Segment Addition Postulate

1.) Find YZ. 2. Point B is between A and C on AC . Use the following information to

(a) draw a diagram, (b) write an equation in terms of x, (c) solve the equation, and then (d) find AB and BC.

AB = 2x – 5 BC = 6x AC = 27

Midpoint, Segment Bisector, Distance

3.) Line CD bisects AB at point C. 4.) Find XM, if M is the midpoint of the segment. Find AC if AB = 56 feet. (Draw the diagram.)

5) Find the coordinates of the midpoint. 6) Use the given endpoint Y and midpoint M R(3, 1) and S ( -3, 7) of to find the coordinates of the other endpoint. Y ( -1, -3) , M ( 5, 9) 7) Find the segment length of . 8) Find AB. J ( -1, 5) k ( -6, -9) 9. Given the coordinates of A(-6, 5) and B(1, 3). 10. If LM = 3x + 3 and KM = 36, find the Find the distance from C(5, 6) to the midpoint of AB. value of x. Round to the nearest tenth. Point L bisects KM. KJ ≅ LJ

YZ

JK

Angle Addition Postulate, Angle Bisector, Linear Pair, Vertical Angles, Complementary & Supplementary Angles

11. In the diagram, bisects ∠ABC. 12. Given m∠ADC = 135°, find m∠BDC. Find m∠ABC.

Use the diagram to answer questions 13 & 14

13. If mL3 = 2x – 10 and mL1= 110, what is the value of x?

14. If mL2 = 7x – 12 and mL4 = 5x + 12 what is the mL2?

Find m DEG∠ and m GEF∠ .

15. 16.

BD

Conditional Statement, Converse, Inverse, Contrapositive, Counterexample, Biconditional 1. Provide the condtional, converse, inverse, and contrapositive for the given statement.

An angle whose measure is less than 90 is an acute angle. conditional - ________________________________________________________________ converse - __________________________________________________________________ inverse - ____________________________________________________________________ contrapositive - _______________________________________________________________

2. Provide all required valid statements to write the following as a biconditional statement. If it is possible to write as a biconditional then do so, if not, provide a counterexample. Conditional: If angle measures 134, then it is obtuse. ________________________________________________________________________ Law of Detachment and Law of Syllogism

Determine if a conclusion can be made from each set of statements. If possible, write the conclusion AND the law that was used. If not possible, write INVALID. 3. If two angles are vertical angles, then 4. If Kevin goes to college, then he will major in Math . they are congruent. If Kevin majors in Math, then he will be a H.S. Math Teacher. < 1 and <2 are vertical angles Conclusion: Conclusion: Properties, Postulates, Theorems, and Definitions; Proofs Use the property to complete the statement for 5 - 9. 5.) Substitution Property of Equality: If AB = 20, then AB + CD = _____________. 6.) Symmetric Property of Equality: If m∠ 2 = m∠ 4, then _______________. 7.) Addition Property of Equality: If AB = CD, then ________ + EF = ________ + EF. 8.) Distributive Property: If 6(x+8) = 2, then _________x + ________ = 2. 9.) Transitive Property of Equality: If m∠ 1 = m∠ 2 and m∠ 2 = m∠ 3, then _________________.

Use one of the Properties, postulates, theorems, or definitions to justify each statement.

1. If a ⊥ b, then ∠ 2 is a right angle. 1. ____________________

2. If m∠ 3 = 90, then ∠ 3 is a right angle. 2. ____________________

3. If m∠ 1 + m∠ 2= 180, then ∠ 1 is supplementary to ∠ 2. 3. ____________________

4. If ∠ 1 and ∠ 4 form a linear pair, then ∠ 1 is supplementary to ∠ 4. 4. ____________________

5. If m∠ 2 = m∠ 3, then ∠ 2 ≅ ∠ 3. 5. ____________________

6. If ∠ 1 and ∠ 4 are right angles, then ∠ 1 ≅ ∠ 4. 6. ____________________ 7. m∠ 1 + m∠ 2 = m∠ XYZ 7. __________________

