Download Honors Geometry Final Exam Study Guide Please complete this

Honors Geometry Final Exam Study Guide
Please complete this study guide in your HW NB; we will do one part each day!
The types of problems on this study guide are the only types of problems you will see on the final!
As an added review of the first semester, we will be going over your midterm exams.
Part 1: Chapter 1: Tools of Geometry
Vocabulary: You will be responsible for the following terms.
1. point 2. line 3. plane 4. complementary angles 5. supplementary angles 6. ray
7. opposite rays 8. adjacent angles 9. vertical angles 10. inductive reasoning
11. bisector 12. midpoint 13. undefined term 14. skew lines 15. postulate
Constructions: You will be responsible for the following constructions.
1. congruent segments
2. congruent angles 3. angle bisector 4. perpendicular bisector
Conjectures:
5. Make a conjecture about the sum of the cubes of the first 25 counting numbers. Show your reasoning. (You do not have to find
the answer, just show the formula.)
6. Please write a counterexample for each conjecture.
a. If a student is in eighth grade, then he or she is in Geometry.
b. If two integers have a product of eleven, then the two integers are one and eleven.
Algebraic Problems Using Geometry Skills
7. S is a point on segment RT. If RS = 5(x - 1), ST = 6x + 5, and RT = 33, find RS, ST, and RT.
8. Use a GOOD DRAWING to solve the following dilemma!
Ray BC bisects <ABD
Ray BE bisects <CBD
Ray BF bisects <EBD
m<EBD = 28,
What is m<ABF?
Using the Midpoint and Distance Formulas
9. Given A (1, 5) and B (-2, -9):
a. Please find the distance between these two points. (If not a whole number, please leave your answers in terms of a mostreduced square root.)
b. Please find the midpoint of AB.
c. If B is the midpoint of AC, please find the coordinates of point C.
Part 2: Chapter 2: Reasoning and Proof
Vocabulary: You are responsible for the following terms.
Be sure to know their forms when applicable. (ex: conditional: p ---> q)
1. hypothesis 2. conclusion 3. conditional
4. biconditional 5. converse 6. deductive reasoning 7. Law of Detachment
8.Law of Syllogism 9. theorem 10. truth value
Conditionals and Biconditionals
1. Given the conditional: If x2 = 25, then x = 5.
a. Write the hypothesis. b. Write the conclusion. c. Write the converse.
d. Determine the truth values of the conditional and the converse. If a truth value is FALSE, please provide a
COUNTEREXAMPLE. e. If the statements are able to be written as a biconditional, write the biconditional. If not, explain why.
2. Given the conditional: If two angles are adjacent angles, then they share a vertex and a side and have no points in common.
a. Write the hypothesis. b. Write the conclusion. c. Write the converse.
d. Determine the truth values of the conditional and the converse. If a truth value is FALSE, please provide a
COUNTEREXAMPLE.
e. If the statements are able to be written as a biconditional, write the biconditional. If not, explain why.
Good Definitions
3. What are the three characteristics of a good definition?
4. Are the following definitions good definitions? Why/why not? Be specific.
a. A median of a triangle is a segment whose endpoints are at a vertex and the midpoint of the opposite side.
b. A perpendicular bisector bisects a segment.
c. A resultant is the sum of two vectors and can show the result of vector actions.
d. Hypotenuse-Leg Theorem is a theorem used to prove triangle congruence and helps CPCTC.
e. Adjacent angles share parts.
f. A Great Dane is a large dog that weighs over one hundred pounds.
Laws of Detachment and Syllogism
5. Use either the Law of Detachment and/or the Law of Syllogism to make conclusions. Write LD if you used the Law of
Detachment and LS if you used the Law of Syllogism. If a conclusion is impossible, write IMPOSSIBLE and write the FALLACY
that explains why.
a. If a person likes all Nicholas Sparks novels, then he or she likes Dear John. Neshama likes Dear John.
b. If a person runs a red light, he or she is disobeying the law. Maddie ran a red light.
c. If the public pools are open, then Danny is swimming. If it is after Memorial Day, then the public pools are open. If Danny is
swimming, then he is wearing a bathing suit.
d. (two conclusions and two laws) If John Grisham writes a novel, then it will become a movie. If the novel becomes a movie, then
Skye will go to that movie. If Skye goes to the movie, she will buy popcorn. John Grisham writes a novel.
