CCSS

Content StandardsG.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.Mathematical Practices4 Model with mathematics.6 Attend to precision.

Then/Now

You used basic geometric concepts and properties to solve problems.

• Identify and model points, lines, and planes.

• Identify intersecting lines and planes.

Vocabulary

• undefined term

• point

• line

• plane

• collinear

• coplanar

• intersection

• definition

• defined term

• space

Concept

Example 1

Name Lines and Planes

A. Use the figure to name a line containing point K.

B. Use the figure to name a plane containing point L.

Example 1a

A. Use the figure to name a line containing the point X.

A. line X

B. line c

C. line Z

D.

Example 1b

B. Use the figure to name a plane containing point Z.

A. plane XY

B. plane c

C. plane XQY

D. plane P

Example 2

Model Points, Lines, and Planes

A. Name the geometric shape modeled by a 10 × 12 patio.

Answer:

Example 2

Model Points, Lines, and Planes

B. Name the geometric shape modeled by a button on a table.

Answer:

Example 2a

A. point

B. line segment

C. plane

D. none of the above

A. Name the geometric shape modeled by a colored dot on a map used to mark the location of a city.

Example 2b

A. point

B. line segment

C. plane

D. none of the above

B. Name the geometric shape modeled by the ceiling of your classroom.

Example 3

Draw Geometric Figures

Example 3

Draw Geometric Figures

A. B.

C. D.

Example 3a

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.

Example 3b

A. B.

C. D.

Example 3b

Example 4

Interpret Drawings

A. How many planes appear in this figure?

Answer:

Example 4

Interpret Drawings

B. Name three points that are collinear.

Answer:

Example 4

Interpret Drawings

C. Are points A, B, C, and D coplanar? Explain.

Answer:

Example 4

Interpret Drawings

Answer:

Example 4a

A. one

B. two

C. three

D. four

A. How many planes appear in this figure?

Example 4b

A. B, O, and X

B. X, O, and N

C. R, O, and B

D. A, X, and Z

B. Name three points that are collinear.

Example 4c

A. yes

B. no

C. cannot be determined

C. Are points X, O, and R coplanar?

Example 4d

A. point X

B. point N

C. point R

D. point A

CCSS

Content StandardsG.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).Mathematical Practices6 Attend to precision.

Then/Now

You identified and modeled points, lines, and planes.

• Measure segments.

• Calculate with measures.

Vocabulary

• line segment

• betweenness of points

• between

• congruent segments

• construction

Example 1

Length in Metric Units

A. Find the length of AB using the ruler.

Answer:

Example 1

Length in Metric Units

B. Find the length of AB using the ruler.

Answer:

Example 1a

A. 2 mmB. 1.8 mmC. 18 mmD. 20 mm

Example 1b

A. 1 cmB. 2 cmC. 2.5 cmD. 3 cm

B.

Example 2

Length in Standard Units

A.

Example 2

Length in Standard Units

B.

Example 2a

A.

A.

B.

C.

D.

Example 2a

A.

B.

C.

D.

B.

Concept

Example 3

Find Measurements by Adding

Find XZ. Assume that the figure is not drawn to scale.

XZ is the measure of XZ. Point Y is between X and Z. XZ can be found by adding XY and YZ.

___

Example 3

A. 16.8 mm

B. 57.4 mm

C. 67.2 mm

D. 84 mm

Find BD. Assume that the figure is not drawn to scale.

B

C

D

16.8 mm

50.4 mm

Example 4

Find Measurements by Subtracting

Find LM. Assume that the figure is not drawn to scale.

Point M is between L and N.

LM + MN = LN Betweenness of points

Example 4

Find TU. Assume that the figure is not drawn to scale.

TU

V3 in

A.

B.

C.

D.

in.

in.

in.

in.

Example 5

Write and Solve Equations to Find Measurements

ALGEBRA Find the value of x and ST if T is between S and U, ST = 7x, SU = 45, and TU = 5x – 3.

ST + TU = SU Betweenness of points

Draw a figure to represent this situation.

Example 5

A. n = 3; WX = 8

B. n = 3; WX = 9

C. n = 9; WX = 27

D. n = 9; WX = 44

ALGEBRA Find the value of n and WX if W is between X and Y, WX = 6n – 10, XY = 17, and WY = 3n.