8. If ∠ 1 ≅ ∠ 2, then YAbisects ∠ XYZ. 8. __________________ 9. FW + WT = FT 9. ___________________

10. If W is the midpoint of FT, then FW= WT. 10. ___________________

11. If BCAB ≅ , then AB = BC. 11. ___________________

12. If TYABthenTYDFandDFAB ≅≅≅ , . 12. _____________________

13. If ∠C is complementary to ∠D and

∠ K is complementary to ∠D, then ∠ K ≅ ∠C. 13. _____________________

14. If AB + GH = RT and AB = 56, then 56 + GH = RT. 14. _____________________

15. If ½ x = 45, then x = 90. 15. _____________________

16.

1. _________ 2.__________ 3.__________ 4.__________ 5.__________ 6.__________ 7.__________

#9 – 10 l

T F W

#1 – 6

#7 – 8

aaaa

1

bbbb

2

3 4

. . . .

X

Y Z

A

1 2

17. Statements Reasons

1. 2x – 15 – x = 21 + 10x 1. Given

1.) 2.) If l ll m find the value of x and y and justify your answer by stating the theorem/postulate used. Identify a set of parallel lines: ____________ Identify a set of skew lines: ____________ Identify a set of perpendicular lines: ________

x=________ y = __________ Reason: ___________ Reason: ________ 3) If a ll b find the value of x and y. Then ____4.) In the diagram l ll m. Identify the value of x justify your answer by stating the theorem/ postulate used.

x = ________ Reason: ________________ ______ 5)

y = ________ Reason: ________________

80

a

b

(x-18)°

24°

m

n

32

3−−= xy

_____6.) 7.) Find the value of x that would make m ll n.

Then justify your reasoning by stating the post/thm

used.

x = _____ Thm/Post: _________________

9.) Write the equation of a line parallel to

y = -2x + 5 and passes through the point (-1, 6).

8.) Find the slope of the line that passes through

(1, -3) and (7, 5).

10.) Write the equation of a line perpendicular

Slope=_______ and passes through ( 5, -1).

Slope of a line parallel to the line=_______

Slope of a line perpendicular to the line= ______

11.) Two of these three lines have a special relationship (parallel or perpendicular).

Line 1: Line 2: containing points Line 3:

2y x 7

5= − + ( ) ( )6 ,4 and 1,3− + =14x 2y 8

a. Which two lines have the special relationship? ___________ & ____________

b. What is the special relationship they have? _____________________________

12.) Part A: Given AB����

and point P not on AB����

, determine the

equation of the line perpendicular to AB����

that goes through P.

Answer: _______________________

Part B: Determine another point on the new line (other than P).

Answer: ____________

13.) Part A: Graph the line y = -3 and label it as g.

Part B: Graph the line parallel to g that goes through ( )2,3 .

Label this line as n. State the equation of the new line.

Answer: _______________________

P A

B

AAS ASA HL not possible

SAS SSS

1.) 2.) Find the perimeter of the triangle.

Part 1: What is the measure of the smallest angle? Part 1: Solve for x. x = _______ Part 2: Then completely classify the triangle. Part 2: Perimeter of ABC:______

Part 3: Find the measures of < DBC.

3.) Solve for x and y. 4) Use the given information listed, state what extra piece is needed to prove the triangles congruent by the method indicated?

Given: J P∠ ≅ ∠ and K Q∠ ≅ ∠

Prove: JKL PQR∆ ≅ ∆ by ASA

x = ______ and y = ______ Missing Piece: ___________

5-9.)Determine whether enough information is given to prove that the triangles are congruent. If so, state all the congruence postulates/theorems you would use, if not state not possible.

5.)____________ 6.)____________ 7.)___________

8.) ___________ 9.)____________

10.) Given: <QAB ≅ <PBA and QA PB≅

Prove: QAB PBA∆ ≅ ∆

Statements Reasons

11.)Given: AC is a bisector of BD 12.) Given: M is the midpoint of AD and BC

< BCA = <DCA

Prove: AC bisects ∠ BAD Prove: AB CD�

Statements Reasons Statements Reasons

C B D

A

M

A

B

C

D

1.) Circle all of the following that could be 2.) In In ∆ABC , m∠A = 100°, m∠B = 30°, and

the lengths of the sides of a triangle m∠C = 50°. List the sides from largest to

smallest.

