Download Chapter 4 Notes - Stevenson High School

Name
Date
Hour
Geometry
Chapter 4 Notes
Ch.




4 Prerequisite Skills
I can classify triangles based on sides and angles
I can apply the Triangle Sum Theorem to solve problems.
I can apply the Exterior Angle Sum Theorem to solve problems.
I can apply properties of isosceles and equilateral triangles to solve problems.
G-CO.10. A. Prove measures of interior angles of a triangle sum to 180°.
 I can prove a theorem that the interior angles of a triangle sum to 180°.
G-CO.7. Use the definition of congruence to show that two triangles are congruent if and only if
corresponding pairs of sides and corresponding pairs of angles are congruent.
 I can identify corresponding angles and sides of two triangles.
 I can write a congruence statement for two congruent figures.
G-SRT.5.A. Use congruence criteria for triangles to solve problems and to prove relationships in
geometric figures.
 I can apply the concepts of congruence to solve problems.
 I can prove relationships in geometric figures (i.e. prove triangles and corresponding parts are
congruent).
 I can determine and justify when two triangles are congruent (including ASA, SAS, AAS, HL and
SSS Theorems).
G-CO.10.B. Prove base angles of isosceles triangles are congruent.
Chapter 4 Proofs
Proof Writing Process/Hints
Here is a process that you should be thinking through for each proof that was modeled in class.
1. Write what is given.
Be sure to mark those in the diagram using tick marks or arcs in the angles.
2. Determine what you know from what they give you. If they give you one of these, use the table below to continue.





Midpoint – show two segments are congruent (mark them)
Angle bisector – show two angles are congruent (mark them)
Segment bisector – show two segments are congruent (mark them).
Segments are perpendicular – show right angles are formed, then show right triangles (mark them.)
Parallel segments – show alternate interior angles are congruent (mark them).
3. Determine what else the diagram shows.


Look for vertical angles (mark them)
Look for angles/segments that are in both triangles. Use the reflexive property to show they are congruent to
themselves.
4. Use all the markings to determine which congruence postulate to use. (ASA, AAS, SAS, SSS, HL). Write that the two
triangles are congruent (order of vertices is critical). The postulate is the reason.
5. If you are going beyond congruent triangles to show angles or segments are congruent (it will be in the “prove” part),
then use CPCTC.
Proof Reasons – Memorize!
USE IF MIDPOINT, BISECOR,
PERPENDICULAR OR
PARALLEL IS IN GIVEN
Reasons to use in proofs
Definition of Midpoint
Alternate Interior Angle
Theorem
Definition of perpendicular
segments
When you would use
Given states that a point is a midpoint.
 Next step you write shows which segments are = and the reason is
“definition of midpoint.” (Don’t forget to mark them on your diagram
with tick marks.)
Given states that two lines are parallel.
 Draw in the arc marks for the alternate interior angles to identify them.
Then write that the two angels are congruent in the step and “Alternate
Interior Angle Theorem” as the reason.
Given states that two lines are perpendicular.
 Next write that the angles (use letters to show which ones) are right
angles. Use “definition of perpendicular segments” as the reason.
 Next you must follow with the triangles are right triangles (use letters to
specify which triangle). Use “definition of right triangle” as the reason.
Definition of segment
bisector
FIND THESE IN
DIAGRAM
Definition of angle bisector
Reflexive property of
congruence
Vertical Angle Theorem
Triangle congruence
postulates:
ASA, AAS, SSS, SAS, HL
CPCTC: Corresponding
Parts of Congruent
Triangles are Congruent
 (Be sure to mark the right angles in the diagram.)
Given states that a segment is bisected by another segment.
 Next write the segments that are congruent. Use “definition of
segment bisector” as the reason. (Be sure to mark the segments
congruent in the diagram.”
Given statement states that an angle is bisected by a segment.
 Next write the angles that are congruent. Use “definition of angle
bisector” as the reason. (Be sure to mark the angles congruent in the
diagram.”
Diagram shows that one segment is the side of two different triangles. Or it
shows that one angle is included in two different triangles.
 Write the segments or angles that are congruent (it will be the same
segment or angle letters). Then write “reflexive property” as the
reason.
(Don’t forget to mark the sides or angles congruent on your diagram.)
Diagram shows vertical angles.
 State the congruent angles and the reason is “Vertical Angle Theorem”
(Don’t forget to mark them on your diagram with arcs.)
Once your angles and sides are marked up, then decide which postulate to
use to show the triangles are congruent. Write the congruence statement
(using triangle marks and letters) for the step and in the reason write one of
the abbreviations (ASA, SAS, AAS, SSS, or HL)
Use when the “prove” stays to prove either angles or segments are
congruent – first use all the above to show that two triangles are congruent
using the postulates above. Then, write the angle or segment that are
congruent as the step and “CPCTC” as the reason.
4.1—Triangles and Angles
PSa
I can classify triangles based on sides and angles
PSb
I can apply the Triangle Sum Theorem to solve problems.
PSc
I can apply the Exterior Angle Sum Theorem to solve problems.
G-CO.10. A.
I can prove a theorem that the interior angles of a triangle sum to 180°.
Triangle:
Triangle Angle Sum Theorem
The sum of the measures of the interior angles
of a triangle is _______.
mA + mB + mC = ________
Interior Angles vs. Exterior Angles
B
B
A
C
C
A
Proof of Triangle Angle Sum Theorem
Statements
B

