Side, side and the angle opposite to the major side - Examples, Exercises and Solutions

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

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

Exercise #1

Look at the triangles in the diagram.

Which of the following statements is true?

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Step-by-Step Solution

According to the existing data:

EF=BA=10 EF=BA=10 (Side)

ED=AC=13 ED=AC=13 (Side)

The angles equal to 53 degrees are both opposite the greater side (which is equal to 13) in both triangles.

(Angle)

Since the sides and angles are equal among congruent triangles, it can be determined that angle DEF is equal to angle BAC

Answer

Angles BAC is equal to angle DEF.

Exercise #2

Look at the triangles in the diagram.

Determine which of the statements is correct.

343434343434555444444555AAABBBCCCDDDEEEFFF

Step-by-Step Solution

Let's consider that:

AC=EF=4

DF=AB=5

Since 5 is greater than 4 and the angle equal to 34 is opposite the larger side in both triangles, then the angle ACB is equal to the angle DEF

Therefore, the triangles are congruent according to the SAS theorem, as a result of this all angles and sides are congruent, and all answers are correct.

Answer

All of the above.

Exercise #3

Look at the triangles in the diagram.

Which of the following statements is true?

242424242424444666666444AAACCCBBBEEEFFFDDD

Step-by-Step Solution

This question actually has two steps:

In the first step, you must define if the triangles are congruent or not,

and then identify the correct answer among the options.

 

Let's look at the triangles: we have two equal sides and one angle,

But this is not the angle between them, therefore, it cannot be proven according to the S.A.S theorem

Remember the fourth congruence theorem - S.A.A
If the two triangles are equal to each other in terms of the lengths of the two sides and the angle opposite to the side that is the largest, then the triangles are congruent.

 

But the angle we have is not opposite to the larger side, but to the smaller side,

Therefore, it is not possible to prove that the triangles are congruent and no theorem can be established.

Answer

It is not possible to calculate.

Exercise #4

Are the triangles in the image congruent?

If so, according to which theorem?

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Step-by-Step Solution

Although the lengths of the sides are equal in both triangles, we observe that in the right triangle the angle is adjacent to the side whose length is 7, while in the triangle on the left side the angle is adjacent to the side whose length is 5.

Since it's not the same angle, the angles between the triangles do not match and therefore the triangles are not congruent.

Answer

No.

Exercise #5

What data must be added so that the triangles are congruent?

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Step-by-Step Solution

Let's consider that:

DF = AC = 8

DE = AB = 5

8 is greater than 5, therefore the angle DEF is opposite the larger side and is equal to 65 degrees.

That is, the figure we are missing is the angle of the second triangle.

We will examine which angle is opposite the large side AC.

ABC is the angle opposite the larger side AC so it must be equal to 65 degrees.

Answer

Angle ABC equals 65.

Exercise #1

Which of the triangles are congruent?

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Step-by-Step Solution

Let's observe the angle in each of the triangles and note that each time it is opposite to the length of a different side.

Therefore, none of the triangles are congruent since it is impossible to know from the data.

Answer

It is not possible to know based on the data.

Exercise #2

Are the triangles in the drawing congruent?

303030303030X+2X+2X+23333332X+4

Step-by-Step Solution

For triangles to be congruent, it is necessary to demonstrate that the S.A.S theorem is satisfied

We have a common side whose length in both triangles is equal to 3.

Now we will look for the lengths of the other sides:

2X+4=X+2 2X+4=X+2

We proceed with the sections accordingly:24=2XX 2-4=2X-X

2=X -2=X

We place it in the right triangle and will find the length of the side:2+2=0 -2+2=0

Since it is not possible for the length of a side to be equal to 0, the triangles are not congruent.

Answer

No

Exercise #3

What data must be added so that the triangles are congruent?

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Step-by-Step Solution

It is not possible to add data for the triangles to be congruent since the corresponding angles are not equal to each other and therefore the triangles could not be congruent to each other.

Answer

Data cannot be added for the triangles to be congruent.

Exercise #4

Look at the triangles in the diagram.

Which of the statements is true?

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Video Solution

Answer

Angle E is equal to angle B.

Exercise #5

ABCD is a kite.

E and F are extensions of diagonal BD.

semicircles are drawn with BE and FD as their bases.

BE = 2X

AF = AE

Calculate the sum of the areas marked in blue.

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Video Solution

Answer

πx2 \pi x^2

Exercise #1

ABCD is a parallelogram.

Express the area of the square GHFB in terms of X.

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Video Solution

Answer

x2 x^2