Utah State Standards - Wasatch



Geometry

Wasatch High School

2011-2012

Student Name _______________________

Teacher Name _______________________

Geometry

State Standards

Standard 1: Students will use algebraic, spatial, and logical reasoning to solve geometry problems.

Objective 1: Use inductive and deductive reasoning to develop mathematical arguments.

a. Write conditional statements, converses, and inverses, and determine the truth value of these statements.

b. Formulate conjectures using inductive reasoning.

c. Prove a statement false by using a counterexample.

Objective 2: Analyze characteristics and properties of angles.

a. Use accepted geometric notation for lines, segments, rays, angles, similarity, and congruence.

b. Identify and determine relationships in adjacent, complementary, supplementary, or vertical angles and linear pairs.

c. Classify angle pairs formed by two lines and a transversal.

d. Prove relationships in angle pairs.

e. Prove lines parallel or perpendicular using slope or angle relationships.

Objective 3: Analyze characteristics and properties of triangles.

a. Prove congruency and similarity of triangles using postulates and theorems.

b. Prove the Pythagorean Theorem in multiple ways, find missing sides of right triangles using the Pythagorean Theorem, and determine whether a triangle is a right triangle using the converse of the Pythagorean Theorem.

c. Prove and apply theorems involving isosceles triangles.

d. Apply triangle inequality theorems.

e. Identify medians, altitudes, and angle bisectors of a triangle, and the perpendicular bisectors of the sides of a triangle, and justify the concurrency theorems.

Objective 4: Analyze characteristics and properties of polygons and circles.

a. Use examples and counterexamples to classify subsets of quadrilaterals.

b. Prove properties of quadrilaterals using triangle congruence relationships, postulates, and theorems.

c. Derive, justify, and use formulas for the number of diagonals, lines of symmetry, angle measures, perimeter, and area of regular polygons.

d. Define radius, diameter, chord, secant, arc, sector, central angle, inscribed angle, and tangent of a circle, and solve problems using their properties.

e. Show the relationship between intercepted arcs and inscribed or central angles, and find their measures.

Objective 5: Perform basic geometric constructions, describing and justifying the procedures used.

a. Investigate geometric relationships using constructions.

b. Copy and bisect angles and segments.

c. Construct perpendicular and parallel lines.

d. Justify procedures used to construct geometric figures.

e. Discover and investigate conjectures about geometric properties using constructions.

Objective 6: Analyze characteristics and properties of three-dimensional figures.

a. Identify and classify prisms, pyramids, cylinders and cones based on the shape of their base(s).

b. Identify three-dimensional objects from different perspectives using nets, cross-sections, and two-dimensional views.

c. Describe the symmetries of three-dimensional figures.

d. Describe relationships between the faces, edges, and vertices of polyhedra.

Standard 2: Students will use the language and operations of algebra to explore geometric relationships with coordinate geometry.

Objective 1: Describe the properties and attributes of lines and line segments using coordinate geometry.

a. Verify the classifications of geometric figures using coordinate geometry to find lengths and slopes.

b. Find the distance between two given points and find the coordinates of the midpoint.

c. Write an equation of a line perpendicular or a line parallel to a line through a given point.

Objective 2: Describe spatial relationships using coordinate geometry.

a. Graph a circle given the equation in the form (x − h)2 + ( y − k)2 = r 2, and write the equation when given the graph.

b. Determine whether points in a set are collinear.

Standard 3: Students will extend concepts of proportion and similarity to trigonometric ratios.

Objective 1: Use triangle relationships to solve problems.

a. Solve problems using the properties of special right triangles, e.g., 30°, 60°, 90° or 45°, 45°, 90°.

b. Identify the trigonometric relationships of sine, cosine, and tangent with the appropriate ratio of sides of a right triangle.

c. Express trigonometric relationships using exact values and approximations.

Objective 2: Use the trigonometric ratios of sine, cosine, and tangent to represent and solve for missing parts of triangles.

a. Find the angle measure in degrees when given the trigonometric ratio.

b. Find the trigonometric ratio given the angle measure in degrees, using a calculator.

c. Find unknown measures of right triangles using sine, cosine, and tangent functions and inverse trigonometric functions.

Standard 4: Students will use algebraic, spatial, and logical reasoning to solve measurement problems.