Proofs!
6. Write the proof of the Vertical Angles Theorem. (Note: You may not use the Vert <'s Thm in your proof or any of the theorems
AFTER the Vert <'s Thm.)
Given: <1 and <2 are vertical angles.
Prove: <1 = <2.
7. Proof of Congruent Supplements Theorem: If two angles are supplementary to the same angle, then the two angles are
congruent. (Note: You may not use the Congruent Supplements Thm in your proof or any of the theorems AFTER the Congruent
Supplements Thm.)
Given: <1 and <2 are supp; <3 and <2 are supp.
Prove: <1 = <3
Part 3: Chapter 3: Parallel and Perpendicular Lines
Constructions: Be able to do the following constructions.
1. parallel lines
2. given a line and a point not on a line, construct a line perpendicular to the given line that passes through the given point
Parallel Lines and Transversals
1. Please use the diagram above. Note: figure is not drawn to scale.
a. Please list all of the angles congruent to <1.
b. Please list all of the angles congruent to <3.
c. Please name a pair of corresponding angles.
d. Please name a pair of alternate interior angles.
e. Please name a pair of same-side interior angles.
f. Please name a pair of alternate exterior angles.
g. Please name a pair of same-side exterior angles.
h. Please name a pair of vertical angles.
i. If m<1 = 3x + 6 and m<6 = 10x + 70, find x, m<8 and m<2.
Using Geometry and Algebra to Solve Problems
2. A hexagon has interior angle measures of: x + 11, 2x – 9, 3x + 5, 4x – 10, 15, and 18. Find the value of x.
3. Find the value of each variable in the diagrams below. Note: Figures are not drawn to scale.
a.
∠SRT ≅∠STR
∠SRT = 30, ∠STU = 9x
b. m || l, m∠ 3 = 2x + 106, and m∠ 4 = 4x + 38
4. Proof of the Converse of the Alternate-Exterior Angles Theorem (Note: you cannot use any definitions, postulates, properties or
theorems that come AFTER the Converse of the Alt Ext <'s Thm - or the Converse of the Alt Ext Angles Thm itself).
Given: <1 = <4 Prove: a ll b
5. An equation of a line is 3x – 6y = 12.
a) What is the x-intercept?
b) What is the y-intercept?
c) What is the slope?
d) Transform this equation into slope-intercept form.
e) Write the equation of the line (in slope-intercept form) of the line that contains the point (-4,2) and is parallel to our given line.
f) Write the equation of the line (in slope-intercept form) of the line that contains the point (0,11) and is perpendicular to our given
line.
Part 4: Chapter 4: Triangle Congruence
1. List the five ways to prove triangles congruent.
2. Can the triangles be proven congruent using only the given information, the Reflexive Property, and Vertical Angles Theorem? If
so, write a congruence statement and the theorem. If no, just write no.
3. Proof of AAS Theorem: Note: You may not use ASA Thm or any theorems that come AFTER it!
Given: <A = <B, AC = BD, <C = <D
Prove: Triangle ACE = Triangle BDF
4. Proof of ITBAT: Note: You may not use ITBAT or any theorems that come AFTER it!
Given: KH = KJ; KM bisects HJ
Prove: <H = <J
5. Proof: Given: Triangle APQ = Triangle BQP ; AP perp PQ; BQ perp PQ
Prove: X is the midpoint of AQ
Part 5: Chapter 5: Relationships within Triangles
In Triangle GHI, R, S, and T are midpoints.
a. Name all the pairs of parallel sides of Triangle GHI and RST.
b. If GH = 20 and HI = 18, find RT.
c. If RH = 7 and RS = 5, find ST.
d. If m<G = 60 and m<I = 70, find m<GTR.
e. If m<H = 50 and m<I = 66, find m<ITS.
f. If m<G = m<H = m<I and RT = 15, find the perimeter of Triangle GHI.
