GOALS of High School GEOMETRY MEASUREMENT Remember the Grade School Rules of how to MEASURE and add to/connect them EXTENSION Learn the METHODS & LOGIC of Extending the Measurement Rules to New Ideas ABSTRACTION Learn how to change geometric measurement problems to algebra problems and vice versa

GOALS of High School GEOMETRY ABSTRACTION A group of seven weary men once arrived at a small hotel and asked for accommodations for the night, specifying that they wanted separate rooms.  The manager admitted that he had only six rooms left, but thought he might be able to put up his guests as they desired.  He took the first man to the first room and asked one of the other men to stay there for a few minutes.  He then took the third man to the second room, the fourth man to the third room, the fifth man to the fourth room and the sixth man to the fifth room.  Then he returned to the first room, got the seventh man, and showed him to the sixth room.  Everyone was thus nicely taken care of.  Or was he?

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Identify and model points, lines, and planes. Identify collinear and coplanar points and intersecting lines and planes in space. undefined term plane coplanar space locus intersection point line collinear Lesson 1 MI/Vocab

Undefined Terms – Words, usually readily understood, that are NOT formally explained by means of more basic words and concepts. The basic undefined terms of Geometry are point, line and plane. POINT – a location in space with NO LENGTH or WIDTH. LINE – an infinite set of points with INFINITE LENGTH, but NO WIDTH. PLANE – an infinite SET of points with INFINITE LENGTH, INFINITE WIDTH, but NO THICKNESS. 22

SPACE – The boundless 3-dimensional set of of ALL POINTS. COLLINEAR – Points that lie on the SAME LINE. COPLANAR – Points that line in the SAME PLANE. LOCUS – The set of points that SATISFY a GIVEN CONDITION. INTERSECTION - the SET of POINTS that are CONTAINED in BOTH Figures

Lesson 1 KC1

Lesson 1 KC1

Lesson 1 KC1

Lesson 1 KC1

Lesson 1 KC1

A. Use the figure to name a line containing point K. Name Lines and Planes A. Use the figure to name a line containing point K. Answer: The line can be named as line a. There are three points on the line. Any two of the points can be used to name the line. Lesson 1 Ex1

B. Use the figure to name a plane containing point L. Name Lines and Planes B. Use the figure to name a plane containing point L. Answer: The plane can be named as plane B. You can also use the letters of any three noncollinear points to name the plane. plane JKM plane KLM plane JLM Lesson 1 Ex1

Name Lines and Planes The letters of each of these names can be reordered to create other acceptable names for this plane. For example, JKM can also be written as JMK, MKJ, KJM, KMJ, and MJK. There are 15 different three-letter names for this plane. Lesson 1 Ex1

A. Use the figure to name a line containing the point X. A. line X B. line c C. line Z D. A B C D Lesson 1 CYP1

B. Use the figure to name a plane containing point Z. A. plane XY B. plane c C. plane XQY D. plane P A B C D Lesson 1 CYP1

Model Points, Lines, and Planes A. Name the geometric shape modeled by a 10  12 patio. Answer: The patio models a plane. Lesson 1 Ex2

Model Points, Lines, and Planes B. Name the geometric shape modeled by a drop of water on a table. Answer: The drop on the table models a point on a plane. Lesson 1 Ex2

A. VISUALIZATION Name the geometric shape modeled by a colored dot on a map used to mark the location of a city. A. point B. line segment C. plane D. none of the above A B C D Lesson 1 CYP2

B. VISUALIZATION Name the geometric shape modeled by the ceiling of your classroom. A. point B. line segment C. plane D. none of the above A B C D Lesson 1 CYP2

Draw Geometric Figures Draw a surface to represent plane R and label it. Lesson 1 Ex3

Draw Geometric Figures Draw a line anywhere on the plane. Lesson 1 Ex3

Draw Geometric Figures Draw dots on the line for point A and B. Label the points. Lesson 1 Ex3

