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1
Unit 4 Day by Day
Day Sections and Objectives Homework
Monday
October 26
U4D1
4.2 and 4.9 Packet Pages 1-3
Types of triangles, isosceles and
equilateral triangles
Page 228 (23-31, 35-37)
Page 288 (5-10, 17-20, 22-26)
Wednesday
October 28
½ Day
U4D2
4.3 and 4.4 Packet Pages 4-7
Sum of interior angles of a triangle
Exterior Angel Theorem
Congruent Triangles
Page 236 (19-24, 41-44)
Page 242 (11, 17-19, 23-25, 31-34)
Friday
October 30
U4D3
4.5-4.7 Packet Pages8-15
SSS, SAS, AAS, ASA, HL, CPCTC
Finish Packet Pages 11-15
Tuesday
November 3
U4D4
Quiz 4.2-4.4 and 4.9
Review
Page 247 (5-17
Page 288 (13-16, 42-44)
Wednesday
November 4
U4D5
Review Packet Pages8-15
Friday
November 6
U4D6
Quiz 4.2-4.9
Review
Tuesday
November 10
U4D7
Test Unit 4
None
2
Chapter 4 – Congruent Triangles
4.2 and 4.9 – Classifying and Angle Relationships within Triangles.
Isosceles triangles are triangles with two congruent sides.
The two congruent sides are called legs.
The third side is the base.
The two angles at the base are called base angles.
3
Match the letter of the figure to the correct vocabulary word in Exercises 1–4.
1. right triangle __________
2. obtuse triangle __________
3. acute triangle __________
4. equiangular triangle __________
Classify each triangle by its angle measures as acute, equiangular, right, or obtuse. (Note: Give two
classifications for Exercise 7.)
5. 6. 7.
For Exercises 8–10, fill in the blanks to complete each definition.
8. An isosceles triangle has ____________________ congruent sides.
9. An ____________________ triangle has three congruent sides.
10. A ____________________ triangle has no congruent sides.
Classify each triangle by its side lengths as equilateral, isosceles, or scalene. (Note: Give two
classifications in Exercise 13.)
11. 12. 13.
Find the side lengths of the triangle.
14. AB ____________________ AC ___________________ BC ______________
15. Given: ABC is isosceles with base AB ; EBDA
Prove: EBCDAC
16. In isosceles ∆PQR, P is the vertex angle. If mQ = 8x – 4 and mR = 5x + 14, find the mP.
17. In isosceles triangle CAT, C is the vertex angle. If A = 8x – 4 and mT = 5x + 14, then what is the
measure of C?
4
3. mX __________ 4. BC __________ mA __________
5. PQ __________ 6. mK __________ t __________
5
4.3 and 4.4 – Angle Relationships and Congruent Triangles.
The interior is the set of all points inside the figure. The exterior is the set of all points
outside the figure.
An interior angle is formed by two sides of a triangle.
An exterior angle is formed by one side of the triangle and extension of an adjacent side.
Each exterior angle has two remote interior angles. A remote interior angle is an
interior angle that is not adjacent to the exterior angle.
6
Congruent Triangles: Two ’s are if their vertices can be matched up so that corresponding angles and sides of the ’s are .
Congruence Statement: RED FOX
List the corresponding ’s: corresponding sides:
R ___ RE ____
E ___ ED ____
D ___ RD ____
Examples:
1. The two ’s shown are .
a) ABO _____ b) A ____
c) AO _____ d) BO = ____
2. The pentagons shown are .
a) B corresponds to ____ b) BLACK _______
c) ______ = mE d) KB = ____ cm
e) If CA LA , name two right ’s in the figures.
3. Given BIG CAT, BI = 14, IG = 18, BG = 21, CT = 2x + 7. Find x.
The following ’s are , complete the congruence statement:
4. YWZ_______
5. MQN _______
6. WTA ________
Parts of a Triangle in terms of their relative positions.
7. Name the opposite side to C.
8. Name the included side between A and B.
9. Name the opposite angle to BC .
D C
O
B A
B
L A
C
K
H
O
R
S
Y X
Z W M N O
P Q W
A
C
H T
A
B C
4 cm
E
7
10. Name the included angle between AB and AC .
State whether the pairs of figures are congruent. Explain.
Exterior Angles: Find each angle measure.
37. mB ___________________ 38. mPRS ___________________
39. In LMN, the measure of an exterior angle at N measures 99.
1m
3L x
and 2
m3
M x . Find mL, mM, and mLNM. ____________________
40. mE and mG __________________ 41. mT and mV ___________________
42. In ABC and DEF, mA mD and mB mE. Find mF if an exterior
angle at A measures 107, mB (5x 2) , and mC (5x 5) . _______________
43. The angle measures of a triangle are in the ratio 3 : 4 : 3. Find the angle measures of the triangle.
____________________
44. One of the acute angles in a right triangle measures 2x°. What is the measure of the other acute angle?
___________________
8
45. The measure of one of the acute angles in a right triangle is 63.7°. What is the measure of the other acute angle?
_______________________
46. The measure of one of the acute angles in a right triangle is x°. What is the measure of the other acute angle?
_________________________
47. Find mB 48. Find m<ACD
49. Find mK and mJ 50. Find m<P and m<T
Use the figure at the right for problems 1-3.
1. Find m3 if m5 = 130 and m4 = 70.
2. Find m1 if m5 = 142 and m4 = 65.
3. Find m2 if m3 = 125 and m4 = 23.
Use the figure at the right for problems 4-7.
4. m6 + m7 + m8 = _______.
5. If m6 = x, m7 = x – 20, and m11 = 80,
then x = _____.