Objective 1: Find measurements of plane and solid figures.

a. Find linear and angle measures in real-world situations using appropriate tools or technology.

b. Develop surface area and volume formulas for polyhedra, cones, and cylinders.

c. Determine perimeter, area, surface area, lateral area, and volume for prisms, cylinders, pyramids, cones, and spheres when given the formulas.

d. Calculate or estimate the area of an irregular region.

e. Find the length of an arc and the area of a sector when given the angle measure and radius.

Homework (This list is subject to change).

Logic Notebook Pages 6-9

Homework: p.8 #17-23 odd, 27, 33, 35, 38. Explain your reasoning.

Homework: p.27 #11,13,15,22,23,25,28,31

Points, Lines, Planes, Segments, Measurements Notebook Pages 10-13

Homework: p.16 #9-12,14,16,17,18,23, 24-29 all

p.59 #13-27 odds, 31

p.71 #15-23 odds, 25-33 odds

Angles Notebook Pages 13-20

Homework: p. 93 #9-19 odds, 24,27,28 p.100 #11,17,23,27

Homework: p. 108 #11-23 odds, 27

Homework: p.113 #8,9,10,14,16,23 p.120 #13,14,16,19,22,25,28

p.126 #9,11,13,14,15,17,21,24

Coordinate Geometry Notebook Pages 20-25

Homework: p.266 #13-19 odds, 27 p.80 #21-27 odds, 30, 36-40

Homework: p.172 #11,15,18,25,28 p.178# 13,15,17,25

Parallel Lines Notebook Pages 25-29

Homework: p.178 #14,16,18,26-30

Homework: p.152 #13-37 odds, 46

Homework: p.166 #9-19, 25,26

Triangles Notebook Pages 30-37

Homework: p.191 #9-21 odds, 26,27,32

Homework: p.196 #9-19 odds, 22, 26-29 p.249 #7,9,11,15

Homework: p.231 #9,11,13,17 p.238 #9-15 odds p.242 #7-11, 17,18

Homework: p.286 #12,14,18,19 p.293 #9-21 odds, 22 p.299 #9,11,13

Similar Triangle Notebook Pages 37-40

Homework: p.353 #17-37 odds, 41,42

Homework: p.359 #9-19 odds, 24, 25,30

Homework: p.366 #7,8,9,11,16,20 p.372 #15-21 odds, 24,28

Congruent Triangles Notebook Pages 41- 42

Homework: p.213 #12-15,17,19,22,23

p.218 #15,17,19-22,25-28

Trigonometry Notebook Pages 49-55

Homework: p.260 #17-35 odds,

Homework: p.557 #7-13 odds,14,16 p.562 #7-13 odds, 16,17

Homework: p.568 #9-19 odds, 20 p.576 #13-27 odds, 31, 36

Polygon and Polyhedron Notebook Pages 49-55

Homework: p.405 #13-33 p.437 #13-17 odds

Homework. p. 499 #15,17,21-29 odds, 32,33

Quadrilateral, Polygons Notebook Pages 56- 62

Homework: p.411 #9-17 odds, 18

Homework: p.320 #11-23 odds, 28,29 p.325 #7-12, 14,15,16

Homework: p.330 #16-21, 23-37 odds, 39-44

Homework: p.337 #10,13-21 odds, 30,32,33,34

Circles Notebook Pages 62- 70

Homework: p.457 #13-27,29,30 p.481 #11,13,17,19,22,26

Homework: p.466 #13-23 odds, 31,41 p.590 #9,11,12-14,19,21,27

Homework: p.596 #9-14, 19,27 p.604 #9-17 odds

Homework: p.616 #8-13

Homework: p.621 #15-25 odds,26,27,34,36

Area and Volume Notebook Pages 71- 83

Homework: p.416 #9,12,13,15,21 p.423 #11,12,14,15

Homework: p.429 #7,8,9,11,12,15 p.486 #9,11,13,17

Homework: p.487 #19-21, 24,25,27,29 p.482 #27

Homework: p.508 #7,9,11,16,18,19

Homework: p.509 #13-15,17

Homework: p.520 #7-14 all p.531 #7-13 (surface area only)

Homework: p.513 #12,14,15,18,19,20 p.525 #9-17 odds, 22 p.531 #8,11,16,21

Utah State Standards

1.1.a Write conditional statements, converses, and inverses, and determine the truth value of theses statements.

1.1.b Formulate conjectures using inductive reasoning.

1.1.c Prove a statement false by using a counterexample.