Draw Geometric Figures Lesson 1 Ex3

Draw Geometric Figures Draw dots on this line for point D and E. Label the points. Lesson 1 Ex3

Draw Geometric Figures Label the intersection point of the two lines as P. Lesson 1 Ex3

Draw Geometric Figures Answer: Lesson 1 Ex3

Draw Geometric Figures Answer: There are an infinite number of points that are collinear with Q and R. In the graph, one such point is T(1, 0). Lesson 1 Ex3

A. Choose the best diagram for the given relationship A. Choose the best diagram for the given relationship. Plane D contains line a, line m, and line t, with all three lines intersecting at point Z. Also point F is on plane D and is not collinear with any of the three given lines. A. B. C. D. A B C D Lesson 1 CYP3

Lesson 1 CYP3

A. B. C. D. A B C D Lesson 1 CYP3

A. How many planes appear in this figure? Interpret Drawings A. How many planes appear in this figure? Answer: There are two planes: plane S and plane ABC. Lesson 1 Ex4

B. Name three points that are collinear. Interpret Drawings B. Name three points that are collinear. Answer: Points A, B, and D are collinear. Lesson 1 Ex4

C. Are points A, B, C, and D coplanar? Explain. Interpret Drawings C. Are points A, B, C, and D coplanar? Explain. Answer: Points A, B, C, and D all lie in plane ABC, so they are coplanar. Lesson 1 Ex4

Answer: The two lines intersect at point A. Interpret Drawings Answer: The two lines intersect at point A. Lesson 1 Ex4

Intersecting Terms Intersection Possibilities Point Line Plane

A. How many planes appear in this figure? A. one B. two C. three D. four A B C D Lesson 1 CYP4

B. Name three points that are collinear. A. B, O, and X B. X, O, and N C. R, O, and B D. A, X, and Z A B C D Lesson 1 CYP4

C. Are point X, O, and R coplanar? A. yes B. no C. cannot be determined A B C Lesson 1 CYP4

A. point X B. point N C. point R D. point A A B C D Lesson 1 CYP4

DEFINE the BIG Three Undefined Terms You should be able to: DEFINE the BIG Three Undefined Terms DESCRIBE the Attributes of each Undefined Term DESCRIBE the resulting INTERSECTION of Undefined Terms as a point, line or plane DETERMINE how many COLLINEAR points are needed to define a Line and a Plane Lesson 1 CYP4

Five-Minute Checks Image Bank Math Tools Animation Menu Archeology and the Coordinate Plane CR Menu

Lesson 1-2 (over Lesson 1-1) Lesson 1-3 (over Lesson 1-2) 5Min Menu

1. Exit this presentation. To use the images that are on the following three slides in your own presentation: 1. Exit this presentation. 2. Open a chapter presentation using a full installation of Microsoft® PowerPoint® in editing mode and scroll to the Image Bank slides. 3. Select an image, copy it, and paste it into your presentation. IB 1

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1-4-1 Using a Protractor to Measure Three Types of Angles 1-4-2 Copy an Angle 1-4-3 Bisect an Angle 1-5 Constructing Perpendiculars 1-7 Orthographic Drawings Animation Menu

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What is the value of x2 + 3yz if x = 3, y = 6, and z = 4? B. 75 C. 78 D. 81 A B C D 5Min 1-1

Solve 2(x – 7) = 5x + 4. A. B. –6 C. D. 6 A B C D 5Min 1-2

Is (–2, 5) a solution of 3x + 4y = 14? A. yes B. no A B 5Min 1-3

Factor 9x2 – 25y2. A. (3x + 5y)2 B. 3x2 + 5y2 C. (3x – 5y)2 D. (3x + 5y)(3x – 5y) A B C D 5Min 1-4

Which choice shows the graph of y = 3x + 2? A. B. C. D. A B C D 5Min 1-5

Find all of the solutions of 2x2 + 5x – 3 = 0. A. –3 only B. C. D. no solution A B C D 5Min 1-6

