Download Packet 3 for Unit 3 M2 Geo

M2 GEOMETRY PACKET 3 FOR UNIT 3 – SECTIONS 4-4 TO 4-6
DUE
NUMBER
3G
ASSIGNMENT
TOPICS
4-6:
Vocabulary: legs of an isosceles triangle, vertex
and base angles of an isosceles triangle
Use algebra to solve problems involving
isosceles and equilateral triangles
p. 289-290 #10-22 even
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M2 GEOMETRY PACKET 3 FOR UNIT 3 – SECTIONS 4-4 TO 4-6
Investigation: Isosceles and Equilateral Triangles
Use this link:
https://www.geogebra.org/m/FEaNhaPn#material/wYbGNx59 (page 1 only)
F
1. In an isosceles triangle, the sides and angles have specific
names.
E
a. The two equal sides of an isosceles triangle are called the legs,
and the third side is the base. Identify the legs and the base of DEF :
Legs: ______ and ______
Base: ______
D
b. The vertex angle of an isosceles triangle is the angle between the two equal sides, and
the other two angles are called the base angles. Identify the vertex angle and the
base angles of DEF :
Vertex angle: __________
Base angles: __________ and __________
c. Identify the legs, base, vertex angle, and base angles of
Geogebra app on your screen.
ABC shown in the
Legs: ______ and ______
Base: ______
Vertex angle: __________
Base angles: __________ and __________
2. In the Geogebra app, drag points A, B, and C to other locations around the screen. What
do you notice about the measures of the base angles?
Complete the conjecture:
Isosceles Triangle Conjecture
If two sides of a triangle are congruent, then the angles
opposite those sides are congruent. For example, in
ABC , if AB  AC , then __________  __________.
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M2 GEOMETRY PACKET 3 FOR UNIT 3 – SECTIONS 4-4 TO 4-6
3. Write the Converse of the Isosceles Triangle Conjecture. Is this statement true?
4. What does the Isosceles Triangle Conjecture imply about the angles in an equilateral
triangle? Explain.
Complete the conjecture:
If a triangle is equilateral, then ______________________________________. Conversely, if
__________________________________________________________________________________.
Find the value of each variable.
5.
6.
7.
8.
9.
10.
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M2 GEOMETRY PACKET 3 FOR UNIT 3 – SECTIONS 4-4 TO 4-6
Practice for 4-4 to 4-6
In SNO at right, name the following:
S
1. two sides and their included angle (SAS)
O
N
2. two angles and their included side (ASA)
3. two sides and a non-included angle (SSA)
4. two angles and a non-included side (SAA)
Determine whether △ABC ≅ △KLM. Explain.
5. A  3,3 , B  1,3 , C  3,1 , K (1, 4), L 3, 4  , M 1,6 
6. A  4, 2  , B  4,1 , C  1, 1 , K (0, 2), L  0,1 , M  4,1
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M2 GEOMETRY PACKET 3 FOR UNIT 3 – SECTIONS 4-4 TO 4-6
Answer the planning questions, and write a two-column proof.
7. Given: PR  DE, R  E, T  F
Prove: PRT  DEF
a. Planning questions:
Which parts are we told are congruent? Use slashes or curves to mark this in the diagram.
Which parts can we conclude are congruent? Use slashes or curves to mark this in the diagram.
Which triangle congruence postulate do these parts describe?
b. Proof:
Statements
Reasons
8. Given: AB  CB , D is the midpoint of AC .
Prove: ABD  CBD
a. Planning questions:
Which parts are we told are congruent? Use slashes or curves to mark this in the diagram.
Which parts can we conclude are congruent? Use slashes or curves to mark this in the diagram.
Which triangle congruence postulate do these parts describe?
b. Proof:
Statements
Reasons
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M2 GEOMETRY PACKET 3 FOR UNIT 3 – SECTIONS 4-4 TO 4-6
9. Given: N  L , JK  MK
Prove: JKN  MKL
a. Planning questions:
Which parts are we told are congruent? Use slashes or curves to mark this in the diagram.
Which parts can we conclude are congruent? Use slashes or curves to mark this in the diagram.
Which triangle congruence postulate do these parts describe?
b. Proof:
Statements
Reasons
10. Given: AB  CB , A  C , DB bisects ABC .
Prove: AD  CD
a. Planning questions:
Which parts are we told are congruent? Use slashes or curves to mark this in the diagram.
Which parts can we conclude are congruent? Use slashes or curves to mark this in the diagram.
Which triangle congruence postulate do these parts describe?
b. Proof:
Statements
Reasons
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M2 GEOMETRY PACKET 3 FOR UNIT 3 – SECTIONS 4-4 TO 4-6
In #11-14, refer to the figure at the right.
