Difference between revisions of "Congruent (geometry)"
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Without loss of generality, we can derive <math>\triangle ABC\cong\triangle AB'C</math> from the fact that <math>C</math> is strictly between <math>A</math> and <math>D</math>. | Without loss of generality, we can derive <math>\triangle ABC\cong\triangle AB'C</math> from the fact that <math>C</math> is strictly between <math>A</math> and <math>D</math>. | ||
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=== SAS Congruence === | === SAS Congruence === | ||
The SAS Congruence theorem is derived from the sixth axiom of congruence. In short, the sixth axiom states that when given two triangles, if two corresponding side congruences hold and the angle between the two sides is equal on both triangles, then the other two angles of the triangle are equal. | The SAS Congruence theorem is derived from the sixth axiom of congruence. In short, the sixth axiom states that when given two triangles, if two corresponding side congruences hold and the angle between the two sides is equal on both triangles, then the other two angles of the triangle are equal. | ||
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=== ASA Congruence === | === ASA Congruence === | ||
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=== AAS Congruence === | === AAS Congruence === | ||
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Since the third angle of a triangle is always the difference of <math>180^\circ</math> and the other two angles, two triangles with two pairs of congruent angles would give another pair of congruent angles. When we have a pair of congruent sides and two pairs of congruent angles adjacent to the side, it is ASA congruence. However, if we're given a pair of congruent angles opposite from the side that is congruent and a pair of congruent angles adjacent to that side, everything is congruent. Therefore, AAS congruence can be proven with ASA congruence. | Since the third angle of a triangle is always the difference of <math>180^\circ</math> and the other two angles, two triangles with two pairs of congruent angles would give another pair of congruent angles. When we have a pair of congruent sides and two pairs of congruent angles adjacent to the side, it is ASA congruence. However, if we're given a pair of congruent angles opposite from the side that is congruent and a pair of congruent angles adjacent to that side, everything is congruent. Therefore, AAS congruence can be proven with ASA congruence. | ||
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=== HL Congruence === | === HL Congruence === | ||
− | If | + | If both the hypotenuse and leg of one right triangle are congruent to that of another, the two triangles are congruent. |
Consider right <math>\triangle ABC</math> and right <math>\triangle XYZ</math> shown in the diagram below. We are given that <math>AB=XY</math> and <math>AC=XZ</math>. | Consider right <math>\triangle ABC</math> and right <math>\triangle XYZ</math> shown in the diagram below. We are given that <math>AB=XY</math> and <math>AC=XZ</math>. | ||
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draw(rightanglemark(X,Y,Z)); | draw(rightanglemark(X,Y,Z)); | ||
</asy> | </asy> | ||
− | Since <math>\angle B=90^\circ</math>, the Pythagorean theorem gives us <math>AB^2+ | + | Since <math>\angle B=90^\circ</math>, the Pythagorean theorem gives us <math>AB^2+BC^2=AC^2</math>. Similarly, using the Pythagorean theorem on <math>\triangle XYZ</math> gives us <math>XY^2+YZ^2=XZ^2</math>. However, since <math>AB=XY</math> and <math>AC=XZ</math>, substitution in the second equation gives <math>AB^2+YZ^2=AC^2</math>. Subtracting this from our first equation plus a little manipiulation gives us <math>AC^2=YZ^2</math>. Therefore, since all lengths are positive, taking the square root of both sides gives <math>AC=YZ</math>, so, by the SSS congruence theorem, we have <math>\triangle ABC\cong\triangle XYZ</math>. |
Since the congruent angle given is not between the two equivalent sides, this may be seen as SSA congruence, which is not necessarily correct. However, this form of SSA congruence holds true for right triangles. | Since the congruent angle given is not between the two equivalent sides, this may be seen as SSA congruence, which is not necessarily correct. However, this form of SSA congruence holds true for right triangles. | ||
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=== LL Congruence === | === LL Congruence === |
Latest revision as of 21:43, 24 March 2022
Congruency is a property of multiple geometric figures.