1
m
A
4
2 3
5
C
Reasons
1.  m
1.
2. m1  m2  m3  180
2.
3. m1  m4; m5  m3
3.
4. m4  m2  m5  180
4.
Triangle Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the
measures of the two __________________ ___________________ angles.
m3 = _____________________
Example One:
Find the measure of 2 in each triangle below.
a)
b)
2
3
1
Example Two:
Write an equation and solve for x.
Then find the measure of the exterior
angle.
(3x+7)°
51°
(6x–11)°
Classifying Triangles
Classification by Sides:
Scalene
Isosceles
Classification by Angles:
Right
Equilateral
Obtuse
Acute
Equiangular
Example Three:
Classify each of the following triangles as specific as possible.
a.
b.
15
c.
17
65°
8
d.
e.
28°
89°
22°
Example Four:
The expressions below represent the measures of the interior angles of ΔABC.
Write an equation and solve for x. Then find the measure of each angle and classify the triangle by
its angle name.
m A  (2x + 5)°
m B  (3x – 15)°
m C
 (4x + 10)°
4.2—Congruence and Triangles
G-CO.7.
 I can identify corresponding angles and sides of two triangles.
 I can write a congruence statement for two congruent figures.
G-SRT.5. A.
I can apply the concepts of congruence to solve problems.
3rd Angles Theorem
1) Find the measure of
C .
2) Find the measure of
F .
3) What do you notice about the measure of
C
and
F ?
This demonstrates the 3rd Angles Theorem, which states:
If two angles in one triangle are _________________________ to two angles in another triangle,
then the _______________________ pair of angles must also congruent.
Congruence and Triangles
Two geometric figures are congruent if they have the same size and shape.
When two figures are congruent, there is a correspondence between their angles and sides such that corresponding
angles are congruent and corresponding sides are congruent.
When naming congruent triangles, you must name them in corresponding order.
Congruence Statement:
Corresponding Parts
Example One:
Corresponding Angles
Corresponding Sides
BAC  EDF
AB  DE
_____  DEF
___  EF
BCA  _____
AC  ___
Give the congruence statement three different ways.
E
B
D
A
C
F
Given ∆ABC  ∆RST, list all of the corresponding sides and angles that are congruent.
Corresponding Angles
Corresponding Sides
Example Two:
Example Three:
If ∆ABC  ∆TUV, then complete the following statements.
a. A 
U
B
59°
b. VT 
V
c. VTU 
8 cm
55° C
D
d. BC =
T
A
e. mA 
Example Four:
∆KJL  ∆DCB Solve for x and y.
K
25
J
B
34
35°
L
Write the congruence statement for the
triangles.
A
B
(4x–20)°
85°
D
5y–5
C
C
4.6—Isosceles & Equilateral Triangles
PSd
I can apply properties of isosceles and equilateral triangles to solve problems.
Properties of Isosceles and Equilateral Triangles
Base Angles Theorem
If two ________ of a triangle are congruent, then the _________ opposite them are congruent.
A
If AB  AC , then ABC  ACB .
B
C
Converse of the Base Angles Theorem
If two ___________ of a triangle are congruent, then the ________ opposite them are congruent.
A
If ABC  ACB , then AB  AC
B
Example One:
Solve for x and/or y in each of the following.
a.
b.
y
5x + 3
28
43
x°
17
Equilateral Triangle Theorem
A triangle is _________________ if and only if it is __________________.
Example Two:
Solve for x and/or y in each of the following.
a.
b.
3y–13
4x°
y+9
C
4.3—Proving Triangles are Congruent: SSS and SAS
G-SRT.5. A.
Theorems).
I can determine and justify when two triangles are congruent (including ASA, SAS, AAS, HL and SSS
Included Angles
1. Name the included angle between the pair of sides given.
a) GK & KJ
b)
KJ & JG
c)
HG & GJ
G
H
K
J
d) GJ & JH
SIDE-SIDE-SIDE (SSS) POSTULATE
If three _________ of one triangle are congruent to three _________ of a second triangle, then the two triangles are
_______________.
If Side
AB  DE
Side
BC  EF
Side
AC  DF
B
E
A
Then Δ______  Δ_______
C D
F
SIDE-ANGLE-SIDE (SAS) POSTULATE
