Side, Side, Angle

🏆Practice side, side and the angle opposite to the major side

Fourth Congruence Theorem: Side-Side-Angle

In summary: SSA
It means that:
if two triangles have two pairs of equal sides and the angle opposite the larger of these two pairs is also equal, then the triangles are congruent.

SAS image

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Test yourself on side, side and the angle opposite to the major side!

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Look at the triangles in the diagram.

Determine which of the statements is correct.

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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 pairs of sides of the same length and the angle opposite the larger of these two pairs is also the same, then the triangles are congruent.
What does this mean?


Let's see it in an illustration:

SSA illustration

If we have:
AB=DEAB=DE
and also:
AC=DFAC=DF

That is, the triangles have two equal sides,

and also:
B=E∠B=∠E

when
​​​​​​​AC>AB​​​​​​​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>ABAC>AB
but, since we have a previous statement that says:
AB=DEAB=DE
and also:
AC=DFAC=DF

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

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

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

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


Let's highlight certain features of the fourth congruence theorem:

Remember that there are 3 requirements and one condition:
The 3 requirements 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 (in both triangles).
  • If all the circumstances and the condition are met, we will be able to prove that the triangles are indeed congruent.

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How can we prove that one side is greater than another in a triangle?

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=5AC=5 and side AB=4AB=4 , then AC>ABAC>AB

    as long as the angle in question is opposite the longer side, in our case ACAC , 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 90o 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

On the Tutorela blog, you'll find a variety of mathematics articles.


Side-Side and the Angle Opposite the Largest Side Exercises

Exercise 1

Assignment

Given: the quadrilateral ABCD ABCD is a parallelogram.

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

the quadrilateral ABCD is a parallelogram

Solution

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

BC=AD BC=AD

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

BCO=DAO \sphericalangle BCO=\sphericalangle DAO

Vertically opposite angles are equal, and therefore:

O1=O2 \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).


Do you know what the answer is?

Exercise 2

Assignment

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

DE is not a side in any of the triangles

Solution

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

Answer

True


Exercise 3

Assignment

In the given drawing:

In the given drawing AB equals CD

AB=CD AB=CD

BAC=DCA \angle BAC=\angle DCA

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

Solution

Given that AB=CD AB=CD

Given that BAC=DCA \angle BAC=\angle DCA

AC=AC AC=AC is the common side

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

Answer

Congruent by S.A.S


Check your understanding

Exercise 4

A rectangle ABCD with side AB measuring 4.5 cm and side BC measuring 2 cm

Prompt

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

4.5×2=9 4.5\times2=9

Answer

9cm2 9 cm²

Exercise 5

Assignment

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

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

The segments BE and AC intersect at point D

Solution

BE BE and AC AC

Intersect at a point D D

AD=DC AD=DC

D D intersects BE BE

ADB=EDC \angle ADB=\angle EDC

Angles opposite by the vertex

The triangles are congruent according to S.A.S S.A.S

Answer

Congruent by S.A.S S.A.S

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