2. Use the figure to the right to answer the following questions.
a. Find BD.
b. C is equidistant from?
c. Can you conclude that CN = DN? Explain.
3. Fill in the blanks.
a. If three or more lines intersect at one point, that point is called the point of __________________.
b. The point of concurrency for the angle bisectors of a triangle is called _____________________.
c. The point of concurrency for the medians of a triangle is called _____________________.
d. The point of concurrency for the altitudes of a triangle is called _____________________.
e. The point of concurrency for the perpendicular bisectors of a triangle is called _____________________.
f. An altitude of a right triangle is _________________.
g. An altitude of an acute triangle is _________________.
h. An altitude of an obtuse triangle is _________________.
4. Find the center of the circle that circumscribes Triangle BOR with B (2,1), O (8,1) and R (2,10).
5. C is the centroid of Triangle DEF. If GF = 9x – 3y, what expression represents CF?
6. Proof of Perpendicular Bisector Theorem: Note: You may not use anything AFTER the Perp Bis Thm or the Perp Bis Thm itself.
Given: AC is perp bis of DB
Prove: AD = AB
7. Use a Coordinate Proof to Prove Thm 5-1.
Given: B is mdpt of AE; D is mdpt of AC
Prove: BD is parallel to EC; BD = ½ EC
8. What is the first step of an indirect proof?
9. Use indirect reasoning to write a convincing argument for: In a plane, two points form exactly one line.
10. Identify the two statements that contradict each other. EXPLAIN!
I. In obtuse triangle VAN, <V = 120.
II. In obtuse triangle VAN, <A = 20.
III. In obtuse triangle VAN, <N = 60.
11. The lengths of two sides of a triangle are given. Write an inequality for the possible lengths of the third side.
a. 3, 3
b. 1, 22
c. 10, 11
12. In triangle QUE, QU > UE > EQ. Write the angles in order from least to greatest.
13. Is it possible for a triangle to have sides with the given lengths? Explain by showing your inequalties.
a. 3, 4, 4
b. 1, 2, 3
c. 8, 8, 8
14. Please explain why m<1 is greater than m<2 in each diagram.
Name the second largest angle in the figure if the side between ∠1 and ∠2 is 17 cm, the side between ∠2 and ∠3 is 15 cm, and the
side between ∠3 and ∠1 is 14 cm. (not drawn to scale)
15. Write an Indirect Proof for Larger Angle ---> Longer Side
Given: m<A > m<B
Prove: BC > AC
16. Proof of Triangle Inequality Theorem
Given: Triangle ABC, DC = AC Prove: AC + CB > AB
Statements
0.
1. <D = <CAD
2. m< CAD + m<CAB = m<DAB
3.
Justifications
0.
1.
2.
3. Comparison Prop of Inequality
4. m<DAB > m<D
4.
5. DB >AB
5.
6. DC + CB = DB
6.
7. DC + CB > AB
8.
8.
Part 6: Chapter 6: Quadrilaterals
Most Helpful Chart EVER! Fill in the boxes with a CHECKMARK if the property applies to the quadrilateral.
Isosceles
Property
Parallelogram Rhombus
Rectangle Square
Trapezoid
Kite
Trapezoid
4 sides
both pairs of
opp sides ll
four = sides
four right <'s
two pairs of
adjacent
sides =
no opposite
sides =
exactly one
pair of ll sides
nonparallel
opposite
sides =
both pairs of
opp sides =
both pairs of
opp <'s =
diagonals
bisect each
other
diagonal
bisects two
angles
diagonals are
perp
diagonals are
=
base <'s =
Finding Unknowns in Quadrilaterals
1. ______If ON = 9x – 4, LM = 8x + 7, NM = x – 3, and OL = 4y – 8, find the values of x and y for which LMNO must be a
parallelogram.
2. ____Find AM if PN = 8 and AO = 5.
3. ______Find values of x and y for which ABCD must be a parallelogram.
4._______ Find the values of the variables for the rectangle. Then find the lengths of the sides.
