Congruent triangles

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Compass & Straightedge Construction and the Impossible Constructions
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Congruence

Congruent is a geometric term that means the same, and is represented symbolically by \cong. Congruence is similar to equality, however, the term congruence is used for shapes. Put simply, two or more shapes, or components of shapes, are congruent if photo-copying one would yield the other. Rotating or flipping a shape does not change whether it is congruent to another shape. For instance, two 8.5" x 11" sheets of white paper are congruent, and are still so if one is flipped over or tuned sideways.


  • Line segments are said to be congruent if they have the same length.
  • Angles are said to be congruent if they have the same measurement.
  • Shapes are said to be congruent if:
    • Their corresponding sides are of the same lengths.
    • Their corresponding angles are of the same measurements.


In diagrams, such as Image 1, the congruence of two segments is denoted by making small cross marks in each of the congruent segments. If there are different sets of congruent segments, the one set is marked with one cross mark, the next, with two, and so on. In Image 1, a \cong e; b \cong f; and c \cong d.

The congruence of two angles is denoted either by making a small cross mark in little circlular segments drawn in each of the congruent angles, or by drawing the same number of segments in each angle that is congruent. These marks are also doubled, tripled, and so on depending on the number of sets of congruent angles. For example, ∠ A \cong ∠ E; ∠ B \cong ∠ F; and ∠C \cong ∠ D.


Congruent triangles

There are four basic ways to tell if triangles are congruent.

  • SSS
    • Side-Side-Side: If the measurements of the three sides of one triangle are the same as those of another, then the triangles are congruent.
  • ASA
    • Angle-Side-Angle: If two angles and the side in between them are the same as those of another triangle, then the triangles are congruent.
  • SAS
    • Side-Angle-Side: If two sides and the angle in between them are the the same as those of another triangle, then the triangles are congruent.
  • AAS
    • Angle-Angle-Side: If two angles and the side opposite of one them are the same as those of another triangle, then the triangles are congruent.


We will prove each of the above statements using the Law of Sines and the Law of Cosines. For an in depth discussion on solving triangles see: Solving Triangles.


In all of the images that follow:

  • Black will indicate sides and angles of known value.
  • Green will indicate angles of unknown measure.
  • Red will indicate sides of unknown length.



Side-Side-Side

Image 2. Side-Side-Side
Image 2. Side-Side-Side

If the measurements of the three sides of one triangle are the same as those of another, then the triangles are congruent.

Image 2 depicts the case where the lengths of all three sides of a triangle are known, while the measurements of the three angles are not. Unknown angles are shown in green. Notice that there is only one possibility for the measurement of each of the three angles that will allow the sides of the triangle to connect to each other.

Using the Law of Cosines, the measurements of the one of the angles can be found. Once the measurement of one of the angles is determined, the simpler Law of Sines can be used to find the remaining two. Sub sequentially, the measurements of all three sides and angles of the triangle will be known, and two triangles can conclusively be proven congruent.


We have just proved that it is sufficient to know that the measurements of the three sides of one triangle are the same as those of another to know that the two triangles are congruent.


Angle-Side-Angle

Image 3. Angle-Side-Angle
Image 3. Angle-Side-Angle

If two angles and the side in between them are the same as those of another triangle, then the triangles are congruent.

Image 3 illustrates that, if given the measurements of two angles and the side in between them, there is only one length possible for each of the remaining two sides such that they will connect and form a triangle. Unknown sides are shown in red, and unknown angles are shown in green. Also, given that the angles of a triangle always add to 180°, there is only one possibility for the measurement of the remaining angle between these two sides.

The unknown angle can be found by subtracting the measurements of the known angles from 180°, yielding the measurement needed so that the three angles add up to the necessary 180°.

Then the Law of Sines can be used to find the lengths of the remaining sides of the triangle. once the lengths of the two unknown sides are found, all of the elements of the triangle will be known, and two triangles can conclusively be proven congruent.


We have just proved that it is sufficient to know that two angles and the side in between them are the same as those of another triangle to know that the two triangles are congruent.


Side-Angle-Side

Image 4. Side-Angle-Side
Image 4. Side-Angle-Side

If two sides and the angle in between them are the the same as those of another triangle, then the triangles are congruent.

As in Image 4, notice that, if given the length of a side, an adjacent angle, and the other adjacent side, there is only one length possible for the remaining side of the triangle.

The length of the remaining side can be found using the Law of Cosines, after which the measurements of the unknown angles can be determined with the Law of Sines. Once these are found, the measurements of all the sides and angles will be known, and two triangles can conclusively be proven congruent.


We have just proved that it is sufficient to know that two sides and the angle in between them are the the same as those of another triangle to know that the two triangles are congruent.