Conjecture

Inductive reasoning-

Ex. Make a conjecture about “Pascal’s Triangle”. Explain your reasoning.

Ex. Make a conjecture about the next number based on the pattern. 1, 3, 6, 10, 15 Explain your reasoning.

Ex. Make a conjecture about the next number based on the pattern. 2, 4, 12, 48, 240

Explain your reasoning.

Ex. Make a conjecture about the next number based on the pattern. 1,1,2,3,5,8,…

Explain your reasoning.

Ex. Make a conjecture about the next number based on the pattern. 1,4,9,16,…

Explain your reasoning.

True means always true.

It only takes one false example to show that a conjecture is not true.

Counterexample

Ex. Find a counterexample to the statement: All geometry students have blue eyes.

Example p.6 #4

Ex. Based on the table showing unemployment rates for various counties in Kansas, find a counterexample for the following statement: The unemployment rate is highest in the counties with the most people. Source: Labor Market Information Services-Kansas Dept. of Human Resources

|County |Civilian Labor Force |Rate |

|Shawnee |90,254 |3.1% |

|Jefferson |9,937 |3.0% |

|Jackson |8,915 |2.8% |

|Douglas |55,730 |3.2% |

|Osage |10,182 |4.0% |

|Wabaunsee |3,575 |3.0% |

|Pottawatomie |11,025 |2.1% |

Homework: p.8 #17-23 odd, 27, 33, 35, 38. Explain your reasoning.

Worksheet p.44 Exercise Set 1.2 evens. Explain your reasoning.

Conditional Statement

If p, then q.

p

q

Ex. Write your own example of a conditional statement.

Ex. Identify the hypothesis and conclusion.

Ex. Put the given information in a conditional statement. Get $1500 cash back when you buy a new car.

Converse statement

If , then .

Negation ~

Inverse statement

If , then .

Contrapositive statement

If , then .

Ex. Write the converse, inverse, and contrapositive of the statement:

If a shape is a square, then it is a rectangle.

Ex. Write the converse, inverse, and contrapositive of the statement and determine the truth value for each. If false, give a counterexample.

If a person lives in Heber, then that person lives in Utah.

Truth value:

Converse:

Truth value:

Inverse:

Truth value:

Contrapositive:

Truth value:

Building a logical argument:

Ex. If a person buys worms, then they can go fishing.

If a person can go fishing, then they can catch dinner.

New logical statement:

Ex. If x=7, then 4x=28.

If 4x=28, then 20x=140.

New logical statement:

Homework: p.27 #11,13,15,22,23,25,28,31

CDAS

Utah State Standards

1.2.a Use accepted geometric notation for lines, segments, congruence.

1.5.b Copy segments using constructions.

2.2.b Determine whether points in a set are collinear.

4.1.a Find linear measures in real-world situations using appropriate tools or technology.

Reading p.77-79 Introducing Geometry

point-

*No shape or size

Diagram

Named by

Line-

*No thickness or width

Diagram

Named by

Ray-

Diagram

Named by

Plane-

*No thickness

Diagram

Named by

Collinear-

Coplanar-

Practice. Model and study real life objects.

What term would we use to model the floor?

What term would we use to model the hand on a clock?

What term would we use to model the corner of the driveway where the concrete meets the road?

Intersection-

2 planes

2 lines

a plane and a line

*Points, lines, and planes have no real measurement. In real life, we have things with a certain shape and size. Things we can measure!

Line segment or Segment-

Diagram

Named by

Practice p.15 #3-7

Length of a segment

Practice with rulers!

centimeter side

inches side

Practice measuring the following segments

1. Use cm.

2. Use inches.

3. Use inches. [pic]

Segment Addition

Draw a diagram of a point that lies between two others.

Practice. p.59 #4,6

Homework. p.16 #9-12,14,16,17,18,23, 24-29 all

p.59 #13-27 odds, 31

p.71 #15-23 odds, 25-33 odds

Utah State Standards

1.2.a Use accepted geometric notation for rays, angles.

1.5.b Copy and bisect angles using constructions.

1.5.d Justify procedures used to construct geometric figures.

4.1.a Find angle measures in real-world situations using appropriate tools or technology.

Claudius __________ first used the unit of measure we think of as a ______________.

Rotation: once around ______, half a turn ______

[pic]= ________ of a turn around a circle.

