Download Unit 5 notes congruence

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5.1 Apply Congruence to Triangles
Vocabulary
Congruent figures
Congruent
Similar
Example 1: Identify congruent parts
1. Identify all pairs of congruent corresponding parts. Then write a congruence statement.
Sides
Angles
_______ ≌ _______
_______ ≌ _______ Congruence Statement
__________ ≌ __________
_______ ≌ _______
_______ ≌ _______
_______ ≌ _______
_______ ≌ _______
Example 2: Use properties of congruent figures
1. DEFG≌ SPQR. Find the values of x and y.
WRITING A CONGRUENCE STATEMENT
1. Pick a marked angle (or mark two ≅ angles)
2. Name 1st triangle in any order (start with the marked angle)
3. Match the second triangle (start with the corresponding marked angle)
Example 3: Congruence statements
1. Write a congruence statement
2. Write a congruence statement
for the congruent triangles
for the congruent triangles
below.
below.
E
R
C
S
W
D
M
O
P
B
Z
3. Choose the correct congruence statement for
the congruent Δ’s. R
J
H
T
M
A. ΔMYJ ΔRTH
C. ΔJYM
K
NOTE: Order matters!!
ΔHTR
Y
B. ΔYJM
ΔRHT
D. ΔMYJ
ΔHRT
4. Write a congruence statement for the two congruent triangles shown below. Then solve for x.
1
5 Different Methods to Prove Triangles are Congruent
SSS
ASA
AAS
Whats the difference between these?
Simple
and easy…
SAS
HL (RASS)
What doesn’t work???
The 1 exception to the
_______ ________
**So basically…if you’ve got 3 pairs of ≅ angles or sides you’ve got congruent triangles except
for two exceptions: ________ and ________
Example 4: How are the triangles congruent? Write a congruence statement.
1.
̅̅̅̅
is the midpt. of ̅̅̅̅,
̅̅̅̅ , ̅̅̅̅ ̅̅̅̅
2. ̅̅̅̅
̅̅̅̅ ,and
Q
S
T
3.
57°
D
28°
A
C
Q
S
For SIDES
G
N
T
K
N
U
I
K
B
D
̅̅̅̅ bisects ̅̅̅̅
D
L
M
C
Therefore:
Angle Bisector
B
K
E
B
Therefore:
G
M
P
D
L
L
O
C
H
Therefore:
Therefore:
Therefore:
Alternate Interior Angles ≅
Q
N
̅̅̅̅ bisects
F
C
Vertical angles
A
J
E
M
J
Right Angle ≅ TH
Reflexive Property
A
D
C
Therefore:
D
Y
̅̅̅ ,
̅̅̅̅,
H
Tips to Help You Prove Triangles are Congruent…
J
J
Midpoint
Bisect
Reflexive property L
A
is the
midpoint of ̅̅̅̅
K
̅̅̅̅, ̅̅̅̅
̅̅̅̅ , ̅̅̅̅
R
Therefore:
For ANGLES
4. ̅̅̅̅
̅̅̅̅
,
O
P
E
,
̅̅̅̅,
̅̅̅̅
B
T
P
R
̅̅̅̅ , ̅̅̅̅
A
Corresponding Angles ≅
D
W
V
T
B
U
S
Therefore:
S
R
Therefore:
Therefore:
2
U
T
Example 4: Decide whether the triangles are congruent. If so, write a congruence
statement. Name all postulates or theorems used to reach your conclusion.
2.
,
, and
3.
and
1. ̅̅̅̅ ̅̅̅̅ ,
̅̅̅̅
E
F
G
̅̅̅̅
B
H
Q
C
S
A
Q
R
P
R
4. ̅̅̅̅
̅ and ̅̅̅̅
H
̅̅̅̅
5. ̅̅̅̅
6. ̅̅̅̅ bisects ̅̅̅̅ , ̅̅̅̅
̅̅̅̅
A
K
S
̅̅̅̅
R
B
S
Q
J
C
D
L
O
P
7.̅̅̅̅̅
̅̅̅̅, ̅̅̅̅̅
̅̅̅̅
8. ̅̅̅̅̅
̅̅̅̅ , ̅̅̅̅
̅̅̅̅ , ̅̅̅̅̅
9.