Angle-Angle-Side

Image 5. Angle-Angle-Side
Image 5. Angle-Angle-Side

If two angles and the side opposite of one them are the same as those of another triangle, then the triangles are congruent.

Image 5 depicts the case where the measurements of two angles and the side opposite of one them are known. Similar to Angle-Side-Angle, we can use our knowledge that the measurements of the angles of a triangle always add to 180° to find the unknown angle. By subtracting the measurements of the known sides from 180°, we find the measurement that the remaining angle must be so that the three add to 180°.

Once the measurements of all the angles are known, the problem is reduced to an Angle-Side-Angle case. The Law of Sines can be used to find the lengths of the two unknown sides. Subsequently, the measurements of all the sides and angles will be known, and two triangles can conclusively be proven congruent.


We have just proved that it is sufficient to know that two angles and the side opposite of one them are the same as those of another triangle to know that the two triangles are congruent.


Cases Where Congruence Cannot be Determined

There are Two Cases where the congruence of two triangle cannot be determined.

  • AAA
    • Angle-Angle-Angle: If the measurements of the three angles of one triangle are the same as those of another, then the triangles cannot be determined congruent. The two triangles can, however, be proven '''similar'''.
  • SSA
    • Side-Side-Angle: If two sides and the angle opposite one of them are the same as those of another triangle, then the triangles cannot be determined congruent. This is known as the Ambiguous Case.



Angle-Angle-Angle

Image 6. Angle-Angle-Angle
Image 6. Angle-Angle-Angle


Image 6 depicts the case where the measurements of all three angles of a triangle are known, while the lengths of the three sides are not. By the Law of Sines, we know that:

\frac{a_1}{\sin \theta_{1}} = \frac{b_1}{\sin \theta_2} = \frac{c_1}{\sin \theta_3}

And

\frac{a_2}{\sin \theta_1} = \frac{b_2}{\sin \theta_2} = \frac{c_2}{\sin \theta_3}


We want to show that the ratio of the length of a1 to a2 is the same as that of b1 to b2, which is the same as that of c1 to c2. Written algebraically:

 \frac{a_1}{a_2}= \frac{b_1}{b_2}= \frac{c_1}{c_2}


We will first prove:

 \frac{a_1}{a_2}= \frac{b_1}{b_2}


We begin with the first two parts of the Law of Sines for our two triangles:

\frac{a_1}{\sin \theta_1} = \frac{b_1}{\sin \theta_2} \frac{a_2}{\sin \theta_1} = \frac{b_2}{\sin \theta_2}
Then we divide both sides by b1 and multiply both sides by sin θ1: Then we divide both sides by b2 and multiply both sides by sin θ1:
\frac{a_1}{b_1}=\frac {\sin \theta_1}{\sin \theta_2} \frac{a_2}{b_2}=\frac {\sin \theta_1}{\sin \theta_2}


Because we have two expressions that are both equal to \tfrac {\sin \theta_1}{\sin \theta_2}, they can be set equal to each other, yielding:

 \frac{a_1}{b_1}= \frac{a_2}{b_2}


Then we multiply both sides of the equation by b1, yielding:

a_1= \frac{a_2~b_1}{b_2}


Finally, we divide both sides of the equation by a2, which results in:

 \frac{a_1}{a_2}= \frac{b_1}{b_2}


We have just proven that the ratio of the length of a1 to a2 is the same as that of b1 to b2. This means that the proportions of side a1 relative to a2 are the same as b1 to b2. The above proof can also be conducted with the pair   \frac{c_1}{\sin \theta_3}   and   \frac{c_2}{\sin \theta_3} When that result is combined with the one we just arrived at, we find the following:


 \frac{a_1}{a_2}= \frac{b_1}{b_2}= \frac{c_1}{c_2}


Thus the lengths of all three sides of one triangle are proportional to the lengths of the corresponding sides of the other, and the triangles have been proven to be similar.

We have just proved that it is sufficient to know that the measurements of the three angles of one triangle are the same as those of another to know that the two are similar.

Nevertheless, we were only able to prove that the lengths of the sides are proportional, not that they were equal. Consequently, we cannot definitively conclude that the sides are the same length, meaning we cannot be certain that the triangles are congruent. This point is best illustrated by Image 6 in which the two triangles have equal corresponding angles, but the sides connecting the angles in the top triangle are larger than the sides connecting those same angles in the bottom one.

Things This Page Needs in the Future

  • Explicit proofs using Law of Sines and Law of Cosines for all of the cases, Angle-Angle-Side is covered when Angle-Side-Angle is proven.
  • Better integration with Solving Triangles and Ambiguous Case.
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