Concept

Example 6

Congruent Segments

FONTS The Arial font is often used because it is easy to read. Study the word time shown in Arial type. Each letter can be broken into individual segments. The letter T has two segments, a short horizontal segment, and a longer vertical segment. Assume that all segments overlap where they meet. Which segments are congruent?

TIME

Example 6

A. barbecuing and beachB. board games and museumsC. beach and picnicD. zoo and board games

LEISURE ACTIVITIES The graph shows the percent of adults who participated in selected activities. Suppose a segment was drawn along the height of each bar. Which categories would have segments that are congruent?

CCSS

Content StandardsG.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).Mathematical Practices2 Reason abstractly and quantitatively.7 Look for and make use of structure.

Then/Now

You graphed points on the coordinate plane.

• Find the distance between two points.

• Find the midpoint of a segment.

Vocabulary

• distance

• irrational number

• midpoint

• segment bisector

Concept

Example 1

Find Distance on a Number Line

Use the number line to find QR.

Answer:

Example 1

A. 2

B. 8

C. –2

D. –8

Use the number line to find AX.

Concept

Example 2

Find Distance on a Coordinate Plane

Find the distance between E(–4, 1) and F(3, –1).

A. 4

B.

C.

D.

Example 2

Find the distance between A(–3, 4) and M(1, 2).

Concept

Example 3

Find Midpoint on a Number Line

DECORATING Marco places a couch so that its end is perpendicular and 2.5 feet away from the wall. The couch is 90” wide. How far is the midpoint of the couch back from the wall in feet?

Example 3

A. 330 ft

B. 660 ft

C. 990 ft

D. 1320 ft

DRAG RACING The length of a drag racing strip is mile long. How many feet from the finish line is

the midpoint of the racing strip?

Concept

Example 4

Find Midpoint in Coordinate Plane

Answer:

Example 4

A. (–10, –6)

B. (–5, –3)

C. (6, 12)

D. (–6, –12)

Example 5

Find the Coordinates of an Endpoint

Example 5

A. (3.5, 1)

B. (–10, 13)

C. (15, –1)

D. (17, –11)

Find the coordinates of R if N (8, –3) is the midpointof RS and S has coordinates (–1, 5).

Example 6

Use Algebra to Find Measures

Example 6

A. 1

B. 10

C. 5

D. 3

CCSS

Content StandardsG.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).Mathematical Practices5 Use appropriate tools strategically.6 Attend to precision.

Then/Now

You measured line segments.

• Measure and classify angles.

• Identify and use congruent angles and the bisector of an angle.

Vocabulary

• ray

• opposite rays

• angle

• side

• vertex

• interior

• exterior

• degree

• right angle

• acute angle

• obtuse angle

• angle bisector

Example 1

Angles and Their Parts

A. Name all angles that have B as a vertex.

Answer:

Example 1

Angles and Their Parts

Answer:

B. Name the sides of ∠5.

Example 1

Angles and Their Parts

C.

Example 1a

A.

A.

B.

C.

D.

Example 1b

B.

A.

B.

C.

D. none of these

A.

B.

C.

D.

Example 1c

C. Which of the following is another name for ∠3?

Concept

Example 2

Measure and Classify Angles

A. Measure ∠TYV and classify it as right, acute, or obtuse.

Answer:

Example 2

Measure and Classify Angles

Answer:

Example 2

Measure and Classify Angles

Example 2a

A. 30°, acute

B. 30°, obtuse

C. 150°, acute

D. 150°, obtuse

A. Measure ∠CZD and classify it as right, acute, or obtuse.

Example 2b

A. 60°, acute

B. 90°, acute

C. 90°, right

D. 90°, obtuse

B. Measure ∠CZE and classify it as right, acute, or obtuse.

Example 2c

A. 30°, acute

B. 30°, obtuse

C. 150°, acute

D. 150°, obtuse

C. Measure ∠DZX and classify it as right, acute, or obtuse.

Example 3

Measure and Classify Angles

INTERIOR DESIGN Wall stickers of standard shapes are often used to provide a stimulating environment for a young child’s room. A five-pointed star sticker is shown with vertices labeled. Find m∠GBH and m∠HCI if ∠GBH ≅ ∠HCI, m∠GBH = 2x + 5, and m∠HCI = 3x – 10.