5. _______ One side of a kite is 5 cm less than 2 times the length of another. If the perimeter is 14 cm, find the length of each side
of the kite.
A. 4 cm, 3 cm
B. 4.2 cm, 3.4 cm
C. 6.3 cm, 7.7 cm
D. 5 cm, 5 cm
Can the Quadrilaterals be Proved to be Parallelograms?
1. _______ I and II are descriptions of two different quadrilaterals. Are I and II parallelograms? Choose the best option.
I. Two adjacent angles are right angles, but the quadrilateral is not a rectangle.
II. All of the angles are congruent.
A. I. impossible ; II. parallelogram
B. I. parallelogram; II. parallelogram
C. I. parallelogram; II. impossible
D. I. impossible; II. impossible
2. Must the figure be a parallelogram? Explain!
3. ________ Must the figure be a parallelogram? Explain.
Proofs
1. Proof of Thm 6.2: Opposite angles of a parallelogram are congruent.
Note: You may NOT use any Theorem AFTER 6.1 OR 6.1 itself.
Given: Parallelogram ABDC
A
B
Prove: <A = <D
C
Statements
Justifications
0.
0.
1. AB is parallel to DC; CA is parallel to BD
1.
2.
2. AIA
3.
3.
4.
4. ASA
5.
5.
D
2. Given: Triangle TRS = Triangle RTW
Prove: RSTW is a parallelogram
S
R
T
W
3. Proof of Trapezoid Midsegment Theorem: Complete a COORDINATE PROOF, using the guidelines below.
Given: MN is the midsegment of trapezoid TRAP.
Prove: MN is parallel to TP; MN is parallel to RA; MN = 1/2(TP + RA)
Use the Midpoint Formula to Find the Coordinates of M and N:
Use the Slope Formula to Find the Slopes of MN, TP, and RA: (Show that these prove the 1st 2 prove statements.)
Use the Distance Formula to Find the Lengths of MN, TP, and RA: (Show that these prove the 3rd prove statement.)
4. Flowchart Proof of Thm 6.10: The diagonals of a rhombus are perpendicular.
Given: ABCD is a rhombus
Prove: AC is perpendicular to BD.
Part 7: Chapter 7: Triangle Similarity
1. Solve for a and b.
2. Solve for a and b.
3. Find the value of x. Leave your answer as an exact square root.
4. Use the Side-Splitter Theorem to find x given that
||
.
5. Find the value of x. Leave your answer as an exact square root.
6. In ΔRST, RS = 12, RT = 11, and ST = 13. In ΔUVW, UV = 24, UW = 22, and VW = 26. Write a similarity statement (if possible).
7. The width of a golden rectangle is 3 m, which is shorter than the length. What is the length?
8. Solve for a and b. Leave your answers as reduced fractions.
9. Find the geometric mean of 75 and 3.
10. The polygons below are similar, but not necessarily drawn to scale. Find the values of x and y.
11. Are the triangles similar? If so, by which postulate or theorem?
12. Derivation of the Golden Ratio
Given: ABCD is a golden rectangle, so ABCD is similar to BCFE.
Justify each step!
1. AB = BC
BC CF
1. ____________________________
2. x = 1
1 x–1
2. ____________________________
3. x2 – x = 1
3. ____________________________
4. x2 – x – 1 = 0
4. ____________________________
5. x = 1.618
5. ____________________________
So, the ratio of l:w = 1.618:1!
2. Proof of SAS Similarity Theorem: Note: You may not use this theorem or any theorem AFTER the theorem in this proof!
Given: <A = <Z; AB = AC
ZY ZX
Prove: Triangle ABC is similar to Triangle ZYX
Statements
Justifications
0.
0.
1. Draw DE so that ZD = AB and DE is parallel to YX
1. congruent segment and parallel lines construction
2.
2. AIA
3. <Z = <Z
3.
4.
4. AA Sim Post
5. ZD = ZE
ZY ZX
5.
6. AC = ZD
ZX ZY
6.
7. AC = ZE
ZX ZX
7.
8.
8. XPP (Cross Product Prop)
9. AC = ZE
9.
10. Triangle ABC = Triangle ZDE
10.