̅̅̅̅
W
L
K
̅̅̅̅
and
are right angles,
̅̅̅̅ , and ̅̅̅̅ ̅̅̅̅,
T
T
N
V
R
F
M
U
X
E
H
10. ̅̅̅̅ bisects
̅̅̅̅
̅̅̅̅
11.
is the midpt. of ̅̅̅̅; ̅̅̅̅̅
̅̅̅̅ ,
12. ̅̅̅̅̅
̅̅̅̅ ,
and
angles
W
X
Y
Z
are right
1. Are there any orders of sides and angles that DON’T prove triangles are congruent?
2. Circle which methods below COULD be used to prove two triangles are congruent?
A. Prove all three corresponding angles are congruent.
B. Prove that two angles and their included side are congruent.
C. Prove all three corresponding sides are congruent.
D. Prove two corresponding sides and one pair of corresponding angles are congruent.
Similar VS Congruent?
1. Which if the following is true about the following
2. Which if the following is true about the following
triangles.
triangles.
A. similar but not congruent
A. similar but not congruent
B. congruent but not similar
B. congruent but not similar
C. both similar and congruent
C. both similar and congruent
D. neither similar nor congruent
D. neither similar nor congruent
3
5.2 Prove Triangles Congruent by SSS, SAS, and HL
SIDE-SIDE-SIDE Congruence Postulate (SSS)
If
Side ̅̅̅̅
If three sides of one triangle are
congruent to three sides of a second
triangle, then the two triangles are
congruent.
Side ̅̅̅̅
Then ________ ≌ _________
Example 1: Use SSS Congruence Postulate
1. Given: ̅̅̅ ̅̅̅̅
Statements
is the midpoint of ̅̅̅̅
1. ̅̅̅ ̅̅̅̅ is the midpoint of ̅̅̅̅
Prove: ∆FGJ ≌ ∆HGJ
2. Given: ̅̅̅̅ ̅̅̅̅ ; ̅̅̅̅
Prove: ∆ABC ≌ ∆DCB
Side ̅̅̅̅
̅̅̅̅
Reasons
1.
Statements
Reasons
1.
1.
When ∆’s overlap, look for
______________________
SIDE-ANGLE-SIDE Congruence Theorem (SAS)
If two sides and the included angle of one triangle are congruent to
two sides and the included angle of a second triangle, then the two
triangles are congruent.
The angle must be right in between
the two sides!
If
Side
̅̅̅̅
Angle
Side
̅̅̅̅
,
Then ________ ≌
Example 2: SAS vs. Potty Mouth
Decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence
Postulate. Please state any theorems or postulates you use.
a. True or False: ∆PQT ≌ ∆RQS?
b. True or False: ∆WXY ≌ ∆ZXY?
c. True or False: ∆NKJ ≌ ∆LKM?
W
X
Y
Z
4
Example 3: Proving triangles congruent using SAS
1. Given: ̅̅̅̅ ̅̅̅̅̅, ̅̅̅̅ bisects ̅̅̅̅
Statements
Prove: ∆
≌ ∆
1. ̅̅̅̅ ̅̅̅̅ , ̅̅̅̅ bisects ̅̅̅̅
Reasons
1.
The ONE Exception to Potty Mouth…
HYPOTENUSE-LEG Congruence Theorem (HL) <also known as “RASS”>
If
If the hypotenuse and a leg of a right triangle
are congruent to the hypotenuse and a leg of
a second triangle, then the two triangles are
congruent.