Example 3

A. m∠BHC = 105, m∠DJE = 105

B. m∠BHC = 35, m∠DJE = 35

C. m∠BHC = 35, m∠DJE = 105

D. m∠BHC = 105, m∠DJE = 35

Find m∠BHC and m∠DJE if ∠BHC ≅ ∠DJE, m∠BHC = 4x + 5, and m∠DJE = 3x + 30.

CCSS

Content StandardsPreparation for G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. Mathematical Practices2 Reason abstractly and quantitatively. 3 Construct viable arguments and critique the reasoning of others.

Then/Now

You measured and classified angles.

• Identify and use special pairs of angles.

• Identify perpendicular lines.

Vocabulary

• adjacent angles

• linear pair

• vertical angles

• complementary angles

• supplementary angles

• perpendicular

Concept

Example 1

Identify Angle Pairs

A. ROADWAYS Name an angle pair that satisfies the condition two angles that form a linear pair.

Example 1

Identify Angle Pairs

B. ROADWAYS Name an angle pair that satisfies the condition two acute vertical angles.

Example 1a

A. ∠CAD and ∠DAE

B. ∠FAE and ∠FAN

C. ∠CAB and ∠NAB

D. ∠BAD and ∠DAC

A. Name two adjacent angles whose sum is less than 90.

Example 1b

A. ∠BAN and ∠EAD

B. ∠BAD and ∠BAN

C. ∠BAC and ∠CAE

D. ∠FAN and ∠DAC

B. Name two acute vertical angles.

Concept

Example 2

Angle Measure

ALGEBRA Find the measures of two supplementary angles if the measure of one angle is 6 less than five times the measure of the other angle.

Example 2

A. 1°, 1°

B. 21°, 111°

C. 16°, 74°

D. 14°, 76°

ALGEBRA Find the measures of two complementary angles if one angle measures six degrees less than five times the measure of the other.

Concept

Example 3

Perpendicular Lines

ALGEBRA Find x and y so thatKO and HM are perpendicular.

Example 3

A. x = 5

B. x = 10

C. x = 15

D. x = 20

Concept

Example 4

Interpret Figures

A. Determine whether the following statement can be justified from the figure below. Explain.m∠VYT = 90

Example 4

Interpret Figures

B. Determine whether the following statement can be justified from the figure below. Explain.∠TYW and ∠TYU are supplementary.

Example 4

Interpret Figures

C. Determine whether the following statement can be justified from the figure below. Explain.∠VYW and ∠TYS are adjacent angles.

Example 4a

A. yes

B. no

A. Determine whether the statement m∠XAY = 90 can be assumed from the figure.

Example 4b

A. yes

B. no

B. Determine whether the statement ∠TAU is complementary to ∠UAY can be assumed from the figure.

Example 4c

A. yes

B. no

C. Determine whether the statement ∠UAX is adjacent to ∠UXA can be assumed from the figure.

CCSS

Content StandardsG.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.Mathematical Practices2 Reason abstractly and quantitatively.6 Attend to precision.

Then/Now

You measured one-dimensional figures.

• Identify and name polygons.

• Find perimeter, circumference, and area of two-dimensional figures.

Vocabulary

• polygon

• vertex of a polygon

• concave

• convex

• n-gon

• equilateral polygon

• equiangular polygon

• regular polygon

• perimeter

• circumference

• area

Concept

Example 1

Name and Classify Polygons

A. Name the polygon by its number of sides. Then classify it as convex or concave and regular or irregular.

Answer:

Example 1

Name and Classify Polygons

B. Name the polygon by its number of sides. Then classify it as convex or concave and regular or irregular.

Answer:

Example 1a

A. triangle, concave, regular

B. triangle, convex, irregular

C. quadrilateral, convex, regular

D. triangle, convex, regular

A. Name the polygon by the number of sides. Then classify it as convex or concave and regular or irregular.

Example 1b

A. quadrilateral, convex, irregular

B. pentagon, convex, irregular

C. quadrilateral, convex, regular

D. quadrilateral, concave, irregular

B. Name the polygon by the number of sides. Then classify it as convex or concave and regular or irregular.

Concept

Example 2

Find Perimeter and Area

A. Find the perimeter and area of the figure.

Answer:

