Download Geometry Notes - Unit 4 Congruence

Geometry Notes - Unit 4
Congruence
Triangle is a figure formed by three noncollinear points.
Classification of Triangles by Sides
Equilateral triangle is a triangle with three congruent sides.
Isosceles triangle is a triangle with at least two congruent sides
Scalene triangle is a triangle with no congruent sides.
Classification of Triangles by Angles
Acute triangle is a triangle with three acute angles.
Right triangle is a triangle with one right angle.
Obtuse triangle is a triangle with one obtuse angle.
Adjacent sides of a triangle are two sides sharing a common vertex.
Adjacent sides
Common Vertex
Hypotenuse of a right triangle is the side opposite the right angle.
Hypotenuse
Right Angle
pg. 1 If I asked an entire class to draw a triangle on a piece of paper, then had each person cut out their triangle, we might
see something interesting happen.
Let’s label the angles 1, 2, and 3 as shown.
2
3
1
By tearing each angle from the triangle, then placing them side by side, the three angles always seem to form a
straight line.
3
2
1
That might lead me to believe the sum of the interior angles of a triangle is 180º.
While that’s not a proof, it does provide me with some valuable insights. The fact is, it turns out to be true, so we
write it as a theorem.
Triangle Sum Theorem:
The sum of the interior angles of a triangle is 180º.
Example
Find the measure of ∠3.
60°
50°
Since the sum of the two angles given is 110˚, ∠3 must be 70˚.
3
pg. 2 Drawing a triangle and cutting out the angles suggests the sum of the interior angles is 180˚ is not a proof. Let’s see
what that proof might look like.
Theorem Proof - The sum of the measures of the angles of a triangle is 180º.
F
R
4
2
S
5
Given: ΔDEF
Prove: m ∠1 + m ∠2 + m ∠3 = 180°
3
1
E
D
The most important part of this proof will be our ability to use the geometry we have already learned. If
we just looked at the three angles of the triangle, we’d be looking for an awfully long time without much
to show for it. What we will do is use what we just learned – we were just working with parallel lines,
so what we will do is put parallel lines into our picture by constructing RS parallel to DE and labeling
the angles formed. Now we have parallel lines being cut by transversals, we can use our knowledge of
angle pairs being formed by parallel lines.
Statements
Reasons
1.
ΔDEF
Given
2.
HJJG
Draw RS &DE
Construction
3.
∠4 & ∠DFS are supp.
Ext. sides of 2 adj. ∠'s form a line
4.
m ∠4 + m ∠DFS = 180°
Def Supp ∠'s
5.
m ∠DFS = m ∠2 + m ∠5
∠ Add Postulate
6.
m ∠4 + m ∠2 + m ∠5 = 180°
Substitution
7.
8.
∠1 ≅ ∠4
∠3 ≅ ∠5
m ∠1 + m ∠2 + m ∠3 = 180°
2 & lines cut by trans., alt. int. ∠'s ≅
Substitution
pg. 3 Notice how important it is to integrate our knowledge of geometry into these problems. Step 3 would not
have jumped out at you. It looks like ∠4, ∠2 & ∠5 form a straight angle, but we don’t have a theorem to
support that so we have to look at ∠4 & ∠DFS first.
A theorem that seems to follow directly from that theorem is one about the relationship between the exterior angle of
a triangle and angles inside the triangle. If we drew three or four triangles and labeled in their interior angles, we
would see a relationship between the two remote interior angles and the exterior angle.
Theorem:
The exterior ∠ of a ∆ is equal to the sum of the 2 remote interior ∠'s
C
Given:
ΔABC
Prove: m ∠1 = m ∠A + m ∠C
2
1
B
A
Statements
Reasons
1.
m ∠A + m ∠C + m ∠2 = 180°
Int ∠ ‘s of ∆ = 180˚
2.
