The Triangles Shown Below Must Be Congruent

Understanding Coinciding Triangles

Nosotros will talk over a number of weather condition that tin exist used to evidence that 2 triangles are congruent (that is, prove that they are the “same” triangle), and we nowadays intuitive geometric proofs for why these conditions work.

Key Terms

o


Congruent

o


Similar

o


Side-side-side (SSS) status

o


Side-angle-side (SAS) condition

o


Angle-side-angle (ASA) condition

o


Hypotenuse-leg (HL) condition

Objectives

o


Know the difference betwixt congruent and similar triangles

o


Recognize the various conditions under which 2 triangles are congruent and know how to justify these conditions

o


Learn how to prove that two triangles are coinciding

Congruence and Similarity

When analyzing groups of figures or figures composed of several smaller parts, it is sometimes helpful to evidence that 2 item figures or parts of a figure are the same or similar. Doing so, in some cases, allows united states to acquire boosted information for our analysis. In the case of triangles, nosotros can apply various techniques to show that two triangles are the aforementioned or that they are similar. If two triangles have sides of the same length and angles of the aforementioned measure, then we say that they are

congruent

triangles. The term

congruent

also applies to other figures: for case, if two line segments are the same length, they are congruent, and if two angles are of the same mensurate, they are coinciding. A pair of congruent triangles is shown below.

Notation that each side and angle of the triangle on the left has a corresponding congruent side or angle in the triangle on the right. Thus, these are congruent triangles. If two triangles only share three coinciding angles (merely non sides), then the triangles are

similar
. Two similar triangles are shown below.

Note that although the sides are of unlike lengths, the same three interior angle measures are shared by both triangles. Thus, these ii triangles are similar.

Proving Congruence

As mentioned to a higher place, coinciding triangles must have all sides of equal length and all angles of equal mensurate. Even so, we don’t need to demonstrate all six of these relationships to prove that triangles are congruent; a number of shortcuts are available. For each shortcut, we will demonstrate intuitive reasons why information technology works.

Our get-go shortcut to proving that 2 triangles (we’ll use triangles ABC and XYZ for the purposes of discussion) are congruent is called the

side-side-side (SSS) condition
. 2 triangles are congruent if their sides are all congruent. Allow’south take a little look at why this works. Consider triangle ABC below, and permit’s construct line segments that are equal to each side of ABC.

Now, consider the line segment congruent to side BC; permit’south attach the other two sides at either end of this line segment, but we’ll otherwise get out them dangling.

Now, we want to connect these two dangling sides of our newly synthetic triangle. Imagine that these sides, which are continued at the points depicted in the diagram, are able to swing freely from that signal. The dashed arcs shown in the diagram below testify where the endpoints of the dangling segments can go. Note that they can meet at only one possible indicate (there are actually two possible points, merely the other point–which is non shown–actually leads to the same issue).

So what does this mean? Given three line segments of specific lengths, merely one triangle tin can be formed. No matter how you motion around the two dangling sides, they volition only e’er form ane item triangle. As a result, if ii triangles have sides of the aforementioned length, they must be the same triangle (that is, they must be congruent)! As a consequence, triangles ABC and XYZ below are coinciding.

Our 2nd shortcut is the

side-bending-side (SAS) condition
. Two triangles are congruent if two sides and the bending between them are the aforementioned for both triangles. We can demonstrate this congruence condition with a little more ease. Consider once again some triangle ABC, and let’s copy ii sides (sides AB and BC) and the angle between them (bending B) and construct the corresponding figure.

Notation that figure XYZ corresponds exactly to the portion of the triangle ABC that omits side Ac: if we were to set XYZ on top of the triangle, they would overlap perfectly. Simply if we connect points X and Z, we cease up with a line segment that is exactly the same length as segment Ac. As such (by SSS, if you like!), the triangle that is formed, XYZ, and the triangle ABC are congruent.

Our third shortcut to proving triangle congruence is the

bending-side-bending (ASA) condition
. Two triangles are coinciding if ii angles and the included side are the same for both triangles. Allow’s accept a expect once once again at a triangle ABC, and let’s construct a congruent side with two coinciding angles.