Example 2

Find Perimeter and Area

B. Find the circumference and area of the figure.

Example 2a

A. P = 12.4 cm, A = 24.8 cm2

B. P = 24.8 cm, A = 34.83 cm2

C. P = 34.83 cm, A = 69.66 cm2

D. P = 24.4 cm, A = 32.3 cm2

A. Find the perimeter and area of the figure.

Example 2b

A. C ≈ 25.1 m, A ≈ 50.3 m2

B. C ≈ 25.1 m, A ≈ 201.1 m2

C. C ≈ 50.3 m, A ≈ 201.1 m2

D. C ≈ 201.1 m, A ≈ 402.1 m2

B. Find the circumference and area of the figure.

Example 3

Terri has 19 feet of tape to mark an area in the classroom where the students may read. Which of these shapes has a perimeter or circumference that would use most or all of the tape?A square with side length of 5 feetB circle with the radius of 3 feetC right triangle with each leg length of 6 feetD rectangle with a length of 8 feet and a width of 3 feet

Largest Area

Example 3

A. a rectangle with a length of 26 inches and a width of 18 inches

B. a square with side length of 22 inches

C. a right triangle with each leg length of 26 inches

D. a circle with radius of 14 inches

Each of the following shapes has a perimeter of about 88 inches. Which one has the greatest area?

Example 4

Perimeter and Area on the Coordinate Plane

Find the perimeter and area of a pentagon ABCDE with A(0, 4), B(4, 0), C(3, –4), D(–3, –4), and E(–3, 1).

Example 4

A. 17.9

B. 22

C. 13.3

D. 9.1

Find the perimeter of quadrilateral WXYZ with W(2, 4), X(–3, 3), Y(–1, 0), and Z(3, –1).

CCSS

Content StandardsG.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Mathematical Practices2 Reason abstractly and quantitatively.6 Attend to precision.

Then/Now

You identified and named two-dimensional figures.

• Identify and name three-dimensional figures.

• Find surface area and volume.

Vocabulary

• polyhedron

• face

• edge

• vertex

• prism

• base

• pyramid

• cylinder

• cone

• sphere

• regular polyhedron

• Platonic solid

• surface area

• volume

Concept

Example 1

Identify Solids

A. Determine whether the solid is a polyhedron. Then identify the solid. If it is a polyhedron, name the bases, faces, edges, and vertices.

Example 1

Identify Solids

B. Determine whether the solid is a polyhedron. Then identify the solid. If it is a polyhedron, name the bases, faces, edges, and vertices.

Example 1

Identify Solids

C. Determine whether the solid is a polyhedron. Then identify the solid. If it is a polyhedron, name the bases, faces, edges, and vertices.

Example 1

A. triangular pyramid

B. pentagonal prism

C. rectangular prism

D. square pyramid

A. Identify the solid.

Example 1

A. cone

B. cylinder

C. pyramid

D. polyhedron

B. Identify the solid.

Example 1

A. triangular prism

B. triangular pyramid

C. rectangular pyramid

D. cone

C. Identify the solid.

Concept

Concept

Example 2

Find Surface Area and Volume

Find the surface area and volume of the cone.

Example 2

A. surface area = 288 ft2

volume = 336 ft3

B. surface area = 336 ft2

volume = 288 ft3

C. surface area = 26 ft2

volume = 60 ft3

D. surface area = 488 ft2

volume = 122 ft3

Find the surface area and volume of the triangular prism.

Example 3

Surface Area and Volume

A. CONTAINERS Mike is creating a mailing tubewhich can be used to mail posters andarchitectural plans. The diameter of the base is

inches, and the height is feet. Find the

amount of cardboard Mike needs to make the tube.

Example 3

Surface Area and Volume

B. CONTAINERS Mike is creating a mailing tubewhich can be used to mail posters andarchitectural plans. The diameter of the base is

inches, and the height is feet. Find the

volume of the tube.

Example 3

A. surface area = 2520 in2

B. surface area = 18 in2

C. surface area = 180 in2

D. surface area = 1144 in2

A. Jenny has some boxes for shipping merchandise. Each box is in the shape of a rectangular prism with a length of 18 inches, a width of 14 inches, and a height of 10 inches. Find the surface area of the box.

Example 3

A. volume = 1144 in3

B. volume = 14 in3

C. volume = 2520 in3

D. volume = 3600 in3

B. Jenny has some boxes for shipping merchandise. Each box is in the shape of a rectangular prism with a length of 18 inches, a width of 14 inches, and a height of 10 inches. Find the volume of the box.