∠1 & ∠2 are supp. ∠'s
Ext. sides 2 adj. ∠'s
3.
m ∠1 + m ∠2 = 180°
Def. supp. ∠'s
4.
m ∠A + m ∠C + m ∠2 = m ∠1 + m ∠2
Substitution
5.
m ∠A + m ∠C = m ∠1
Subtr. Prop. of Equality
Corollary – Triangle Sum Theorem:
The acute angles of a right triangle are complementary.
pg. 4 30°
Example)
Find the value of x.
The sum of the interior angles is 180˚
3x°
(2x + 10)°
30˚ + (2x + 10)˚ + (3x)˚ = 180˚
Example)
(combine like terms)
5x + 40 = 180
(subtract 40)
5x = 140
(divide by 5)
x = 28˚
Find the value of x.
60°
x
(2x + 10)°
Exterior angle is equal to the sum of the two remote interior angles
(2x+10)˚ = x˚ + 60˚
(subtract x)
x + 10 = 60
(subtract 10)
x = 50˚
Congruence
In math, the word congruent is used to describe objects that have the same size and shape. When you
traced things when you were a little kid, you were using congruence. Neat, don’t you think?
Stop signs would be examples of congruent shapes. Since a stop sign has 8 sides, they would be
congruent octagons.
We are going to look specifically at triangles. To determine if two triangles are congruent, they must
have the same size and shape. They must fit on top of each other, they must coincide.
Mathematically, we say all the sides and angles of one triangle must be congruent to the corresponding
sides and angles of another triangle.
pg. 5 A
D
B
C
E
F
In other words, we would have to show angles A, B, and C were congruent ( ≅ ) to angles D, E and F,
then show AB , BC , and AC were ≅ to DE , EF , and DF respectively.
We would have to show those six relationships.
Ah, but there is good news. If I gave everyone reading this three sticks of length 10”, 8”, and 7”, then
asked them to glue the ends together to make triangles, something interesting happens. When I collect
the triangles, they all fit very nicely on top of each other, they coincide. They are congruent!
Why is that good news? Because rather than showing all the angles and all the sides of one triangle are
congruent to all the sides and all the angles of another triangle (6 relationships), I was able to determine
congruence just using the 3 sides. A shortcut!
That leads us to the Side, Side, Side congruence postulate.
SSS Postulate:
If three sides of one triangle are congruent, respectively, to three sides of another
triangle, then the triangles are congruent.
A
B
D
C
E
F
pg. 6 Using the same type of demonstration as before, we can come up with two more congruence postulates.
The Side, Angle, Side postulate is abbreviated SAS Postulate.
SAS Postulate:
If two sides and the included angle of one triangle are congruent to two sides and
the included angle of another triangle, respectively, then the two triangles are
congruent.
A
B
D
C
E
F
A third postulate is the Angle, Side, Angle postulate.
ASA Postulate:
If two angles and the included side of one triangle are congruent to the
corresponding parts of another triangle, the triangles are congruent.
A
B
D
C
E
F
If you are going to be successful, you need to memorize those three postulates and be able to visualize
that information.
Combining this information with previous information, we will be able to determine if triangles are
congruent. So study and review!
pg. 7 Proofs: Congruent
Δ 's
To prove other triangles are congruent, we’ll use the SSS, SAS and ASA congruence postulates. We also need
to remember other theorems that will lead us to more information.
For instance, you should already know by theorem the sum of the measures of the interior angles of a triangle is
180º. A corollary to that theorem is if two angles of one triangle are congruent to two angles of another
triangle; the third angles must be congruent. OK, that’s stuff that makes sense. Let’s write it formally as a
corollary.
Corollary:
If 2 angles of one triangle are congruent to two angles of another triangle, the third angles are
congruent.
Using that information, let’s try to prove this congruence theorem.
If two angles and the non included side of one triangle are congruent to the corresponding
parts of another triangle, the triangles are congruent.
AAS Theorem:
First let’s draw and label the two triangles.