Right Angle
Side (Hyp)
Side (leg)
A& D
are right s
̅̅̅̅
̅̅̅̅
,
Then ________ ≌
Example 4: SAS vs HL
Which pairs of triangles are congruent and why?
A
1.
2.
M
6
X
3.
6
I
4. Given: ̅̅̅̅ ̅̅̅̅ ,
JUN and UNK are rt. s
Prove: ∆JUN ≌ ∆KNU
J
K
U
Statements
1. ̅̅̅̅
Reasons
̅̅̅̅ , , ̅̅̅̅ bisects ̅̅̅̅
1.
N
QUICK REFRESHER: Perpendicular Lines
a
1 2
b
4
right angles.
Statements
Reasons
1. a  b
1. Given
2.
2.
3.
3.
5
Example 5
1. Given: ̅̅̅̅
̅̅̅̅
Prove: ∆
̅̅̅̅ ̅̅̅̅
̅̅̅̅
∆
̅̅̅̅ ;
2. Given: ̅̅̅ ̅̅̅̅;
is the midpt of ̅̅̅;
is a right angle; ̅̅̅
Prove: ∆
∆
3. Given: ̅
̅̅̅̅̅ ̅̅̅ ̅̅̅̅;
is a rt. angle;
is the midpoint of ̅̅̅̅
Prove ∆
∆
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
Statements
̅̅̅
1. ̅̅̅
̅̅̅̅; is the midpt of ̅̅̅;
is a right angle; ̅̅̅ ̅̅̅
Reasons
1.
2.
2.
3.
3.
4.
4.
5.
5.
Statements
1. ̅ ̅̅̅̅̅ ̅̅̅ ̅̅̅̅;
is a rt.
Angle; is the midpoint of ̅̅̅̅
6
Reasons
1.
5.3 Prove Triangles Congruent by ASA and AAS
ANGLE-SIDE-ANGLE Congruence Postulate (ASA)
If
If two angles and the included side of one
triangle are congruent to two angles and
the included side of a second triangle, then
the two triangles are congruent.
Angle
Side
̅̅̅̅
Angle
,
Then ________ ≌
ANGLE-ANGLE-SIDE Congruence Theorem (AAS)
If two angles and a non-included side of
one triangle are congruent to two angles
and the corresponding non-included side of
a second triangle, then the two triangles are
congruent.
If
Angle
Angle
Side
̅̅̅̅
,
Then ________ ≌
Example 1: Can the triangles be proven congruent with the information given in the diagram? If so,
write a congruence statement and state the postulates/theorems used to reach your conclusion.
a.
b.
c.
Example 2: Proving Triangles similar using ASA and AAS
1. Given: D ≌ R; ̅̅̅̅ ̅̅̅̅̅
Statements
Prove: ∆DOM ≌ ∆RMO
1.
O
R
D
Reasons
1.
M
BHL ≌ AHL;
̅̅̅̅ bisects BLA
Prove: ∆BLH≌ ∆ALH
L
2. Given:
1.
Statements
BHL ≌ AHL; ̅̅̅̅ bisects BLA
A
B
H
7
Reasons
1.
MORE PROOFS (Yay!)
1. Given: ̅̅̅̅
Prove: Δ
̅̅̅̅ , ̅̅̅̅
̅̅̅̅, and ̅̅̅̅ bisects ̅̅̅̅
Δ
S
M
̅̅̅̅, ̅̅̅̅
H
F
J
1. ̅̅̅̅
2.
Statement
̅̅̅̅, ̅̅̅̅ ̅̅̅̅, and ̅̅̅̅ bisects ̅̅̅̅
̅̅̅̅, ̅̅̅̅
3.
4.
4.
5.
F
G
3. Given: ̅̅̅̅
Prove: ∆
1. Given
2.
3.
2. Given: ̅̅̅̅ ̅̅̅̅, ̅̅̅̅ bisects
Prove: ∆
∆
E
1. ̅̅̅̅
Reason
5.
