Unit 0 - Algebra 1 Review
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Honors Geometry Unit 0 - Algebra 1 Review Unit 1 – Introduction to Geometry
Day Objective Packet Pages Homework
U1 Day 1 1.1 Define basic terms of geometry. Pages 2-5 Apply basic facts about points, lines, and planes.
U1 Day 2 1.2 Use length and midpoint of a Pages 6-9 segment. Construct midpoints and congruent segments.
U1 Day 3 Quiz 1.1-1.2 Page 10
Start Angles
Day 4 Pages 10-12 1.3 - 1.4 Name and classify angles. Measure and construct angles and angle bisectors.
Day 5 Pages 13-16 1.5-1.6 Review Angles
Day 6 Quiz 1.3-1.6 Pages 17-19 Review
Day 7 Unit 1 Test
1 Honors Geometry Day 1 – Basic Definitions
Term Definition Labeling Picture Point Dot Place in space
Line Series of points that extends in two directions forever
Plane Flat and infinite surface with no thickness
Collinear points Points that lie on the same line
Coplanar Points that lie in the same plane
Segment Part of a line consisting of two points and all points between them
Endpoint Point at one end of segment or starting point of a ray
Ray Part of a line that starts at an endpoint and extends forever in one direction
Opposite Rays Two rays that have a common endpoint and form a line.
2 4 Parts of the mathematical system:
1.) Postulates: statements assumed to be true (accepted without proof) (axiom) 2.) Theorems: statements that are proven true (can be proved true)
3.) Defined terms: terms that have a definition
4.) Undefined terms: can be explained by using examples and descriptions
IMPORTANT POSTULATES AND THEOREMS ABOUT POINTS, LINES, AND PLANES
1) Postulate: Through any two points, there is exactly one ______.
2) Postulate: Through any three non-collinear points, there is exactly one ______.
3) Postulate: If two points lie in a plane, then the ______containing those two points lies entirely in the plane.
4) Postulate: If two planes intersect, then their intersection is a ______. ______
5) Theorem: If two lines intersect, then their intersection is exactly one ______.
6) Theorem: Given a line and a point that is NOT on that line, there is exactly one ______, which contains both the point and the line.
7) Theorem: If two lines intersect, there is exactly one ______, which contains the two lines.
ALWAYS, SOMETIMES, NEVER
1) Two points lie in exactly one line. Circle the letter(s) that apply: 8) Three different points: 2) Three points lie in exactly one line. a) always determine a plane b) can be collinear 3) Three collinear points lie in exactly one plane. c) can be non-collinear d) always lie in at least one plane 4) Two intersecting planes intersect in a segment. 9) A plane is determined by: 5) Three points determine a plane. a) a line and a point b) two intersecting lines 6) Two intersecting lines determine a plane. c) any three points d) a line and a point not on the line 7) Two non-intersecting lines determine a plane. 10) Three points are ______coplanar. (A/S/N)
3 4 Draw each of the following.
1. AB and point C are 2. Line l intersects plane M in point P. 3. Plane M and plane coplanar, but A, B, and C are non–collinear points PX lies in plane M. N intersect in AB .
4. Points W, X, Y, and Z 5. Plane M contains AB and 6. AB Intersects plane M in point X. are coplanar, but points W, X, Y, and P are not. XY , but AB and XY do not PQ is contained in plane M, but intersect each other. AB and PQ do not intersect each other.
8. Plane M and plane N intersect in XY . 7. Line l intersects plane M in point A. Line k intersects line l at point B. Points X, Y, and Z are coplanar but not Line k does not intersect plane M. collinear. Points X, Y, Z, and W are neither coplanar nor collinear.
5 Day 2 – Segments and lengths Ruler Postulate: points of a line can be put into a one-to-one correspondence with the set of real numbers so that no two points are paired with the same coordinate. letters- points (top) coordinates- numbers (bottom) A B C D E F G H I J K L M · · · · · · · · · · · · · -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
“ AB ” refers to “segment AB” “AB” refers to the length of segment AB
CJ = the distance between points C and J = number value which can never be negative!
Distance: between any two points is the absolute value of the difference of the coordinates.