C
A
F
B
D
E
Given: ∠A ≅ ∠D , ∠C ≅ ∠F , AB ≅ DE
Prove: ΔABC ≅ ΔDEF
pg. 8 Statements
Reasons
∠A ≅ ∠D ,
∠C ≅ ∠F ,
1.
Given
AB ≅ DE
∠B ≅ ∠E
2.
2 ∠'s of a Δ congruent 2
∠'s of another Δ , 3rd
∠'s congruent
ΔABC ≅ ΔDEF
3.
ASA
Now we have 4 ways of proving triangles congruent: SSS, SAS, ASA, and AAS. You need to know these.
Here’s what you need to be able to do. First, label congruences in your picture using previous knowledge.
After that, look to see if you can use one of the four methods (SSS, SAS, ASA, AAS) of proving triangles
congruent. Finally, write those relationships in the body of proof. You are done. Piece of cake!
A
Given: AC & BD
AB bisects CD
X
Prove: ΔACX ≅ ΔBDX
D
C
Remember to mark up your picture with that given information as I did.
B
A
X
D
C
B
pg. 9 Statements
1.
AC & BD
AB bisects CD
Reasons
Given
2.
CX ≅ DX
Def. of bisector
3.
∠C ≅ ∠D
& lines cut by trans., alt. int. ∠'s are ≅
4.
∠AXC ≅ ∠BXD
Vert. ∠'s are ≅
5.
ΔACX ≅ ΔBDX
ASA
I can’t tell you how important it is to fill in the picture by labeling the information given to you and writing
other relationships you know from previous theorems, postulates, and definitions.
In order to prove those triangles congruent, we had to know the definition of a bisector and the subsequent
mathematical relationship. And, even though vertical angles were not part of the information given, we could
see from the diagram vertical angles were formed and we could then use the theorem that all vertical angles are
congruent.
Proofs: CPCTC
When two triangles are congruent, each part of one triangle is congruent to the corresponding part of the
other triangle. That’s referred to as Corresponding Parts of Congruent Triangles are Congruent, thus
CPCTC.
One way you can determine if two line segments or two angles are congruent is by showing they are the
corresponding parts of two congruent triangles.
1. Identify two triangles in which the segments or angles are the corresponding parts.
2. Prove the triangles are congruent.
3. State the two parts are congruent, supporting the statement with the reason; “corresponding parts of
congruent triangles are congruent”.
pg. 10 **That reason is usually abbreviated “CPCTC”.
Let’s see if we can prove two line segments are congruent.
D
A
P
B
C
Given:
AB and CD bisect each other
Prove:
AD ≅ BC
Now the strategy to prove the segments are congruent is to first show the triangles are congruent.
So let’s fill in the picture showing the relationships based upon the information given and other relationships
that exist using our previously learned definitions, theorems, and postulates.
D
A
P
C
B
Using the definition of bisector, I can determine that AP ≅ PB and AP ≅ PB . I also notice I have a pair of vertical
angles. Notice how I marked the drawing to show these relationships. Even though the vertical angles are marked in
my diagram, I must write the statement in my proof.
Let’s go ahead and fill in the body of the proof.
pg. 11 Statements
Reasons
1. AB and CD bisect each other
Given
2. AP ≅ PB , CP ≅ PD
Def. of bisector
3. ∠APD ≅ ∠PBC
Vert. ∠'s ≅
4. ΔAPD ≅ ΔPBC
SAS
5. AD ≅ BC
CPCTC
Filling in the body of the proof is easy after you mark the congruences, in your picture.
The strategy to show angles or segments are congruent is to first show the triangles are congruent, then use CPCTC.
Try this one on your own.
C
Given: AC ≅ BC , AX ≅ BX
1 2
A
X
Prove: ∠1 ≅ ∠2
B
C
1 2
Mark the picture with the parts that are congruent based on what’s
given to you.
Then mark the relationships based upon your knowledge of
geometry.