Opposite rays-

Diagram

Angle-

Diagram

Label the following:

Sides of the angle Interior and exterior of an angle

Vertex

Naming an angle

Practice.

Name the vertex of [pic].

Name the sides of [pic].

Write another name for [pic].

We measure an angle in units of ____________.

Notation for the measure of an angle

A __________________ is a tool to help us measure angles.

Practice. Measure the following angles using a protractor.

Most computer programs will also measure angles for us!

Classifying Angles by their measure

Acute angles

Diagram

Right Angles

Diagram

Obtuse Angles

Diagram

Straight Angles

Diagram

Practice. Classify each angle as right, acute, or obtuse.

[pic]

[pic]

HW: p. 93 #9-19 odds, 24,27,28 p.100 #11,17,23,27

Utah State Standards

1.2.a Use accepted geometric notation for rays, angles.

1.5.b Copy and bisect angles using constructions.

1.5.d Justify procedures used to construct geometric figures.

4.1.a Find angle measures in real-world situations using appropriate tools or technology.

Angle Addition

Draw an angle with vertex Q and a point H in its interior. Draw a ray from Q through H.

Practice. P.108 #7,8

Congruent Angles-

Diagram

Notation

Ex. Finding Angle Measure with algebra.

Given: [pic]

[pic]

Find x

Angle bisector-

Diagram

An angle bisector creates two ___________ angles.

Practice.

[pic]are opposite rays. [pic]bisects [pic]

If [pic]and [pic],

Find [pic].

Homework Problems p. 108 #11-23 odds, 27

Utah State Standards

1.2.b Identify and determine relationships in adjacent, complementary, supplementary, or vertical angles and linear pairs.

1.5.c Construct perpendicular lines

Special Pairs of Angles (two angles)

Adjacent Angles-

Diagram

*Things to watch out for…

Shared interior

No common vertex

Linear Pair-

Diagram

Supplementary Angles-

2 Diagrams

Ex. Supplement of an angle

1. What is the supplement of a 120 degree angle?

2. How could you use algebra to write the “supplement of an angle”?

3. 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.

Complementary Angles-

2 Diagrams

Theorem- If the non-common sides of two adjacent angles form a right angle, then the angles are ___________________ angles.

Ex. Complements of an Angle

1. What is the complement of a 40 degree angle?

2. How could you use algebra to write the “complement of an angle”?

3. Find the measures of two complementary angles if the difference of the measures of the two angles is 12.

Vertical Angles

Diagram

*Must be formed by two intersecting lines!

Vertical angles are _____________.

Perpendicular Lines-

Perpendicular lines intersect to form ________ right angles.

Diagram

Practice.

Find the measure of angle 1.

Find x so that [pic]

*Never assume lines are perpendicular!

HW: p.113 #8,9,10,14,16,23 p.120 #13,14,16,19,22,25,28

p.126 #9,11,13,14,15,17,21,24

CDAS

Utah State Standards

2.1.b Find the distance between two given points and find the coordinates of the midpoint.

1.5.b Bisect segments using constructions.

4.2.b Solve problems using the distance formula.

Coordinate System

Who is Rene Descartes?

Points on the system

Quadrants

Practice. Drawing geometric figures on the coordinate system.

[pic]on a coordinate plane contains Q(-2,4) and R(4, -4). Add point T so that T is collinear with these points.

Practice Square Roots p.552

Investigation. Distance in Coordinate Geometry

Segment length on the coordinate system

We need the coordinates of the endpoints.

The Distance between any two points [pic] and [pic]:

Ex. Using the distance formula to find the length of a segment

1. Find the length of [pic] given the coordinates E (-4, 1) and F(3, -1)

2. Find the length of [pic] given the coordinates G (2, 8) and H (-2, 2)

Perimeter of a closed figure

Ex. Finding perimeter on a coordinate Plane

*Use the distance formula

Find the perimeter of triangle PQR if P(-5,1), Q(-1,4), and R(-6,-8).

Congruent-

Symbol

Notation - _____________________ means AB = CD.

Diagram-

Midpoint of a Segment-

Practice. p.66 #7

Investigation. Midpoint Conjecture

Finding the midpoint on a coordinate system

Ex. Finding the midpoint of a segment.