Distance Formula:
Length: distance between two points
One dimensional distance: To find the distance you subtract the coordinates and then take the absolute value to ensure the answer is positive!
BD = _____ AM = _____ EK = ______HM = ______
Congruent Segments: segments that are equal in length. (have the same lengths)
Use the number line at the top. T or F 1. BG DK 2. HK AD
Midpoint: a point that divides a segment into 2 @ segments.
1.) If V is the midpoint of AG , then _____ @ ______.
2.) C is the midpoint of AB . AC = 2x + 1 and CB = 3x – 4 . Find x, AC, CB, and AB.
3.) Y is the midpoint of XZ . Find the coordinate of the 3rd point for each example. a) coord. Of X is –8 b) coord. of X is 11 c) coord. of Y is -8 coord. Of Z is 6 coord. of Y is 15 coord. of Z is -19 coord. Of Y = ____ coord. of Z = _____ coord. of X = _____ 4) Find the midpoint of AG if A(4, -7) and G(-9, 4). ______6 5) Find B if M (1, 11) is the midpoint of BT and T(7,-14) ______
6) Use the Distance Formula to find the distance between K(-7, -4) and L(-2, 0).______
7)Find the length of F(9, 5) and G(–2, 2). ______
Segment Bisector: A segment, line, or ray that intersects a segment at its midpoint.
Line a bisects MS at point D. What is the first conclusion you can make? ______
What is the second conclusion? ______
H· HL and KT intersect at the midpoint of HL . True or False. ·T 1. KM = MT
2. KT is a bisector of LH . ·M 3. MT bisects LH . 4. HL is a bisector of KT . · 5. M is the midpoint of KT . K ·L
In order for you to say that point B is between two points A and C, all three points must lie on the same line and AB + BC = AC.
SEGMENT ADDITION POSTULATE (Def. of Between) If R is between S and I, then SR + ●RI = SI **note the● segment symbols are not used**● Ex 1. S R I a) If SR = 15 and RI = 22, find SI. b) If SR = 5 and SI = 18, find RI. c) If SI = 60, SR = 2x – 8 , RI = 3x – 12 , find x and then find SR and RI. d) If SR = x2 + 7, RI = 4x – 5 , and SI = 34, find x. 1. What set of numbers does the Ruler Postulate put in one-to-one correspondence with the set of points on a line? 7 2. If 6 is the coordinate of point X on AB , can any other point on AB have 6 as a coordinate? Why?
3. If c is a real number, is there a point on line m paired with c?
4. How many points are contained in a line? Why?
A C D E F G H I 5-8 use the number line: -4 -2 0 1 2 4 5 6
5. Name two points whose distance from E is 3.
6. Name two points whose distance from C is 2.
7. What does CH represent? Find CH.
8. Find IA.
In 9 – 14, the numbers given are the coordinates of two points. Fin the distance between the points.
9. -8 and 4 10. -13 and -14 11. 0 and 7.5
12. 12 and 37 13. -5.6 and 7.4 14. a and b
In 15 – 20, Points R,S, and T are distinct and collinear. In each case draw a segment representing the situation given. Tell which point is between the other two.
15. RS = 7, ST = 3, RT = 10
16. RS = 6, ST = 14, TR = 8
17. ST = 6.3, RT = 4.7, RS = 1.6
18. TR = 1.2, RS = 7, TS = 5.8
Make a number line representing the situation. Find the missing coordinate.
19. RS = 10 and RT = 14. The coordinate of S is 7. The coordinate of T is 3.
20. TS = 7 and TR = 9. The coordinate of S is -2. The coordinate of R is 0.
For the next examples, draw a picture, give the formula, and find a solution.
8 Example 1 -In the figure below, B is the midpoint of . AB = 2x + 12 and BC = 5x + 10. Draw the example, come up with an equation, and find x and AC.
Example 2- In the figure, RS = 2x – 4, ST = 3x + 7, and RT = 43. Draw the example, come up with an equation, and find x, RS, and ST.
Example 3 -In the figure below, C is the midpoint of . AC= 5x-6 and CB = 2x. Draw the example, come up with an equation, and find x and AC.