Refer to the figure. Name three collinear points. (over Lesson 1-1) Refer to the figure. Name three collinear points. A. Q, T, B B. Q, T, A C. L, T, B D. A, T, B A B C D 5Min 2-1

Refer to the figure. What is another name for line AB? (over Lesson 1-1) Refer to the figure. What is another name for line AB? A. B. line ℓ, C. D. A B C D 5Min 2-2

Refer to the figure. Name a line in plane Z. (over Lesson 1-1) Refer to the figure. Name a line in plane Z. A. B. line ℓ or C. D. A B C D 5Min 2-3

Refer to the figure. Name the intersection of planes Z and W. (over Lesson 1-1) Refer to the figure. Name the intersection of planes Z and W. A. B. C. D. line ℓ A B C D 5Min 2-4

Refer to the figure. How many lines are in plane Z? (over Lesson 1-1) Refer to the figure. How many lines are in plane Z? A. zero B. two C. four D. infinitely many A B C D 5Min 2-5

(over Lesson 1-1) Three lines are coplanar. What is the greatest number of intersection points that can exist? A. 2 B. 3 C. 4 D. 5 A B C D 5Min 2-6

Find the precision for a measurement of 42 centimeters. (over Lesson 1-2) Find the precision for a measurement of 42 centimeters. A. 40.25 cm to 41.75 cm B. 41 cm to 43 cm C. 41.5 cm to 42.5 cm D. 41.75 cm to 42.75 cm A B C D 5Min 3-1

(over Lesson 1-2) If M is between L and N, LN = 3x – 1, LM = 4, and MN = x – 1, find x and MN. A. x = 2, MN = 5 B. x = 2, MN = 1 C. x = 5, MN =4 D. x = 1.5, MN = 0.5 A B C D 5Min 3-2

Use the figure to find RT. (over Lesson 1-2) Use the figure to find RT. A. B. C. D. A B C D 5Min 3-3

A B Use the figure to determine whether MN is congruent to QM. A. yes (over Lesson 1-2) Use the figure to determine whether MN is congruent to QM. A. yes B. no A B 5Min 3-4

A B Use the figure to determine whether MQ is congruent to NQ. A. yes (over Lesson 1-2) Use the figure to determine whether MQ is congruent to NQ. A. yes B. no A B 5Min 3-5

If AB is congruent to BC, AB = 4x – 2, and BC = 3x + 3, find x. (over Lesson 1-2) If AB is congruent to BC, AB = 4x – 2, and BC = 3x + 3, find x. A. 5 B. 4 C. 3 D. 2 A B C D 5Min 3-6

Use the number line to find the measure of AC. (over Lesson 1-3) Use the number line to find the measure of AC. A. 2 B. 3 C. 4 D. 5 A B C D 5Min 4-1

Use the number line to find the measure of DE. (over Lesson 1-3) Use the number line to find the measure of DE. A. 9 B. 8.5 C. 8 D. 7 A B C D 5Min 4-2

Use the number line to find the midpoint of EG. (over Lesson 1-3) Use the number line to find the midpoint of EG. A. E B. F C. G D. H A B C D 5Min 4-3

Find the distance between P(–2, 5) and Q(4, –3). (over Lesson 1-3) Find the distance between P(–2, 5) and Q(4, –3). A. 10 B. 8.12 C. 9 D. 6.32 A B C D 5Min 4-4

(over Lesson 1-3) Find the coordinates of R if M(–4, 5) is the midpoint of RS and S has coordinates (0, –10). A. (8, 20) B. (8, –20) C. (–8, 20) D. (–8, 0) A B C D 5Min 4-5

(over Lesson 1-3) What is the perimeter of triangle DEF if its vertices are D(–2, –6), E(–2, 6) and F(3, –6)? A. 12 units B. 13 units C. 17 units D. 30 units A B C D 5Min 4-6

Refer to the figure. Name the vertex of 3. (over Lesson 1-4) Refer to the figure. Name the vertex of 3. A. A B. B C. C D. D A B C D 5Min 5-1