1. Find the coordinates of M, the midpoint of [pic] for P (-1, 2) and R (6, 1).

2. Find the coordinates of D if E is the midpoint of [pic]. E (-6,4) and F(-5, -3).

Segment Bisector-

HW: p.266 #13-19 odds, 27 p.80 #21-27 odds, 30, 36-40

Utah State Standards

1.2.e Prove lines parallel or perpendicular using slope or angle relationships

2.1.c Write an equation of a line perpendicular or a line parallel to a line through a given point.

Investigation Equations of Lines

The equation of a line on a coordinate plane describes where it is and how it slants up or down.

Slope

Practice. p.171 #4-6

Positive Slope Negative Slope Slope = 0 Slope undefined

Slope-intercept form of a line.

y = mx + b m is the ___________. b is the _______________.

Practice. p. 177 #6, 3, 7

Ex. Writing the equation of a line on a coordinate plane.

1. Write the equation of a line given the slope = 2, and the y-intercept is (0, 6)

2. Write the equation of a line given the slope =3, and the point (-1, 4) that lies on the line.

3. Write the equation of a line given two points on the line (-2, 6) and (1, 5).

HW: p.172 #11,15,18,25,28 p.178# 13,15,17,25

CDAS

Utah State Standards

1.2.e Prove lines parallel or perpendicular using slope or angle relationships

2.1.c Write an equation of a line perpendicular or a line parallel to a line through a given point.

Parallel Lines-

Symbol

Postulate- Two lines are parallel if and only if their slopes are _____________________.

Perpendicular Lines-

Symbol

Postulate- Two lines are perpendicular if and only if their slopes are ________________.

If we multiplied the two slopes together we get _______.

Ex. A line has a slope of 3, A second line has a slope of [pic].

1. Write the equation of a line perpendicular to the line described by

y = -2x + 5 , given a point on the line (-3, 4).

Check by graphing:

2. Write the equation of a line parallel to the line described by [pic], given a point on the line (-6, 1).

Check by graphing:

HW: p.178 #14,16,18,26-30

Utah State Standards

1.2.c Classify angle pairs formed by two lines and a transversal.

________________ lines are coplanar lines that do not intersect.

Segments and rays on those lines are also ___________.

Notation:

Diagram with symbols:

Symbol for not parallel to

_____________planes never intersect. Example and Diagram:

________________ lines do not intersect and are NOT coplanar (nor parallel)

Segments and rays on those lines are also __________.

Diagram

Practice problems p. 144 #4-11

A line that intersects two or more lines in a plane at different points is called a __________________. Ex.

Special angles formed by transversals

We will say line p is our transversal.

Angles on the same side of the transversal

are called ______________.

Angles on opposite sides of the transversal

are called _____________.

The lines crossed are q and r

The space between lines q and r is known

as ________________.

The space outside lines q and r is know as

________________.

Consecutive Interior Angles:

Sometimes called Same-Side Interior Angles

Alternate Interior Angles:

Alternate Exterior Angles:

Corresponding Angles: in the same relative position.

Investigation. Special Angle Relationships

Practice Problems p. 152 #4-7

**Special angles formed by a transversal are even more special when the lines crossed are ___________________.

Postulate: If two parallel lines are cut by a transversal, then corresponding angles are _________________.

Diagram

Ex. angle measures of special angle pairs.

[pic]

[pic]

Find [pic]

Theorem- If two parallel lines are cut by a ______________, then each pair of alternate interior angles is _______________.

Theorem- If two parallel lines are cut by a _______________, then each pair of consecutive interior (same-side interior) angles is ____________________.

Theorem- If two parallel lines are cut by a _______________, then each pair of alternate exterior angles is ____________________.

Practice. p.153 #38

HW: p.152 #13-37 odds, 46

Utah State Standards

1.2.e Prove lines parallel using angle relationships

If we are not on the coordinate system, how do we prove that lines are parallel?

Recall: slopes of parallel lines are ____________.

Ways to prove lines are parallel

1. Show corresponding angles are ___________.

2. Show alternate exterior angles are ____________.

3. Show consecutive interior angles are ______________.

4. Show alternate interior angles are ____________.

5. Show that two lines are _____________ to the same line.

*We must use one of these 5 rules to prove lines are parallel!!!!

Practice. p. 165 #4,5

HW: p.166 #9-19 (Give a reason with vocabulary words), 25,26

CDAS

Utah State Standards

1.3 Analyze characteristics and properties of triangles

1.3.c. Prove and apply theorems involving isosceles triangles.

2.1.a Verify the classifications of geometric figures using coordinate geometry to find the lengths and slopes.