Example 4 -In the figure, RS = 3x – 12, and ST = 2x – 8, and RT = 60. Draw the example, come up with an equation, and find x, RS, and ST.
Example 5 G is the midpoint of . EG = 3y, and GF = 36 – y. Draw the example, come up with an equation, and find x and EG.
Example 6. If SR = x2 + 7, RI = 4x – 5 , and SI = 34, find x. Draw an example.
9 Day 4 – Angles and Angle Bisectors Angle: the union of 2 non-collinear rays with a common endpoint
A Name this angle in 4 different ways: Vertex: 1 B C Sides:
Interior of an angle:
Exterior of an angle:
Name three of the angles.
Measure: of an angle is usually given in degrees.
Protractor Postulate: Given line AB and a point O on AB, all rays that can be drawn from O can be put into a one-to-one correspondence with the real numbers from 0 to 180.
Acute-
Obtuse-
Right-
Straight-
Perpendicular –
Example - Use the diagram to find the measure of each angle. Then classify each as acute, right, or obtuse. a) 10 Congruent Angles: angles that have the same measure. Angle Bisector: a ray that divides an angle into two congruent angles. Example 1 - Draw a picture. If CD bisects ACB, Then ______ ______. Example 2 –Draw a picture and solve. m Example 3- Draw a picture for the following and write an equation to solve for x. RQ bisects PRS. If m PRQ = x + 40 and m QRS = 3x – 2, solve for x, Example 4 - Write an equation to solve for x. x = ______m BAD = 122, x = ______ Example 5 - Draw a picture for the following problem. E is on the interior of Find mPRS = ______and m SRE = ______ Example 6- Draw a picture for the following and write an equation to solve for x. If SU bisects 11 Example 7 - Draw a picture for the following and write an equation to solve for x. Write the equation for #s 9-13. 12 Pairs of Angles *Two angles are ______if their sides form two pairs of opposite rays. 2 3 1 4 Examples: vertical angles are always ______ *Two angles are ______if they have a common side, a common vertex, and no common interior points. 1 2 3 Examples: *Two angles are ______if the sum of their measures is 90. Each angle is a complement of the other. A 35 Examples: 55 C B *Two angles are ______if the sum of their measures is 180. Each angle is a supplement of the other. Examples: E D Linear Pair: ______angles whose non-common sides form a ______linear pair is always ______and ______ m<1 + m<2 = ______1 2 13 Use this figure for 1 - 6 1. Name all angles shown. G H 2. Name 1 in 2 other ways. 1 2 3 3. Name all angles which have side EF . D E F 4. Name a point in the interior of GEF. 5. Name a point in the exterior of 2. 6. Name 2 pairs of adjacent angles. State whether the numbered angles shown in 9 – 16 are adjacent or not. If NOT, explain. A 7. 8. 9. 10. ABC, ABD 2 D 1 1 2 B 1 2 C 1 2 1 11. 12. 2 13. 1 14. 2 1 2 Verifying Angle Relationships-Label the diagram below so that m∠BGC=33 and m∠DGE=57 1. ∠FGA ≅ ______2. m∠CGD = ______3. ∠BGF and ______are supplementary 4. m∠AGF = ______5. ∠EGC and ______are supplementary 6. m∠AGB = ______A F 7. m∠AGC = ______8. m∠BGD= ______ G E 9a. Are ∠AGF and ∠CGD supplementary?______why/why not? B b. Are they a linear pair?______why/why not? D C Solve the following for the indicated variable. Set up an equation using the angle relationships 1. 2. 3x – 5 6x – 23 x + 16 2x – 16 50 3. 3x – y 4. x x 3x – 8 14 2x – 36 2y – 17 More Angles 1. The supplement of a right angle is a ______angle. 2. The supplement of an obtuse angle is a ______angle. 3. The supplement of an acute angle is a ______angle. 4. Vertical angles are ______. 5. ______lines form right angles. 6. Congruent supplementary angles each have a measure of ______. 7. Congruent complementary angles each have a measure of ______. 8. The angles in a linear pair are ______and______. 9. m<1 = 40. What is the measure of its complement? ______Its supplement? ______ 10. m<2 = 120. What is the measure of its complement? ______Its supplement?______