Refer to the figure. Name a point in the interior of ACB. (over Lesson 1-4) Refer to the figure. Name a point in the interior of ACB. A. G B. D C. B D. A A B C D 5Min 5-2

Refer to the figure. Name the sides of BAC. (over Lesson 1-4) Refer to the figure. Name the sides of BAC. A. B. C. D. A B C D 5Min 5-3

(over Lesson 1-4) Refer to the figure. Name the angles with vertex B that appear to be acute. A. BAC, BDC, BDA B. BDC, BCA, BAC C. CAB, CDB, DCB D. ABC, ABD, DBC A B C D 5Min 5-4

(over Lesson 1-4) Refer to the figure. If bisects ABC, mABD = 2x + 3, and mDBC = 3x – 13, find mABD. A. 70 B. 35 C. 32 D. 16 A B C D 5Min 5-5

(over Lesson 1-4) If P is in the interior of MON and mMOP = mMON, what can you conclude? A. PON  NOM B. MON is an acute angle. C. D. mMOP > mPON A B C D 5Min 5-6

Refer to the figure. Name two acute vertical angles. (over Lesson 1-5) Refer to the figure. Name two acute vertical angles. A. AEB and AED B. AED and BEC C. DEC and CEB D. AEB and DEC A B C D 5Min 6-1

Refer to the figure. Name a linear pair whose vertex is E. (over Lesson 1-5) Refer to the figure. Name a linear pair whose vertex is E. A. AEB, DEC B. AEB, AED C. AED, BEC D. CED, BEA A B C D 5Min 6-2

Refer to the figure. Name an angle supplementary to BEC. (over Lesson 1-5) Refer to the figure. Name an angle supplementary to BEC. A. CED B. DEA C. AED D. AEC A B C D 5Min 6-3

(over Lesson 1-5) 1 and 2 are a pair of supplementary angles, and the measure of 1 is twice the measure of 2. Find the measures of both angles. A. m1 = 60, m2 = 120 B. m1 = 30, m2 = 60 C. m1 = 120, m2 = 60 D. m1 = 60, m2 = 30 A B C D 5Min 6-4

(over Lesson 1-5) If RS is perpendicular to ST and is the angle bisector of RST, what is mTSV? A. 180 B. 90 C. 60 D. 45 A B C D 5Min 6-5

If two angles are both congruent and supplementary, they must be ? . (over Lesson 1-5) If two angles are both congruent and supplementary, they must be ? . A. two right angles B. two acute angles C. two obtuse angles D. an obtuse and an acute angle A B C D 5Min 6-6

Name polygon A by its number of sides. (over Lesson 1-6) Name polygon A by its number of sides. A. pentagon B. nonagon C. heptagon D. octagon A B C D 5Min 7-1

Name polygon B by its number of sides. (over Lesson 1-6) Name polygon B by its number of sides. A. pentagon B. nonagon C. heptagon D. octagon A B C D 5Min 7-2

Find the perimeter of polygon A. (over Lesson 1-6) Find the perimeter of polygon A. A. 35 cm B. 40 cm C. 25 cm D. 30 cm A B C D 5Min 7-3

Find the perimeter of polygon B. (over Lesson 1-6) Find the perimeter of polygon B. A. 40 in. B. 45 in. C. 42 in. D. 43 in. A B C D 5Min 7-4

Classify the polygons as regular or irregular. (over Lesson 1-6) Classify the polygons as regular or irregular. A. polygon A: regular; polygon B: regular B. polygon A: regular; polygon B: irregular C. polygon A: irregular; polygon B: regular D. polygon A: irregular; polygon B: irregular A B C D 5Min 7-5

(over Lesson 1-6) What is the area of rectangle FGHJ if its vertices are F(–3, 2), G(4, 2), H(4, –3), and J(–3, –3)? A. 24 square units B. 35 square units C. 49 square units D. 25 square units A B C D 5Min 7-6

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