If the two sides and the _______________ angle of one triangle are _________________ to two sides and the
________________ angle of a second triangle, then the two triangles are __________________.
Angle
PQ  WX
PQS  WXY
Side
QS  XY
If Side
Then Δ_______  Δ________
Q
P
X
S
W
Y
Always check touching triangles for these “hidden” congruencies:
Overlapping Side
G
I
H
H
K
Examples:
Congruent Vertical Angles
Parallel Lines
G
H
K
J
K
J
J
L
Decide whether it is possible to prove the triangles are congruent. List all of the congruent parts and then state
which postulate we would use to prove the triangles are congruent. Write the triangle congruent statement if
we have enough information. If we cannot prove the triangles congruent, then write “Not Enough Information.”
2.
Congruent Parts
A
Reason : ____________
1.
2.
B
C
D
M
N
3.
3.
Triangle Congruence Statement
Δ_______  Δ________
Congruent Parts
I
Reason : ____________
1.
K
J
2.
L
3.
O
4.
Triangle Congruence Statement
Δ_______  Δ________
Congruent Parts
Reason : ____________
1.
R
P
2.
3.
Triangle Congruence Statement
Δ_______  Δ________
Q
5.
G
H
Congruent Parts
Reason : ____________
1.
K
J
2.
3.
Triangle Congruence Statement
Δ_______  Δ________
4.4—Proving Triangles are Congruent: ASA, AAS, and HL
G-SRT.5. A.
Theorems).
I can determine and justify when two triangles are congruent (including ASA, SAS, AAS, HL and SSS
Included Sides
Name the included side between the pair of angles given.
a) WZX and ZXW
b) YWX and WXY
X
c) WXY and XYW
d) ZXW and XWZ
e) XWZ and WZX
W
Y
Z
ANGLE-SIDE-ANGLE (ASA) POSTULATE
If two angles and the ___________________ side of one triangle are congruent to two angles and
the ___________________ side of a second triangle, then the two triangles are _____________.
If Angle
BAC  EDF
Side
Angle
AC  DF
BCA  EFD
Then Δ______  Δ_______
B
E
A
C D
F
ANGLE-ANGLE-SIDE (AAS) POSTULATE
If the two angles and a ___________________ side of one triangle are congruent to two angles and a
___________________ side of a second triangle, then the two triangles are __________________.
If Angle
QPS  XWY
Angle
PQS  WXY
Side
QS  XY
Then Δ_______  Δ________
Q
P
X
S
W
Y
HYPOTENUSE–LEG (HL) CONGRUENCE THEOREM
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then
D
A
the two triangles are congruent.
If Hypotenuse
Leg
AC  DF
AB  DE
Then Δ_______  Δ________
Examples:
B
C
E
F
Decide whether it is possible to prove the triangles are congruent. List all of the congruent parts and then state
which postulate we would use to prove the triangles are congruent. Write the triangle congruent statement if
we have enough information. If we cannot prove the triangles congruent, then write “Not Enough Information.”
2.
Congruent Parts
P
Reason : ____________
1.
2.
Triangle Congruence Statement
Δ_______  Δ________
3.
R
S
Q
3.
Congruent Parts
Reason : ____________
1.
2.
Triangle Congruence Statement
Δ_______  Δ________
3.
4.
Congruent Parts
Reason : ____________
1.
2.
Triangle Congruence Statement
Δ_______  Δ________
3.
5.
K
L
Congruent Parts
Reason : ____________
1.
N
M
2.
Triangle Congruence Statement
3.
Δ_______  Δ________
4.5—Using Congruent Triangles
G-SRT.5. A.
 I can determine and justify when two triangles are congruent (including ASA, SAS, AAS, HL and SSS
Theorems).
 I can prove relationships in geometric figures (i.e. prove triangles and corresponding parts are congruent).
1. Given the diagram below, list the congruent parts marked in the diagram. What additional information is
needed to prove the two triangles congruent by AAS?
Q
A
S
A
P
R
S
T
Complete the two column proof.
Statements
2. Given: U is the midpoint of RS
UQR  UTS
Prove: UQR  UTS
S
Q
T
U
Reasons
1. U is the midpoint of RS
1.
2.
2. Definition of Midpoint
3. UQR  UTS
3.
4.
4. Vertical Angles Theorem
5. UQR  UTS
5.
R
Statements
3. Given: UR  ST
 URS is a right angle
 UTS is a right angle
Prove: ∆RSU

