Side, Side, and the Angle Opposite the Larger of the Two Sides

It's time to dive into the fourth theorem of triangle congruence: Side, Side, and the Angle Opposite the Larger of the Two Sides,
or simply put:
SSA
This congruence theorem is practical and straightforward, and it will help us prove triangle congruence under certain simple conditions.
What does the Side, Side, and the Angle Opposite the Larger of the Two Sides congruence theorem say?
If two triangles have two sides of the same length and the angle opposite the larger of these two sides is also the same, then the triangles are congruent.
What does this mean?

Let's see it in an illustration:

If we have:
$AB=DE$
and also:
$AC=DF$

That is, the triangles have two equal sides,

and also:
$∠B=∠E$

when
$​​​​​​​AC>AB$

That is, the angle opposite to the larger side is also equal.
We can determine that the triangles are congruent according to the SAS (Side-Angle-Side) theorem

Pay attention that, even though it is given in only one triangle $AC>AB$
but, since we have a previous statement that says:
$AB=DE$
and also:
$AC=DF$

we can determine according to the transitive relation that also: $DF>DE$

Therefore, we will determine that:
$△ABC≅△DEF$

Notice that we have written the congruence in the correct order.
When
$AB=DE$
$AC=DF$
$∠B=∠E$

Since the triangles are congruent, identical in their sides and angles, we can say that:
$AB=DE$
$BC=EF$
$AC=DF$
$∠A=∠D$
$∠B=∠E$
$∠C=∠F$

Remember that there must be 3 circumstances and one condition:
The 3 required circumstances are:

• One side of one of the triangles has to be equal to another side of the second triangle
• Another side of one of the triangles has to be equal to another side of the second triangle
• An angle of one of the triangles has to be equal to another angle of the second triangle

The condition:

• The angle in question must be opposite the longest side among the two sides being referred to in the required circumstances (in both triangles).
• If all the circumstances and the condition are met, we will be able to prove that the triangles are indeed congruent.

Let's look at some ways to do it:

• According to the data given in the question:
  In certain cases, the data can be written as seen in the previous example or with a number.
  Sometimes you will have to deduce it from other information, for instance, if side $AC=5$ and side $AB=4$, then $AC>AB$
  
  as long as the angle in question is opposite the longer side, in our case $AC$, and if the other circumstances are met, we can demonstrate the congruence of the triangles.

• When the length of the sides is not revealed, we will rely on the angles:
  
  Let's look at the following property:
  when a side is opposite an angle of $90^o$ degrees or more, this will be the longest side of the triangle.
  Consequently, we can determine with great confidence that this side is longer than any other side of the triangle.
  
  Additionally, it is very important that you know the following theorem:
  In every triangle, the larger the side, the larger the angle it faces.
  That is to say, if we have angles where one is larger than another, we can conclude that the side opposite the larger angle is longer than the side opposite the smaller angle.
  
  Note:
  The angle in question does not necessarily have to be the largest of all the angles in the triangle, it just needs to be opposite the longest side among the two sides we are examining.
  The side opposite the angle also does not necessarily have to be the longest of all sides, just longer than the other side in question.

If you found this article interesting, you might also be interested in the following articles:

Congruence Criterion: Side, Angle, Side

Congruence Criterion: Angle, Side, Angle

Congruence Criterion: Side, Side, Side

Style of Writing Formal Proof in Geometry

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Assignment

Given: the quadrilateral $ABCD$ is a parallelogram.

According to which congruence theorem do the triangles $ΔADO≅ΔCBO$ overlap?

Solution

Since the quadrilateral $ABCD$ is a rectangle, in the rectangle there are two pairs of opposite equal parallel sides, therefore:

$BC=AD$

Alternate interior angles are equal because they are between parallel lines, therefore:

$\sphericalangle BCO=\sphericalangle DAO$

Vertically opposite angles are equal, and therefore:

$\sphericalangle O_1=\sphericalangle O_2$

We verify that the triangles are congruent according to the side-angle-angle theorem.

Answer:

Congruent according to ASA (Angle-Side-Angle).

Assignment

Is $DE$ not a side of any of the triangles?

Solution

If we look at the graphic, we see that from point $E$ a line goes to point $D$, therefore $DE$ is a straight line that is not a side of any triangle in the drawing.

Answer

True

Assignment

In the given drawing:

$AB=CD$

$\angle BAC=\angle DCA$

According to which theorem of congruence are the triangles $\triangle ABC \cong \triangle CDA$ congruent?

Solution

Given that $AB=CD$

Given that $\angle BAC=\angle DCA$

$AC=AC$ is the common side

We verify that the triangles are congruent by side, angle, side

Answer

Congruent by S.A.S

Prompt

Given rectangle $ABCD$ with side $AB$ measuring $4.5$ cm and side $BC$ measuring $2$ cm.

What is the area of the rectangle?

Solution

The formula to calculate the area of a rectangle is the base times the height; in this case, we replace them

$4.5\times2=9$

Answer

$9 cm²$

Assignment

The segments $BE$ and $AC$ intersect at point $D$.

According to which theorem of congruence do the triangles $ΔABD≅ΔCED$ match?

Solution

$BE$ and $AC$

Intersect at a point $D$

$AD=DC$

$D$ intersects $BE$

$\angle ADB=\angle EDC$

Angles opposite by the vertex

Triangles overlap according to $S.A.S$

Answer

Overlapping $S.A.S$