The side XY is the same length every bit side AB, and the angles A and B take the same measures every bit angles X and Y. The dashed line segments are defined by the angles. Two non-parallel and non-coincident lines only intersect at one betoken, so two angles and an included side determine one and only one triangle.

As a result, triangle XYZ and triangle ABC are congruent.

Our last full general shortcut for proving congruent triangles is the

angle-angle-side (AAS) status
. If two triangles share two angles of the same measure every bit well as one side (not included by the angles) of the same measure out, the triangles are congruent. Let’s consider our triangle ABC and let’s copy two of the angles and a non-included side.

The dashed lines in figure XYZ indicate lines whose lengths we exercise not however know. Withal, because of the angles that must exist as shown, we know that the remaining sides of the triangle must exist parallel to (in the instance of YZ) or coincident with (in the case of XZ) the dashed lines. Let’s extend the line segment from Z. Although it doesn’t intersect Y, we can and then accommodate the length of XZ until it does.

Then,

What we have shown here is that simply one length of segment XZ volition form a triangle, given side XY and angles 10 and Z. As a result, two angles and a not-included side make up one’s mind exactly ane triangle, and therefore, triangles ABC and XYZ above must be congruent.

These illustrations should help demonstrate why each of these iv full general conditions proves the congruence of triangles in general. A special instance for proving congruence involves right triangles: the

hypotenuse-leg (HL) condition
. Think that if we know ii sides of a correct triangle, nosotros tin use the Pythagorean theorem to calculate the length of the third side. As a consequence, if two right triangles both have a hypotenuse and a leg of the aforementioned lengths, the remaining leg must be the same length for both triangles as well. Then, by our SSS condition, the two triangles must be congruent.

Practice Problem:


Using the four general conditions for triangle congruence (SSS, SAS, ASA, SSA), prove the HL condition.

Solution:


We can apply our techniques of geometric reasoning to solve this problem. Let’due south use a two-column proof format to clearly outline our reasoning, and let’s first by drawing 2 triangles, ABC and XYZ. Notice that we label the hypotenuses XZ and Air-conditioning as having length

h
, and the shared legs AB and XY as having length

50
.

The ii-column proof format would then exist equally follows (note that we utilise the symbol



to point congruence).

1. AB

XY Given

two. AC

XZ Given

3. BC =



Pythagorean theorem

four. YZ =



Pythagorean theorem

5. BC

YZ Statements 3 and 4

six. ABC

XYZ SSS condition (statements 1, 2, and 5)

Notation that for statements 3 and 4 we used the Pythagorean theorem to algebraically calculate the lengths of BC and YZ; even though we do not have stock-still numbers, the expressions are still identical given

h

and

l
. Thus, nosotros take proven, using one of our four general conditions (SSS), that the HL condition is a valid fashion to prove the congruence of two correct triangles.

Practice Problem:


Prove that triangles MNO and MPO are congruent.

Solution

:


Once more, allow’s use a two-cavalcade proof to illustrate our reasoning. First, nosotros know that the line segment MO is congruent with itself (thus, triangles MNO and MPO accept a congruent side-we tin call this belongings

identity
). We also know that the triangles take two congruent angles that share the segment MO: angles PMO (which we tin can write as


) and NMO (which we tin can write as


), as well equally angles NOM and POM (


and


).

1. MO

MO Identity

2.



Given (in figure)

iii.



Given (in effigy)

4.


ASA condition

Thus, we have proven using ASA that the two triangles are coinciding. (Note that in statement 4, we use the triangle symbol to betoken a triangle, as opposed to an bending.)

Exercise Problem:


Bold line segments AB and DC are parallel and sides AD and BC are parallel, prove that triangles ABC and ACD are congruent.

Solution

:


Because this figure involves parallel line segments, nosotros can apply what we know about parallel lines cut by transversals. The corresponding angles inside the figure are shown below.

1.



Identity

ii.



Corresponding angles

3.



Corresponding angles

4.


ASA condition

Thus, nosotros run across how nosotros tin apply a number of dissimilar geometric concepts that we accept studied thus far to evidence relationships betwixt figures.