4.7: SAS Triangle Congruence
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 SideAngleSide (SAS) shortcut to prove triangles are congruent.
Watch This
CK12 Foundation: Chapter4SASTriangleCongruenceA
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
James Sousa: Example 1: Prove Two Triangles are Congruent
Guidance
An included angle is when an angle is between two given sides of a triangle (or polygon). In the picture below, the markings indicate that
Consider the question: If I have two sides of length 2 in and 5 in and the angle between them is
Investigation: Constructing a Triangle Given Two Sides and Included Angle
Tools Needed: protractor, pencil, ruler, and paper
 Draw the longest side (5 in) horizontally, halfway down the page. The drawings in this investigation are to scale.
 At the left endpoint of your line segment, use the protractor to measure a
45∘ angle. Mark this measurement.  Connect your mark from Step 2 with the left endpoint. Make your line 2 in long, the length of the second side.
 Connect the two endpoints by drawing the third side.
Can you draw another triangle, with these measurements that looks different? The answer is NO. Only one triangle can be created from any given two lengths and the INCLUDED angle.
SideAngleSide (SAS) Triangle Congruence 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.
The markings in the picture are enough to say that
Example A
What additional piece of information would you need to prove that these two triangles are congruent using the SAS Postulate?
a)
b)
c)
d)
For the SAS Postulate, you need two sides and the included angle in both triangles. So, you need the side on the other side of the angle. In
Example B
Write a twocolumn proof to show that the two triangles are congruent.
Given:
Prove:
Statement  Reason 

1. 
Given 
2. 
Definition of a midpoint 
3. 
Vertical Angles Postulate 
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.
Watch this video for help with the Examples above.
CK12 Foundation: Chapter4SASTriangleCongruenceB
Vocabulary
Two figures are congruent if they have exactly the same size and shape. By definition, two triangles are congruent if the three corresponding angles and sides are congruent. The symbol
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:
Prove:
Statement  Reason 

1.  1. 
2. 
2. 
3. 
3. 
Answers:
1. The pair of triangles is congruent by the SAS postulate.
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
3.
Statement  Reason 

1. 
1. Given 
2. 
2. Vertical Angle Theorem 
3. 
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.
 Use SAS
 Use SAS
 Use SAS
Complete the proofs below.

Given:
B is a midpoint ofDC¯¯¯¯¯¯¯¯ AB¯¯¯¯¯¯¯¯⊥DC¯¯¯¯¯¯¯¯ Prove:△ABD≅△ABC 
Given:
AB¯¯¯¯¯¯¯¯ is an angle bisector of∠DAC AD¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯ Prove:△ABD≅△ABC 
Given:
B is the midpoint ofDE¯¯¯¯¯¯¯¯ andAC¯¯¯¯¯¯¯¯ ∠ABE is a right angle Prove:△ABE≅△CBD 
Given:
DB¯¯¯¯¯¯¯¯ is the angle bisector of∠ADC AD¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯ Prove: \begin{align*}\triangle ABD \cong \triangle CBD\end{align*}
For each pair of triangles, write what needs to be congruent in order for the triangles to be congruent by SAS. Then, write the congruence statement for the triangles.
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.Image Attributions
Here you'll learn how to prove that two triangles are congruent given only information about two pairs of sides and included angles.