Congruence of Right Triangles (using the Pythagorean Theorem) - Examples, Exercises and Solutions

Congruent triangles are triangles that are identical in size, so if we place one on top of the other, they will match exactly.

To prove that a pair of triangles are congruent, we need to prove that they satisfy one of these three conditions:

1. SSS - Three sides of both triangles are equal in length.

2. SAS - Two sides are equal between the two triangles, and the angle between them is equal.

3. ASA - Two angles in both triangles are equal, and the side between them is equal.

If we take an initial look at the drawing, we can already see that there is one equal side between the two triangles (marked in blue),

We don't have information about the other sides, so we can rule out the first two conditions,

And now we'll focus on the last condition - angle, side, angle.

We can see that angle D equals angle A, both equal to 50 degrees,

And now we're focusing on angles E.

At first glance, we might think there's no way to know if these angles are equal, but if we look at how the triangles are positioned,
We can see that these angles are actually corresponding angles, and corresponding angles are equal.

Therefore - if the angle, side, and second angle are equal, we can prove that the triangles are equal using the ASA condition

Since we know that the triangle is isosceles, we can establish that AC=AB and that

AD=AD since it is a common side to the triangles ADC and ADB

Furthermore given that the line AD intersects side BC, we can also establish that BD=DC

Therefore, the triangles are congruent according to the SSS (side, side, side) theorem

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, the angle ACB must be 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.

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 a common angle, 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.

According to the existing data:

$EF=BA=10$(Side)

$ED=AC=13$(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

Let's consider the corresponding congruent triangles letters:

$CBO=ABO$

That is, from this we can determine:

$CB=AB$

$BO=BO$

$CO=AO$

We observe the order of the letters in the congruent triangles and write the matches (from left to right).

$ABC=CDA$

That is:

Angle A is equal to angle C.

Angle B is equal to angle D.

Angle C is equal to angle A.

From this, it is deduced that angle BAC (where the letter A is in the middle) is equal to angle C — that is, to angle DCA (where the letter C is in the middle).

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.

To answer the question, we need to know the fourth congruence theorem: S.A.S.

The theorem states that triangles are congruent when they have an equal pair of sides and an equal angle.

However, there is one condition: the angle must be opposite the longer side of the triangle.

We start with the sides:

DF = CB = 16
GD = AC = 9

Now, we look at the angles:

A = G = 120

We know that an angle of 120 is an obtuse angle and this type of angle is always opposite the larger side of the triangle.

Therefore, we can argue that the triangles are congruent according to the S.A.S. theorem.

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.

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.

In order for triangles to be congruent, one must demonstrate that the S.A.S theorem is satisfied

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

Now let's examine the lengths of the other sides:

$2X+4=X+2$

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

$-2=X$

We place this value in the right triangle we should find the length of the side:$-2+2=0$

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

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.

To answer the question, we need to know all four congruence theorems -

SAS, SAA, ASA, SSA

Now let's see what data we can prove from the question -

AB is parallel to DC

And from this it follows that angle BAC equals angle ACD, because these are equal alternate angles,

Also, we have a shared side AC, so we have a shared angle and side.

We could have proven congruence using AB=DC, and then we would have SAA,

but this is not given to us in the options,

so let's look at the fourth congruence theorem - SSA,

for it to work, we need to show another side, BC=AD,

and this is indeed one of the options!

But it's important to remember that the fourth congruence theorem has a condition,

the theorem is valid only if the angle is opposite to the larger of the two sides,

therefore we need to know that ACAC,

and indeed, one of these options exists!

Thus, we can see that there are two things we need,

and therefore answer D is correct!

We are given: AD=BC

The angle ADB is equal to the angle CBD since AD is parallel to BC and the corresponding angles are equal between parallel lines.

DB=DB since it is a common side.

Therefore, we have two triangles that are congruent according to the S.A.S. (side, angle, side) theorem.