A
X
B
In this case CX ≅ CX
pg. 12 From the picture we can see three sides of one triangle are congruent to three sides of another triangle, therefore
the triangles are congruent by SSS Congruence Postulate.
ΔAXC ≅ ΔBXC
If the triangles are congruent, then the remaining corresponding parts of the triangles are congruent by
CPCTC. That means ∠1 ≅ ∠2 , just what we wanted to prove.
We can develop quite a few relationships based upon knowing triangles are congruent. Let’s look at a
few.
If 2 ∠'s of ∆ are ≅ , the sides opposite those ∠'s are ≅ .
Theorem:
Given: ΔABC , ∠A ≅ ∠B
C
Prove: AC ≅ BC
A
B
C
1 2
Statements
Reasons
1.
Draw ∠ bisector CX
Construction
2.
∠1 ≅ ∠2
Def. ∠ bisector
3.
∠A ≅ ∠B
Given
4.
CX ≅ CX
Reflexive Prop.
5.
ΔCAX ≅ ΔCBX
AAS
6.
AC ≅ BC
CPCTC
A
B
X
pg. 13 The converse of that theorem is also true.
Theorem:
If 2 sides of a triangle are congruent, the angles opposite those sides are
congruent.
Theorem:
A diagonal of a & ogram separates the & ogram into 2 ≅ Δ 's .
Given:
.RSTW
Prove:
ΔRST ≅ ΔTWR
T
W
1
4
2
R
Statements
3
S
Reasons
1.
RSTW is a & ogram
Given
2.
RS &WT
Def. of & ogram
3.
∠1 ≅ ∠2
2 & lines cut by trans., alt. int. ∠'s ≅
4.
RT ≅ RT
Reflexive
5.
RW &ST
Def. of & ogram
6.
∠3 ≅ ∠4
2 & lines cut by trans., alt. int. ∠'s ≅
7.
ΔRST ≅ ΔTWR
ASA
Knowing the diagonal separates a parallelogram into 2 congruent triangles suggests some more
relationships.
pg. 14 Theorem:
The opposite sides of a parallelogram are congruent.
Theorem:
The opposite angles of a parallelogram are congruent.
Both of those can be proven by adding CPCTC to the last proof.
Many students mistakenly think the opposite sides of a parallelogram are equal by definition. That’s not
true, the definition states the opposites sides are parallel. This theorem allows us to show they are also
equal or congruent.
The idea of using CPCTC after proving triangles congruent by SSS, SAS, ASA, and ASA will allow us
to find many more relationships in geometry.
Right Triangles
While the congruence postulates and theorems apply for all triangles, we have postulates and theorems
that apply specifically for right triangles.
HL Postulate:
If the hypotenuse and leg of one right triangle are congruent to the hypotenuse
and leg of another right triangle, then the triangles are congruent.
Since the HL is a postulate, we accept it as true without proof. The other congruence theorems for right
triangles might be seen as special cases of the other triangle congruence postulates and theorems.
LL Theorem:
If two legs of one right triangle are congruent to two legs of another right triangle,
the triangles are congruent.
That’s a special case of the SAS Congruence Theorem and could be considered redundant.
HA Theorem:
If the hypotenuse and an acute angle of one right triangle are congruent to the
hypotenuse and acute angle of another right triangle, the triangles are congruent.
pg. 15 This congruence theorem is a special case of the AAS Congruence Theorem and can also be considered
redundant.
And finally, we have the Leg Angle Congruence Theorem.
LA Theorem:
If a leg and an acute angle of one right triangle are congruent to a leg and an acute
angle of another right triangle, the triangles are congruent.
If you drew and labeled the picture of the LA Congruence Theorem, you would see that could be
derived from either the ASA or AAS congruence theorems depending on which set of legs or angles are
congruent. Once again these theorems could be considered redundant.
As marked in this first set, ASA.
As marked in this second set, AAS.
pg. 16