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

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

In order 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 observe that there is one equal side between the two triangles, as they are both marked in blue,

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

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

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

Let's proceed to the angles E.

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

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

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).

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.