Contents
Intuitive Definition
Two geometric figures are congruent if one of them can be turned and/or flipped and placed exactly on top of the other, with all parts lining up perfectly with no parts on either figure left over. In plain language, two objects are congruent if they have the same size and shape.
Technical Definition
Two geometric objects are congruent if one can be transformed into the other by an isometry, such as a translation, rotation, reflection or some combination thereof.
Axioms
- If
,
are two points on a straight line
, and if
is a point upon the same or another straight line
, then, upon a given side of
on the straight line
, we can always find one and only one point
so that the segment
(or
) is congruent to the segment
. We indicate this relation by writing
Every segment is congruent to itself; that is, we always have
- If a segment
is congruent to the segment
and also to the segment
, then the segment
is congruent to the segment
; that is, if
and
, then
.
- Let
and
be two segments of a straight line
which have no points in common aside from the point
, and, furthermore, let
and
be two segments of the same or of another straight line
having, likewise, no point other than
in common. Then, if
and
, we have
.
- Let an angle
be given in the plane
and let a straight line
be given in a plane
. Suppose also that, in the plane
, a definite side of the straight line
be assigned. Denote by
a half-ray of the straight line
emanating from a point
of this line. Then in the plane
there is one and only one half-ray
such that the angle
, or
, is congruent to the angle
and at the same time all interior points of the angle
lie upon the given side of
. We express this relation by means of the notation
Every angle is congruent to itself; that is,
or
- If the angle
is congruent to the angle
and to the angle
, then the angle
is congruent to the angle
; that is to say, if
and
, then
.
- If, in the two triangles
and
the congruences
hold, then the congruences
also hold.
Triangle Congruence
SSS Congruence
If the three sides of one triangle are congruent to the corresponding sides of of another triangle, then congruence between the two triangles is established.
We start with and
shown in the diagram below where
,
, and
supposing that
.
We construct point
on the opposite side of
from
such that
. Since
is on the opposite side of
from
, segment
must intersect the line
at some point
. Updating our diagram gives us
We now have several cases depending on the location of point
relative to
and
.
is strictly between
and
, as shown in the diagram above.
Therefore,
must be in the interiors of both
and
, but since
and
, we have
and
. From this, we find that
and
are isosceles, and therefore
and
, so
. We have
by SAS congruence.
is strictly between
and
.
Without loss of generality, we can derive
from the fact that
is strictly between
and
.
This article is a stub. Help us out by expanding it.
SAS Congruence
The SAS Congruence theorem is derived from the sixth axiom of congruence. In short, the sixth axiom states that when given two triangles, if two corresponding side congruences hold and the angle between the two sides is equal on both triangles, then the other two angles of the triangle are equal.
This article is a stub. Help us out by expanding it.
ASA Congruence
This article is a stub. Help us out by expanding it.
AAS Congruence
Since the third angle of a triangle is always the difference of and the other two angles, two triangles with two pairs of congruent angles would give another pair of congruent angles. When we have a pair of congruent sides and two pairs of congruent angles adjacent to the side, it is ASA congruence. However, if we're given a pair of congruent angles opposite from the side that is congruent and a pair of congruent angles adjacent to that side, everything is congruent. Therefore, AAS congruence can be proven with ASA congruence.
This article is a stub. Help us out by expanding it.
HL Congruence
If both the hypotenuse and leg of one right triangle are congruent to that of another, the two triangles are congruent.
Consider right and right
shown in the diagram below. We are given that
and
.
Since
, the Pythagorean theorem gives us
. Similarly, using the Pythagorean theorem on
gives us
. However, since
and
, substitution in the second equation gives
. Subtracting this from our first equation plus a little manipiulation gives us
. Therefore, since all lengths are positive, taking the square root of both sides gives
, so, by the SSS congruence theorem, we have
.
Since the congruent angle given is not between the two equivalent sides, this may be seen as SSA congruence, which is not necessarily correct. However, this form of SSA congruence holds true for right triangles.
LL Congruence
LL Congruence is the basically the same as SAS congruence since we are given a leg, a right angle, and the other leg.