6. If m8 = 4x, m7 = 30, and m9 = 6x -20,
then x = _____.
7. m9 + m10 + m11 = _______.
For 8 – 12, solve for x.
8.
9.
1 3
5 2 4
6
8
7 10
11
9
120
x°
(5x)°
x° 140°
35°
9
4.5 – 4.7 Proving Triangles are congruent Ways to Prove ’s :
SSS Postulate: (side-side-side) Three sides of one are to three sides of a second ,
Given: AS bisects PW ; AWPA
SAS Postulate: (side-angle-side) Two sides and the included angle of one are to two sides
and the included angle of another .
Given: PX bisects AXE; XEAX
ASA Postulate: (angle-side-angle) Two angles and the included side of one are to two angles
and the included side of another .
Given: MHAT
THMA
//
//
AAS Theorem: (angle-angle-side) Two angles and a non-included side of one are to two
angles and a non-included side of another .
Given: CAtsbiUZ sec
ZAUZCUUZ ;
HL Theorem: (hypotenuse-leg) The hypotenuse and leg of one right are to the hypotenuse
and leg of another right .
Given: FCAT
Isosceles FAC with legs ACFA,
CPCTC: Corresponding parts of congruent triangles are congruent
A
P W S
A
X
P
E
A
M
T
H
C
R U Z
A
A
F T C
10
∆’s SSS, SAS, ASA, AAS, or HL
State which congruence method(s) can be used to prove the ∆s . If no method applies, write none. All markings must
correspond to your answer.
11
Fill in the congruence statement and then name the postulate that proves the ∆s are . If the ∆s are not , write “not possible” in second blank. (Leave first blank
empty) *Markings must go along with your answer**
12
#1 Given: USUTSRUTSR ;//; Prove: UVST //
1. USUTSRUTSR ;//; 1. _____________________________
2. 1 4 2. __________________________________________
3. ∆RST ∆TUV 3. __________________________________________
4. 3 2 4. __________________________________________
5. UVST // 5. __________________________________________
#2 Given: D is the midpoint of CBCAAB ; Prove: CD bisects ACB.
1. D is the midpoint of CBCAAB ; 1. _________________________________________
2. DBAD 2. __________________________________________
3. CDCD 3. __________________________________________
4. ∆ACD ∆BCD 4. __________________________________________
5. 1 2 5. __________________________________________
6. CD bisects ACB. 6. __________________________________________
#3 Given: AR≅ AQ; RS ≅ QT Prove: AS ≅ AT
1. AR≅ AQ; RS ≅ QT 1. ________________________
2. <R <Q 2. __________________________________________
3. ARS AQT 3. __________________________________________
4. AS ≅ AT 4. __________________________________________
13
Fill in Proofs:
#1
Given: AB CB
AC BD
Prove: Δ ADB Δ CDB
1. AB CB 1. _________________________________________________
2. AC BD 2. _________________________________________________
3. 1 & 2 are right ’s. 3. _________________________________________________
4. 1 2 4. _________________________________________________
5. BD BD 5. _________________________________________________
6. Δ ADB Δ CDB 6. _________________________________________________
#2
Given: AC BD
BD bisects ADC
Prove: AB CB
1. AC BD 1. _________________________________________________
2. 1 & 2 are right ’s 2. _________________________________________________
3. 1 2 3. _________________________________________________
4. BD BD 4. _________________________________________________
5. BD bisects ADC 5. _________________________________________________
6. 3 4 6. _________________________________________________
7. Δ ADB Δ CDB 7. _________________________________________________
8. AB CB 8. _________________________________________________
B
3 4
1 2
D
A C
D
B
3 4
1 2 A C
14
Congruent Triangles Proofs
1. Given: SP ; O is the midpoint of PS
Prove: O is the midpoint of RQ
2. Given: ABCD ; D is the midpoint of AB
Prove: CBCA
3. Given: KRSNNRSK //;//
Prove: KRSNNRSK ;
4. Given: MEADMEAD ;//
M is the midpoint AB
Prove: EBDM //
5. BONUS Given: DACDABACAB ;
Prove: BCD is isosceles
7. BONUS Given: PRMRQRPQRM ;
Prove: PQMQ
A B
C
D
S
K
N
R
1
2 3
4
A M B
E D
A
C B
D
Q
R
P
M
P
O
R S
Q
15
8. Given:
DCDE
BDAD
BCAE
Prove: ADCEDB
9. Given: MKAB
B is the midpoint of MK
Prove: yx
10. Given: 21
FMCD
Prove: CD bisects MCF
11. Given: QVPV
QSPS
//
//
Prove: yx
12. Given:
BDAE
CDCE
BCAC
Prove: 21
13. Given:
ABDB
BEBC
BEDB
BCAB
Prove: AD
A B
C
D
E A
B M K
x y
C D
M
F
1
2 S
Q V P
x Y
C
A D E B
1 2
E
A
B
D
C
16
14. Given: D and C are supplementary
B and C are supplementary
BEDFABAD ,
Prove: yx
15. Given: RSTSSKSL ,
Prove: LK
16. Given: RDDABRBA ,
Prove: RAXDBX
17. Given:
ZOYO
ZXZOYXYO
,
Prove: ZXYX
18. Given: GFBCEFCD
CFGECFBD
,
,
Prove: ACF is isosceles
19. Given: BCAB
BD is a median of ABC
Prove: CBDABD
20. Given: PR bisects QPS
PR is an altitude of QPS
Prove: PQS is isosceles
A B
E
C F D
x
y
S
R
K
T
L
B D
A
X
R
Y
O
Z
X A
B
C D E
F
G
B
C
D
A
P
Q
R
S