11. If AC  AD , name two congruent angles.
12. If BE  BC , name two congruent angles.
13. If EBA  EAB , name two congruent segments.
14. If CED  CDE , name two congruent segments.
In #15-16, find each measure.
15. mABC
16. mEDF
In #17-18, find the value of each variable.
17.
18.
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M2 GEOMETRY PACKET 3 FOR UNIT 3 – SECTIONS 4-4 TO 4-6
Review for 4-1 to 4-6
1. The triangle shown at right is equilateral. Find the value of x.
2. Use the distance formula to find the lengths of the sides of
ABC with vertices A  4,1 , B  2, 1 , and
C  2, 1 . Then classify ABC by the length of its sides.
3. In the figure at right, find the measures of all numbered angles.
m1  ______
m2  ______
m3  ______
4. If DJL  EGS , which segment in EGS is congruent to DL ?
5. Use the distance formula to find the lengths of the sides of DEF and PQR . Are the two triangles
congruent? Explain.
D  6,1 , E 1, 2  , F  1, 4  , P(0,5), Q  7,6  , R 5,0 
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M2 GEOMETRY PACKET 3 FOR UNIT 3 – SECTIONS 4-4 TO 4-6
Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to
prove congruence, write not possible.
6.
7.
8.
Answer the planning questions, and write a two-column proof.
9. Given: RS  TS , V is the midpoint of RT
Prove: RSV  TSV
a. Planning questions:
Which parts are we told are congruent? Use slashes or curves to mark this in the diagram.
Which parts can we conclude are congruent? Use slashes or curves to mark this in the diagram.
Which triangle congruence postulate do these parts describe?
b. Proof:
Statements
Reasons
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M2 GEOMETRY PACKET 3 FOR UNIT 3 – SECTIONS 4-4 TO 4-6
10. Given: S is the midpoint of QT , QR TU
Prove: QSR  TSU
a. Planning questions:
Which parts are we told are congruent? Use slashes or curves to mark this in the diagram.
Which parts can we conclude are congruent? Use slashes or curves to mark this in the diagram.
Which triangle congruence postulate do these parts describe?
b. Proof:
Statements
Reasons
11. Given: D  F , GE bisects DEF
Prove: DG  FG
a. Planning questions:
Which parts are we told are congruent? Use slashes or curves to mark this in the diagram.
Which parts can we conclude are congruent? Use slashes or curves to mark this in the diagram.
Which triangle congruence postulate do these parts describe?
b. Proof:
Statements
Reasons
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M2 GEOMETRY PACKET 3 FOR UNIT 3 – SECTIONS 4-4 TO 4-6
In #12-15, refer to the figure at the right.
12. If RV  RT , name two congruent angles.
13. If RS  SV , name two congruent angles.
14. If SRT  STR , name two congruent segments.
15. If STV  SVT , name two congruent segments.
16. Use the figure at right to find each measure. mKMH  50
mKML =______
mHMG =______
mGHM =______
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M2 GEOMETRY PACKET 3 FOR UNIT 3 – SECTIONS 4-4 TO 4-6
Answers to p. 8-11:
1. x  3
2. AB  2 2 , BC  4 , AC  2 10 , scalene
3. m1  40 , m2  20 , m3  110
4. ES
5. DE  PQ  5 2 , EF  QR  2 10 , DF  PR  5 2 , yes, by SSS
6. not possible
7. SSS or SAS
8. SSS
9. a. told: RS  TS ; conclude: RV  TV and VS  VS ; postulate: SSS
b. (1) RS  TS , V is the midpoint pf RT (Given)
(2) RV  TV (Def. of midpoint)
(3) VS  VS (Reflexive)
(4) RSV  TSV (SSS)
10. a. told: none; conclude: QS  TS and R  U and Q  T ; postulate: AAS
(This one can also be done using QS  TS and Q  T and RSQ  TSU and ASA.)
b. (1) S is the midpoint of QT , QR TU (Given)
(2) QS  TS (Def. of Midpoint)
(3) R  U , Q  T (Alternate interior  s are  )
(4) QSR  TSU (AAS)
11. a. told: D  F ; conclude: DEG  FEG and GE  GE ; postulate: AAS
b. (1) D  F , GE bisects DEF (Given)
(2) DEG  FEG (Def. of  bisector)
(3) GE  GE (Reflexive)
(4) DEG  FEG (AAS)
(5) DG  FG
12. RVT  RTV
13. SRV  SVR
14. SR  ST
16. mKML  60, mHMG  70, mGHM  40
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15. SV  ST