Triangle-

Diagram

Sides:

Vertices:

Angles:

Investigation Triangle Classification

Classifying Triangles by Angles

Acute Triangle

Diagram

*special example: equiangular triangle

Obtuse Triangle

Diagram

Right Triangle

Diagram

Practice problems: classification by angle p. 190 #4-6

Classifying Triangles by Sides

Scalene Triangle

Diagram

Isosceles Triangle

Diagram

Vertex angle-

Base angles-

Practice. p.190 #4-6 by sides

Practice. 190 #7

Homework: p.191 #9-21 odds, 26,27,32

Investigation. Angles of a Triangle p.193 of textbook.

Angle Sum Theorem- The sum of the measures of the angles of a triangle is _________.

Practice Problems:

1. Find x 2. Find the missing angle measures.

Corollary: The acute angles of a right triangle are __________________.

Investigation. Discovering Properties of Isosceles Triangles

Isosceles Triangle Theorem-

Diagram and Abbreviation

Converse of Isosceles Triangle Theorem-

Diagram and Abbreviation

*Look at an equilateral triangle…

Corollary- A triangle is equilateral if and only if it is ___________________.

Corollary- Each angle measure of an equilateral triangle measures ________.

Diagram

Exterior Angle

Diagram

Remote Interior Angles

Exterior Angle Theorem- the measure of an exterior angle of a triangle is equal to the sum of _______________________________________________________________.

Practice.

Find the measure of each numbered angle.

HW: p.196 #9-19 odds, 22, 26-29 p.249 #7,9,11,15

Utah State Standards

1.3.e Identify medians, altitudes, and angle bisectors of a triangle, and the perpendicular bisectors of the sides of a triangle, and justify the concurrency theorems.

1.5.a Investigate geometric relationships using constructions

1.5.e Discover and investigate conjectures about geometric properties using constructions.

Concurrent Lines-

Diagram

Point of Concurrency

Sketchpad Investigation. Segments of a Triangle.

Perpendicular Bisector of a Segment

Are the 3 perpendicular bisectors of a triangle concurrent lines?

Circumcenter-

Equidistant from-

Angle bisector

Are the 3 angle bisectors of a triangle concurrent lines?

Incenter-

Equidistant from-

Medians of a triangle-

Diagram

Are the 3 medians concurrent lines?

Centroid-

Located-

Practice. p.231 #6,7

Altitude-

Diagram

Are the 3 altitudes of a triangle concurrent lines?

Orthocenter-

HW: p.231 #9,11,13,17 p.238 #9-15 odds p.242 #7-11, 17,18

Utah State Standards

1.3.d Apply triangle inequality theorems.

Investigation. Triangle Inequalities.

Inequality-

Exterior Angle Inequality Theorem

Diagram

Ex. p.286 #5

Angle-Side Inequality Relationships

Diagram

Practice.

Determine the relationship between the measures of the given angles.

[pic]

Triangle Inequality Theorem-

Diagram

Example. Determine whether the given measures can be the lengths of the sides of a triangle.

1. 2, 4, 5

2. 6, 8, 14

3. In [pic], PQ=7.2 and QR=5.2. Which measure cannot be PR?

a. 7

b. 9

c. 11

d. 13

Hinge Theorem (SAS Inequality Theorem)

Diagram

HW: p.286 #12,14,18,19 p.293 #9-21 odds, 22 p.299 #9,11,13

CDAS

Utah State Standards

1.2.a Use accepted geometric notation for similarity.

1.3.a Prove congruency and similarity of triangles using postulates and theorems.

Ratio-

Can be expressed as or or said:

The denominator cannot = 0.

Ex. The total number of students who participate in sports programs at Wasatch High School is 440. The total number of students in the school is 1100. Find the athlete-to-student ratio to the nearest tenth.

Proportion-

*Ratios must have the same units!

Ex. [pic]

To Solve a Proportion-

Ex. [pic] Ex. [pic]

Ex. A boxcar on a train has a length of 40 feet and a width of 9 feet. A scale

model is made with a length of 16 inches. Find the width of the model.

HW: p.353 #17-37 odds, 41,42

Utah State Standards

1.2.a Use accepted geometric notation for similarity.

1.3.a Prove congruency and similarity of triangles using postulates and theorems.

Similar Polygons-

Ex.