Statement
̅̅̅̅, ̅̅̅̅ bisects
Reason
1. Given
H
̅̅̅̅ ̅̅̅̅
∆
̅̅̅̅ ; ̅̅̅̅
1. ̅̅̅̅
̅̅̅̅
̅̅̅̅ ̅̅̅̅
Statement
̅̅̅̅; ̅̅̅̅ ̅̅̅̅
Reason
1. Given
4. Decide whether it is possible to show the given triangles are congruent. If it is possible, write a congruence statement
and tell which congruence postulate or theorem you used.
a. Given: ̅̅̅̅ bisects PQR
b. Given: ̅̅̅̅
F
18
̅̅̅̅
c. Given:
E
A
A
C
T
B
18
B
U
8
L
1) If you are given two triangles, Δ
and Δ
,
where
and ̅̅̅̅ ̅̅̅̅ , what additional
information would not be sufficient to prove Δ
Δ
?
A.
B. ̅̅̅̅ ̅̅̅̅
C. ̅̅̅̅ ̅̅̅̅
D.
and
2) Given ̅̅̅̅ ̅̅̅̅, ̅̅̅̅ ̅̅̅̅, is the midpoint of ̅̅̅̅.
Which theorem or postulate can be used to prove
Δ
Δ
?
A
T
A. ASA
B. HL
C. SSA
P
Y
R
D. SAS
are right angles
3) Given ̅̅̅̅ ̅̅̅̅, ̅̅̅̅ ̅̅̅̅ , and
. Which
theorem or postulate can be used to prove Δ
Δ
?
L
O
A. SSA
B. SAS
C. ASA
E
H
D. HL
4) Given ̅̅̅̅ bisects
and ̅̅̅̅
correct congruence statement.
5) If you are given two triangles, Δ
and Δ
,
where ̅̅̅̅̅ ̅̅̅̅ and ̅̅̅̅ ̅̅̅̅, what additional
information would be sufficient to prove Δ
Δ
?
6) If ∆
and ∆
are right triangles and ̅̅̅̅
what theorem or postulate proves ∆
∆
A. Δ
B. Δ
C. Δ
D. Δ
̅̅̅̅ , choose the
E
Δ
Δ
Δ
Δ
S
L
G
̅̅̅̅ ,
?
B
A.
B.
and
are right angles
C. ̅̅̅̅̅ ̅̅̅̅
D.
A.
HL
B.
SAS
C.
SSS
D.
ASA
A
C
D
7) If ̅̅̅̅ ̅̅̅̅ and
, which congruence
postulate or theorem would prove ∆GHJ ∆GKJ? ?
8) Choose the correct congruency statement given the
triangles below.
.
H
A.
SAS
B.
SSS
C.
HL
D.
AAS
J
G
K
9
a. ∆
∆
b. ∆
∆
c. ∆
∆
d. ∆
∆
5.4 Coordinate Geometry Review
The MIDPOINT FORMULA
The DISTANCE (length) FORMULA
Example 1
1. Find the midpoint of a segment with endpoints
and
2. Find the length of the segment whose endpoints are
) and
3. Find the midpoint of ̅̅̅̅ given
G
and
4.
is the midpoint of ̅̅̅̅.
) is one
endpoint. Find the coordinates of the other endpoint D.
SLOPES OF PARALLEL AND PERPENDICULAR LINES
PARALLEL lines have the _______ slopes
PERPENDICULAR lines have slopes whose product is ______
Hint: 2 changes
m1
m2
, then find the slope of line n.
3. Which statement would prove that triangle POQ is a
right triangle?
m2
1)
2)
Example 2
1. Given line n is parallel to line k and line k has a slope
of
m1
2. Given line j line p and line p line m. If the slope of
line j is 12, then find the slope of line m.
4.
