCΔXYZ.

This is called the Side-Angle-Side (SAS) Postulate and it is a shortcut for proving that two triangles are congruent. The placement of the word Angle is important because it indicates that the angle you are given is between the two sides.

f-d_cee2c890f31b8ae0caefb7cd8c4e176f3b464ae58aeca28db82ae6d1+IMAGE_TINY+IMAGE_TINY.png
Figure 4.14.2

B would be the included angle for sides ¯AB and ¯BC.

What if you were given two triangles and provided with only two of their side lengths and the measure of the angle between those two sides? How could you determine if the two triangles were congruent?

Example 4.14.1

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

f-d_852448eaea7e9493d8517a0cb61b8b84faaa41e910fbdc4777a47325+IMAGE_TINY+IMAGE_TINY.png
Figure 4.14.3

Solution

The pair of triangles is congruent by the SAS postulate. ΔCABΔQRS.

Example 4.14.2

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

f-d_bad366f4ad886b8cd01e6d3fec17fa0fcda9af8d300533b9cfb63d8d+IMAGE_TINY+IMAGE_TINY.png
Figure 4.14.4

Solution

We know that one pair of sides and one pair of angles are congruent from the diagram. In order to know that the triangles are congruent by SAS we need to know that the pair of sides on the other side of the angle are congruent. So, we need to know that ¯EF¯BA.

Example 4.14.3

Fill in the blanks in the proof below.

Given:

¯AB¯DC,¯BE¯CE

Prove: ΔABEΔACE

f-d_575a2e0593763e6fb3eb86a98bf20a6b8fd06c9dbcb90c53eb2e838a+IMAGE_TINY+IMAGE_TINY.png
Figure 4.14.5

Solution

Statement Reason
1. 1.
2. AEBDEC 2.
3. ΔABEΔACE 3.
Statement Reason
1. ¯AB¯DC,¯BE¯CE 1. Given
2. AEBDEC 2. Vertical Angle Theorem
3. ΔABEΔACE 3. SAS postulate

Example 4.14.4

What additional piece of information do you need to show that these two triangles are congruent using the SAS Postulate, ABCLKM, ¯ABLK¯AB, ¯BC¯KM, or BACKLM?

f-d_502a8288a0ae68d09d4afc39181fcdf57c1f33b123ed042870cabaed+IMAGE_TINY+IMAGE_TINY.png
Figure 4.14.6

Solution

For the SAS Postulate, you need the side on the other side of the angle. In ΔABC, that is ¯BC and in ΔLKM that is ¯KM. The answer is ¯BC¯KM.

Example 4.14.5

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

f-d_30be11a88a9c033cb6a693afb0fda465fd9b106c80444832c43a1c9e+IMAGE_TINY+IMAGE_TINY.png
Figure 4.14.6

Solution

While the triangles have two pairs of sides and one pair of angles that are congruent, the angle is not in the same place in both triangles. The first triangle fits with SAS, but the second triangle is SSA. There is not enough information for us to know whether or not these triangles are congruent.

Review

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

  1. f-d_a30d374a5da5273af9f11b1f0344c1fc006e1964a3d08318d0aa80a4+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.14.7
  2. f-d_ded93c2b13ce498cb02b6aeca551efad75c28f15ade011504166d0df+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.14.7
  3. f-d_17efe62378d001251fe3259568a026be853b68793e6261d9b554c506+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.14.7

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

  1. f-d_241ceb6317d76cfd44148a92b63368fc3c0dee028f9b5fc2d6859287+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.14.7
  2. f-d_fc90b95065daab8e8cb558cff608472a845b94d58d232372be4e0f39+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.14.7
  3. f-d_ab6f7d67e291acff7083cd17dc65e64bffcddfe85941d6d00d388ce5+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.14.7

Fill in the blanks in the proofs below.

  1. Given:
    • B is a midpoint of ¯DC
    • ¯AB¯DC

    Prove: ΔABDΔABC

    f-d_150648577d2e3777ceaf7e05299f91a609ac26271341acdd63b63182+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.14.8
Statement Reason
1. B is a midpoint of ¯DC,¯AB¯DC 1.
2. 2. Definition of a midpoint
3. ABD and ABC are right angles 3.
4. 4. All right angles are \cong\)
5. 5.
6. ΔABDΔABC 6.
  1. Given:
    • ¯AB is an angle bisector of DAC
    • ¯AD¯AC

    Prove: ΔABDΔABC

    f-d_150648577d2e3777ceaf7e05299f91a609ac26271341acdd63b63182+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.14.9
Statement Reason
1. 1.
2. DABBAC 2.
3. 3. Reflexive PoC
4. ΔABDΔABC 4.
  1. Given:
    • B is the midpoint of¯DE and ¯AC
    • ABE is a right angle

    Prove: ΔABEΔCBD

    f-d_b5831b67d99b1f53e36c306a39698362d66fbc498f121314055c8c65+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.14.10
Statement Reason
1. 1. Given
2. ¯DB¯BE,¯AB¯BC 2.
3. 3. Definition of a Right Angle
4. 4. Vertical Angle Theorem
5. ΔABEΔCBD 5.
  1. Given:
    • ¯DB is the angle bisector of ADC
    • ¯AD¯DC

    Prove: ΔABDΔCBD

    f-d_9a6bcfd03d0be2a3a8b015232f2ee6581d1b9dd687336be3c5dac795+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.14.11
Statement Reason
1. 1.
2. ADBBDC 2.
3. 3.
4. ΔABDΔCBD 4.

Review (Answers)

To see the Review answers, open this PDF file and look for section 4.7.

Resources

Vocabulary

Term Definition
Base Angles The base angles of an isosceles triangle are the angles formed by the base and one leg of the triangle.
Congruent Congruent figures are identical in size, shape and measure.
Equilateral Triangle An equilateral triangle is a triangle in which all three sides are the same length.
Included Angle The included angle in a triangle is the angle between two known sides.
SAS SAS means side, angle, side, and refers to the fact that two sides and the included angle of a triangle are known.
Side Angle Side Triangle A side angle side triangle is a triangle where two of the sides and the angle between them are known quantities.
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: SAS Triangle Congruence Discussion Questions

Study Aids: Triangle Congruence Study Guide

Practice: SAS

Real World: SSS Triangle Congruence


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