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SAS Triangle Congruence

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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? After completing this Concept, you'll be able to use the Side-Angle-Side (SAS) shortcut to prove triangle congruency.

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

James Sousa: Introduction to Congruent Triangles

James Sousa: Determining If Two Triangles are Congruent

Finally, watch this video.

James Sousa: Example 1: Prove Two Triangles are Congruent

Guidance

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 .)

\overline{AC} \cong \overline{XZ}, \ \overline{BC} \cong \overline{YZ} , and \angle{C} \cong \angle{Z} , then \triangle ABC \cong \triangle XYZ .

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.

\angle{B} would be the included angle for sides \overline{AB} and \overline{BC} .

Example A

What additional piece of information do you need to show that these two triangles are congruent using the SAS Postulate?

a) \angle{ABC} \cong \angle{LKM}

b) \overline{AB} \cong \overline{LK}

c) \overline{BC} \cong \overline{KM}

d) \angle{BAC} \cong \angle{KLM}

For the SAS Postulate, you need the side on the other side of the angle. In \triangle ABC , that is \overline{BC} and in \triangle LKM that is \overline{KM} . The answer is c.

Example B

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

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

Prove : \triangle ACB \cong \triangle ECD

Statement Reason
1. 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. \angle{ACB} \cong \angle{DCE} 3. Vertical Angles Postulate
4. \triangle ACB \cong \triangle ECD 4. SAS Postulate

Example C

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

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.

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Guided Practice

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

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

3. Fill in the blanks in the proof below.

Given :

\overline{AB} \cong \overline{DC}, \ \overline{BE} \cong \overline{CE}

Prove : \triangle ABE \cong \triangle ACE

Statement Reason
1. 1.
2. \angle{AEB} \cong \angle{DEC} 2.
3. \triangle ABE \cong \triangle ACE 3.

Answers:

1. The pair of triangles is congruent by the SAS postulate.  \triangle CAB \cong \triangle QRS .

2. 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 \overline{EF} \cong \overline{BA} .

3.

Statement Reason
1. \overline{AB} \cong \overline{DC}, \ \overline{BE} \cong \overline{CE} 1. Given
2. \angle{AEB} \cong \angle{DEC} 2. Vertical Angle Theorem
3. \triangle ABE \cong \triangle ACE 3. SAS postulate

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 SAS
  2. Use SAS
  3. Use SAS

Fill in the blanks in the proofs below.

  1. Given : B is a midpoint of \overline{DC} \overline{AB} \perp \overline{DC} Prove : \triangle ABD \cong \triangle ABC
Statement Reason
1. B is a midpoint of \overline{DC}, \overline{AB} \perp \overline{DC} 1.
2. 2. Definition of a midpoint
3. \angle{ABD} and \angle{ABC} are right angles 3.
4. 4. All right angles are \cong
5. 5.
6. \triangle{ABD} \cong  \triangle{ABC} 6.
  1. Given : \overline{AB} is an angle bisector of \angle{DAC} \overline{AD} \cong \overline{AC} Prove : \triangle ABD \cong \triangle ABC
Statement Reason
1. 1.
2. \angle{DAB} \cong \angle{BAC} 2.
3. 3. Reflexive PoC
4. \triangle ABD \cong \triangle ABC 4.
  1. Given : B is the midpoint of \overline{DE} and \overline{AC} \angle{ABE} is a right angle Prove : \triangle ABE \cong \triangle CBD
Statement Reason
1. 1. Given
2. \overline{DB} \cong \overline{BE}, \ \overline{AB} \cong \overline{BC} 2.
3. 3. Definition of a Right Angle
4. 4. Vertical Angle Theorem
5. \triangle ABE \cong \triangle CBD 5.
  1. Given : \overline{DB} is the angle bisector of \angle{ADC} \overline{AD} \cong \overline{DC} Prove : \triangle ABD \cong \triangle CBD
Statement Reason
1. 1.
2. \angle{ADB} \cong \angle{BDC} 2.
3. 3.
4. \triangle ABD \cong \triangle CBD 4.

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