11. <B = <ZDE
11.
12.
12. Transitive Prop
13.
13.
3. Proof of Side Splitter Theorem
Given: Triangle QXY with RS parallel to XY
Prove: XR = YS
RQ SQ
Statements
Justifications
0.
0.
1. <1 = <3; <2 = <4
1.
2.
2. AA Sim Post
3. XQ = YQ
RQ SQ
3.
4.
4. Segment Add Post
5. XR + RQ = YS + SQ
RQ
SQ
5.
6.
6.
Part 8: Chapter 8: Right Triangles and Trigonometry
1. Do the following triangle side lengths form an acute, right or obtuse triangle? JUSTIFY your answer by stating a theorem!
a) 3, 4, 5
b) 7, 8, 13
c) 6, 8, 9
2. Use Special Right Triangles to find the missing side lengths. Leave your answers as exact (with square roots, if necessary)!
a) A 30-60-90 triangle with shorter leg = 5 cm
b) A 45-45-90 triangle with hypotenuse = 8 ft
c) A 30-60-90 triangle with longer leg = 12 in
3. What does SOHCAHTOA stand for? (Write the ratios.)
4. Use SOHCAHTOA to find the values to the nearest hundredth. Please show your work.
a) B
<C = 67.9
Find AC
11.6
A
b) H
C
I
<J = 51.5; IJ = 18.9; Find HJ
c)
K
L
J
KL = 2.8; LM = 9.9; Find <M
M
5. Kehinde is using a theodolite (surveying instrument) to measure the height of Taiwo's building. The theodolite is 5.5 ft tall, is 180
feet away from the building and the angle of elevation from the top of the theodolite to the top of Taiwo's building is 29.23 degrees.
Find the height of Taiwo's building to the nearest hundredth of a foot.
6. A giraffe and a tree are 100 feet away from each other. From top of the giraffe's head to the top of the tree, the angle of elevation
is 4.8 degrees, and the angle of depression from the top of the giraffe's head to the base of the tree is 79 degrees. How tall is the
giraffe? How tall is the tree tree? Find each height to the nearest hundredth of a foot.
7. Describe the vector below as an ordered pair.
8. Add the vectors.
a. <-12,7> and <0,-8>
b. <-2,-2> and <-8,2>
9. Catherine is flying a plane north at a speed of 350 mph, but the wind is blowing east at a speed of 70 mph. What are Catherine's
resultant speed and direction? Give speed in mph and direction as an angle measure with compass directions. Round your
answers to the nearest hundredth.
10. Guy wants his boat to travel directly east, but the current is flowing at a speed of 9 mph south. Guy can drive his boat at a
speed of 30 mph. At what angle and compass direction should he drive the boat in order to travel directly east? What will his
resultant speed be?
11. Proof of PT
Given: Triangle ABC is a right triangle.
Prove: a2 + b2 = c2
Statements
Justifications
0.
0.
1.
1. Cor 2 to Rt Triangle Alt Thm
(leg is geo mean b/w adjacent piece of hyp
and hyp)
2. a2 = rc; b2 = qc
2.
3. a2 + b2= rc + b2
3.
4. a2+ b2 = rc + qc
4.
5. a2+ b2 = c(r + q)
5.
6. r + q = c
6.
7. a2+ b2 = c(c)
7.
8.
8.
12. Use the Pythagorean Theorem to prove the 45-45-90 Triangle Theorem.
A
Given: ABC is a right triangle; <C = <A = 45
Prove: AC = AB root 2
B
Statements
Justifications
0.
0.
1.(AB)2 + (BC)2 =(AC)2
1.
2. AB = BC
2.
3.
3. Subst
4. 2(AB)2 =(AC)2
4.
5.
5.
C
Part 9: Chapter 10: Areas of Regular Polygons
1. The lengths of the sides of a right triangle are 5 in, 12 in, and 13 in. What is the area of the triangle?
2. What is the area of a parallelogram with sides of 12 in and 18 in and a height to the 18-in side of 5 in?
3. Two sides of a parallelogram are 7-in and 12-in. The height to the 12-in side is 4 in. What is the height to the 7-in side to the
nearest hundredth?