A. slope ̅̅̅̅ = slope̅̅̅̅
B. slope ̅̅̅̅ = slope̅̅̅̅
C. (slope ̅̅̅̅)(slope ̅̅̅̅ ) = 1
D. (slope ̅̅̅̅)(slope ̅̅̅̅ ) =
10
is a rectangle. Find the slope of ̅̅̅̅
5.5 Use Congruent Triangles
Opening Question: What is the length of OD if ∆
∆
CPCTC :
?
∆
∆
therefore:
____ ____
____ ____
____ ____
____ ____
____ ____
____ ____
**When to use CPCTC** Can only be used after you know (or prove) that the triangles are congruent.
Ex:
Statements
1. ∆MAN≌∆GAL
2. M≌ G
Example 1: Using CPCTC
1. Given: B is the midpoint ̅̅̅̅ ;
Prove: ̅̅̅̅
Statements
1. B is the midpoint ̅̅̅̅ ;
̅̅̅̅
A
Reasons
1. (SSS, ASA, etc.)
2. CPCTC
Reasons
1.
C
B
D
E
2. Given: ̅
Prove:
J
̅̅̅̅
Statements
L
Reasons
K
M
3. Given: ̅̅̅̅ ̅̅̅̅,
Prove: ̅̅̅̅ ̅̅̅̅
Statements
A
D
C
B
11
Reasons
4. Given: ̅̅̅̅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅
Prove: ADB ≌ CBD
D
A
̅̅̅̅
1. ̅̅̅̅
Statements
̅̅̅̅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅
Reasons
̅̅̅̅
1. given
C
B
Note: If it doesn’t ask you to
prove triangles are congruent…
1st: Prove ∆
∆
2nd:Use CPCTC!
5. Given: Q ≌ S; ̅̅̅̅ ̅̅̅̅; ̅̅̅̅
Prove: R is the midpoint of ̅̅̅̅
P
̅̅̅̅
1.
R
Statements
Q ≌ S; ̅̅̅̅ ̅̅̅̅,̅̅̅̅
Reasons
̅̅̅̅
1. given
Q
S
T
6. Given: ̅̅̅̅̅ ≌ ̅̅̅̅;̅̅̅̅̅ ≌ ̅̅̅̅
Prove: W ≌ K
O
R
W
Statements
̅̅̅̅̅
̅̅̅̅
̅̅̅̅̅
1.
≌
;
≌ ̅̅̅̅
Reasons
1. given
K
7. Given: ̅̅̅̅ ̅̅̅̅;̅̅̅̅ ̅̅̅̅;
̅̅̅̅ bisects ̅̅̅̅; ̅̅̅̅ ≌ ̅̅̅̅
Prove: ∆ANG≌∆EGL
A
N
E
G
Statements
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅; ̅̅̅̅ bisects ̅̅̅̅;
1.
;
̅̅̅̅ ≌ ̅̅̅̅
L
12
Reasons
1. given
8. Given: ̅̅̅̅ bisects ̅̅̅̅̅ ,
L ≌ W, PL = 6
Prove: WK = 6
K
Statements
1. ̅̅̅̅ bisects ̅̅̅̅̅ , L ≌ W, PL = 6
Reasons
1. given
L
A
W
P
9. Given: ̅̅̅̅
̅̅̅̅ ̅̅̅̅
̅̅̅̅,
1.̅̅̅̅̅
Prove:
Statements
̅̅̅̅ ̅̅̅̅ ̅̅̅̅;
Reasons
1. given
A
E
B
C
D
10. Given: ̅̅̅̅ ̅̅̅, ̅̅̅̅
Prove:
= 12
̅̅̅̅̅,and ̅̅̅
̅̅̅̅̅,
1. ̅̅̅̅
J
R
= 12
Statements
̅̅̅, ̅̅̅̅ ̅̅̅̅̅,and ̅̅̅
= 12
G
A
M
13
Reasons
̅̅̅̅̅,
1. given
5.6 Multiple Choice + Spiral Review
SIMILAR VS CONGRUENT
There are _____ ways to prove triangles are SIMILAR:
There are ____ ways to prove triangles are CONGRUENT:
_________, _________, _________
________, ________, ________, _______, ________.
*Note: Need 3 congruent parts (except no _____________)
NOTE: If two triangles are CONGRUENT, then they are also _________.