A 10m, 13m, 18m

B 1ft, 1ft, 2ft C 7in, 10in, 2in

D 6cm, 5cm, 1cm

E 5.5 in, 7.5 in, 12 in 2. ____________________ ____3.)Two sides of a triangle are 7 cm and 4 cm. 4.) Adam needs a triangular piece of Which measure could NOT be a length of the paper for his Geometry project. His third side? teacher only gave him two sides of the

A 10 cm B 4 cm triangle, which are 11cm and 17cm. C 12 cm D 8.5 cm Between which two numbers must the

third side of the triangular piece fall? State as an Inequality: _________ ______5.) In ∆MNP, MN < NP < MP, 6.) Three land marks are placed on a which statement is true? a map at points H, I and J. A triangle is

A ∠P < ∠N < ∠M B ∠M > ∠P > ∠N formed by connecting these markers

C ∠N > ∠P > ∠M D ∠N > ∠M > ∠P by string so that HI= 150 ft, HJ=245 ft

and IJ = 365 ft. Which statements are true about the measures of the angles

of HIJ?� Circle all that apply.

A m∠ H is the smallest B m∠ H is the largest C m∠ I is the smallest D m∠ J is the smallest

1.) The ratio of the measures of the angles of a 2.) Two similar rectangular rugs have a

triangle is 7:14:15. Find the measure of each angle. scale factor of 2:3. The smaller rug has a perimeter of 24 feet. Part 1: Find the larger rugs perimeter: _____ Part 2: What is the ratio of their areas? ____

3.) The ratio of the perimeter of ∆U to the perimeter of ∆V is 1:2. If the triangles are similar, what is the value of x + y ? x + y = _____

4.) Triangles ABC and DEF are similar and ______5.) Given: ∆RSW ~ ∆TUW. Which have measurements as shown. proportion is ture? What is the measure of EF?

EF = ____________

6.) Given: ∆PQR ~ ∆WUN; m ∠R is 78° 7.) Given: ∆ABC ~ ∆LMN.

What is the measure of ∠W? What is the length of ������?

m<W = ___________ x = ________ and AC = _______

______8.)Which drawing contains a pair ______9.) Which pair of triangles is similar? of similar triangles? State the postulate or State the postulate or theorem used? theorem used?

A. B. C. D.

Post/Thm: ____________ Post/Thm: ____________

61

6

10.) Given DE is parallel to CB, 11.) If ∆BCD ~ ∆BEF , find the value of h? AD = 6, DC = 12 and DE = 4, what is CB?

CB =______ h = _____

_____12.) In the diagram below of , 13.) A triangle has sides whose lengths are

which reason justifies the conclusion 6, 8, and 10. Give the side lengths a similar that ? triangle could have.

A AA B ASA

C SAS D SSS

Side lengths:_____________

14.) A girl 160 cm tall, stands 360 cm from a lamp post 15.) Triangles CDE and NOP are similar.

at night. Her shadow from the light is 90 cm long. The perimeter of the smaller triangle CDE is How high is the lamp post? (draw a diagram and solve 133. The lengths of two corresponding sides on the triangles are 53 and 212. What is the perimeter of NOP?

Height of the lamp post: ______ Perimeter of NOP: _______

C B

D

E

F

14.) Triangles EFG and QRS are similar. 15.) Triangles CDE and NOP are similar.

The length of the sides of EFG are 144, 128, and 112. The perimeter of the smaller triangle CDE is The length of the smallest side of QRS is 280, 133. The lengths of two corresponding What is the length of the longest side of QRS? sides on the triangles are 53 and 212. (draw a diagram and solve) What is the perimeter of NOP?

Length of the longest side: ______ Perimeter of NOP: _______