4. Find the area of a kite with diagonals of 12 ft and 4 ft.
5. Find the area of a rhombus if one diagonal is 8 m and one side is 5 m.
6. Draw a regular octagon. Draw and label an apothem a with a radius, r, that is the apothem's hypotenuse. Then, draw and label a
central angle AND find its measure.
7. A regular hexagon has side length of 12 ft. Find the area of the polygon in simplest radical form.
8. An equilateral triangle has an apothem of 9root3 m. Find the area of the triangle in simplest radical form.
9. The radius of a square is 9 cm. Find the area of the square in simplest radical form.
10. Two similar polygons have corresponding sides in the ratio 8:9. Find the ratio of their perimeters and the ratio of their areas.
11. The corresponding sides of two similar parallelograms are in the ratio 3:7. The area of the smaller parallelogram is 81 sq in.
Find the area of the larger parallelogram.
12. The area of two similar rectangles are 1245 sq ft and 80 sq ft. Find the ratio of their perimeters. Leave your answer in simplest
radical form.
13. Find the area of a regular octagon with perimeter of 80 in. Give the area to the nearest hundredth.
14. Find the area of a regular decagon with side length of 12 in. Give the area to the nearest hundredth.
15. The radius of a regular pentagon is 19 ft. Give the area to the nearest hundredth.
16. A triangle has sides of 18 cm and 20 cm with an included angle of 54 degrees. Give the area to the nearest hundredth.
17 Area of a Triangle Proof! Note: You may not use the Area of a Triangle Theorem or any theorem after it!
Given: Rectangle ABCD with diagonal AC
Prove: Area of Triangle ADC = 1/2bh
Statements
Justifications
0.
0.
1. AC = AC
1.
2.
2. opp sides of rectangle congruent
3.
3. SSS
4. Triangle CBA + Triangle ADC = Rect ABCD
4.
5. Triangle ADC + Triangle ADC = Rect ABCD
5.
6.
6. LT
7. Triangle ADC = ½ Rect ABCD
7.
8. Rect ABCD = bh
8.
9.
9.
18. Proof of Area of a Trapezoid: Note: You may not use the Area of a Trapezoid Theorem or any theorem after it!
Given: Trap ABCD Prove: Area of Trap = 1/2(b1 + b2)h
Statements
Justifications
0.
0.
1. Triangle ADE + Triangle BCF + Rect ABFE = Trap ABCD
1.
2. Triangle ADE = 1/2hx;
Triangle BCF = 1/2h(b2 - b1 - x)
2.
3.
3. Area of Rectangle Theorem
4.
4. Subst
5. 1/2h(x + b2 - b1 – x) + b1 h = Trap ABCD
5.
6.1/2h(b2 - b1 ) + b1 h = Trap ABCD
6.
7. h(1/2(b2 - b1 ) + b1 = Trap ABCD
7.
8. h(1/2b2 - 1/2b1 + b1 ) = Trap ABCD
8.
9. h(1/2b2 +1/2b1) = Trap ABCD
9.
10.
10.
19. SAS Area Theorem Proof: Note: You may not use the SAS Area Theorem or any theorem after it!
Given: Triangle ABC; m<A, b, c
Prove: Formula for Area of ABC w/o height
Statements
Justifications
0.
0.
1. sin<A = h/c
1.
2. h = c(sin<A)
2.
3. A = 1/2bh
3.
4.
4.
Law of Sines and Law of Cosines: Use these laws to answer the questions below:
1. If <A = 62 degrees, <B = 47 degrees, and a = 10, find the length of side b.
2. If a = 16 and b = 33, and <C = 38, find the length of side c.
3. If <B = 29 degrees, <C = 96 degrees and b = 8 find the length of side a.
4. If a = 13, b = 15, and c = 22, find <C.
Areas of Sectors and Circles
1.
2.
Find the area of a sector of a circle whose central angle = 150 degrees with a radius = 8 m. Leave your answer in terms
of pi.
Find the area of the shaded region. Round your answer to the nearest tenth.