UR  ST
 URS is a right angle
 UTS is a right angle
1.
∆TUS
U
T
2. ∆RSU and ∆TUS are right Δs
3.
4. ∆RSU
R
Reasons
1.
2.
3. Reflexive Property

∆TUS
4.
S
Statements
4. Given: PA  PC
PK bisects APC
1. PA  PC
PK bisects APC
Prove: ∆PAK  ∆PCK
A
2.
K
P
C
Reasons
1.
2. Definition of Angle Bisector
3.
3. Reflexive Property
4.
4.
4.5—Using Congruent Triangles cont.
G-SRT.5. A.
 I can determine and justify when two triangles are congruent (including ASA, SAS, AAS, HL and SSS
Theorems).
 I can prove relationships in geometric figures (i.e. prove triangles and corresponding parts are congruent).
G-CO.10. B.
I can prove base angles of isosceles triangles are congruent.
Complete the two column proof.
3. Given:
Prove:
1.
BAD  CDA
BA  CD
∆ BAD  ∆ CDA
B
Statements
BAD  CDA
BA  CD
2.
C
3. ∆ BAD
Reasons
1.
2. Reflexive Property
 ∆ CDA
3.
.
E
A
D
Recall that when two figures are congruent, all of their corresponding sides and angles are also congruent.
After we have shown that two triangles are congruent and want to state that a corresponding part is
congruent, the reason is:
Statements
Example 1 Using CPCTC
Reasons
1.
1.
2. MN  OP
2.
3. NP  PN
3.
4.
4.
5. MP  ON
5.
Example 2 Using CPCTC
Statements
Reasons
1.
1.
2.
2. Given
3. ACB  ECD
3.
4.
4.
5.
B  D
5.
Example 3 Using CPCTC
Statements
1.
1.
2.
2. Given
ABC  EDC
3.
4. ACB  ECD
4.
5.
5. Definition of Bisects
6. ACB   ______
6.
7.
7.
3.
Statements
Example 4 Using CPCTC
Proof of the Base Angles Theorem
Given: AB  AC , AM bisects A
Prove: B  C
Reasons
Reasons
1.
AB  AC ,
AM bisects A
1. Given
2.
2.
3.
AM  AM
4.
ABM   ______
5.
B  C
3.
4.
5.