Example 1
1. Which of the following is true about the triangles below?
N
E
46°
2. Which of the following is true about the triangles below?
R
U
L
46°
U
M
B
A. similar but not congruent
B. congruent but not similar
C. both similar and congruent
D. neither similar nor congruent
3. Which of the following is true about the triangles below?
U
K
Y
P
A. congruent but not similar
5 T 5
B. both similar and congruent
C. neither similar nor congruent
D. similar but not congruent
4. Which of the following is true about the triangles
below?
G
2
Y
36°
36°
6
C
6
M
E
Spiral Review
is the midpoint of ̅̅̅̅̅,
1.
U
4
A. similar but not congruent
B. neither similar nor congruent
C. congruent but not similar
D. both similar and congruent
A. neither similar nor congruent
B. both similar and congruent
C. congruent but not similar
D. similar but not congruent
5
– ,
Find
2.
K
10
T
is the midpoint of ̅̅̅̅̅. Which of the
following is an appropriate statement?
A.
B.
C.
D.
3. (a) Given line j is perpendicular to line k and the slope of
line k is -8. Find the slope of line j.
4. Find the distance between the given coordinates (-2, 46)
and (13, 10)
(b) If the slope of line a is 3, find the slope of line b.
a
b
14
5. Find the length between Rand T given R(-g, j) and
6. If R(-6, -4) is one endpoint of ̅̅̅̅ and M(1, -8) is the
midpoint of ̅̅̅̅, then find the coordinates of T.
A. √
A. (8, -20)
B. (4, -12)
C.(8, -12)
D.(4, -20)
B. √
C.√
D.√
7. Which information below would help prove that
is a right angle?
I
K
A. (slope IM) = (slope MY)
B. (slope IM) = (slope MY)
C. (slope IM) (slope MY) = 1
D. (slope IM) (slope MY) = -1
M
Y
Trig ratio
45° -45° - 90°
30° - 60° - 90°
RIGHT TRIANGLES
1. Which equation should be used to find the length of
̅̅̅̅ ?
A
2. Which equations would help find MT?
H
A.
A.
B.
C.
B.
12
42
36°
M
C.
F
D.
D.
3. An equilateral triangle has an altitude of √ .
(a) Find the perimeter
(b) Find the area
4. A square has a perimeter of 40 inches.
(a) Find the length of the diagonals.
22°
(b) Find the area of the square.
15
M
T
Mini-Review
1. In the diagram, QRST ≌ WXYZ. Find the values of x and y.
2.Choose the correct congruence statement that
states the triangles below are congruent.
A. ∆
B. ∆
C. ∆
D. ∆
R
T
∆
∆
∆
∆
C
3. (a) What does CPCTC stand for?
A
(b) Can you use CPCTC before or after you prove triangles
are congruent?
4. Determine whether or not it is possible to prove the triangles are congruent. If it is possible, then write a congruence
statement and tell which congruence postulate or theorem you used.
c) Given:
and
b) Given: ̅̅̅̅ bisects ̅̅̅̅.
a) Given:
, ̅̅̅̅ ̅̅̅̅
J
C
E
R
5
5
T
∆
___ ∆
5 .Given: ̅̅̅̅̅ bisects
, ̅̅̅̅̅
∆
̅̅̅̅ ,
A
N
Y
A
∆
___ ∆
Prove:
6. Given: ̅̅̅̅̅ ̅̅̅̅̅
is the midpoint of ̅̅̅̅
Prove: ∆
∆
16
___ ∆
I
17