Symbol for Similar

Notation

Corresponding Angles Corresponding Sides

Practice. p.359 #4,5

Scale Factor-

*Depends on the order of comparison

Practice. p.359 #6,7

Ex. ABCDE ~ RSTUV

Find the scale factor of polygon ABCDE to RSTUV.

Find x and y.

HW: p.359 #9-19 odds, 24, 25,30

Investigation Shortcuts.

Utah State Standards

1.2.a Use accepted geometric notation for similarity.

1.3.a Prove congruency and similarity of triangles using postulates and theorems.

In similar triangles, we have three sets of congruent angles and three sets of sides in proportion. But there are shortcuts to prove similarity.

AA ~ (angle angle similarity)-

Diagram

SSS ~ (side side side similarity)-

Diagram

SAS~ (side angle side similarity)-

Diagram

Practice. p.365 #3,4,6

p.372 #6,7

HW: p.366 #7,8,9,11,16,20 p.372 #15-21 odds, 24,28

CDAS

Utah State Standards

1.3.a Prove congruence and similarity of triangles using postulates and theorems.

Congruent Triangles-

Diagram

Notation

Corresponding parts-

*CPCTC-

Practice. p.205 #7-9

Reading and Investigation. Shortcuts.

Ways to Prove Triangles are Congruent

1. All three pairs of corresponding angles are ___________ and all three pairs of corresponding sides are ____________.

2. SSS

3. SAS

4. ASA

5. AAS

*6. HL

SSS- Side-Side-Side Congruence-

Diagram

SAS- Side-Included Angle-Side Congruence-

Diagram

*must be the included angle-

Practice. p.213 #5,6

ASA- Angle-Included Side-Angle Congruence

Diagram

AAS- Angle-Angle-Side Congruence

Diagram

Practice.p.218 #6-9

HL- Hypotenuse and Leg Congruence

*For right triangles only!!!

Hypotenuse-

Diagram

HW: p.213 #12-15,17,19,22,23

p.218 #15,17,19-22,25-28

CDAS

Utah State Standards

1.3.b Prove the Pythagorean Theorem in multiple ways, find missing sides of right triangles using the Pythagorean Theorem, and determine whether a triangle is a right triangle using the converse of the Pythagorean Theorem

Right triangle review

Diagram

Investigation. Pythagorean Theorem.

Pythagorean Theorem-

Formula Diagram

Ex. Finding a missing side of a right triangle.

Ex. p.259#11 Ex. 2 Find x.

Converse of the Pythagorean Theorem-

To test if you have a right triangle, set up the Pythagorean Theorem formula using the longest side given as the hypotenuse.

If it works (makes a true equation) and if the numbers are whole numbers, then the triangle is a right triangle! And the set of the lengths of the sides is called a __________________________

*The value that you put alone (the hypotenuse) is always the longest side.

Ex. problems. Are the given sides, the sides of a right triangle?

8, 15, 16 20, 48, 5

Common Triples.

HW: p.260 #17-35 odds

Utah State Standards

3.1.a Solve problems using the properties of special right triangles

Take a look at this triangle.

Why is it isosceles?

Let’s look at if a= 3, what are the lengths of the other two sides.

Let’s look at if a= 7, what are the lengths of the other two sides.

Let’s look at if a= x, what are the lengths of the other two sides.

Theorem- In a 45-45-90 triangle, the length of the hypotenuse is ________ times the length of a leg.

This theorem is a time-saver! (you don’t have to do the Pythagorean theorem every time)

Diagram

Ex. Find a missing side of a 45-45-90

1. Each leg of a 45-45-90 has a length of 8. What is the length of the hypotenuse?

2. Find the length of a leg of a 45-45-90 triangle, if the length of the hypotenuse is 6.

Take a look at this triangle.

It’s equiangular and equilateral!

We’re going to chop it in half by drawing an altitude.

We’ll look at a side length of 8.

Let’s look at a side length of 10.

Theorem- In a 30-60-90 triangle, the length of the hypotenuse is _________ times the length of the shorter leg, and the length of the longer leg is _________ times the length of the shorter leg.

Another time-saver!

Diagram

Ex. Practice problems 30-60-90 triangles.

1. Find AC.

2. Find QR.

HW: p.557 #7-13 odds,14,16 p.562 #7-13 odds, 16,17

Utah State Standards

3.1.b Identify the trigonometric relationships of sine, cosine, and tangent with the appropriate ratio of sides of a right triangle.

3.1.c Express trigonometric relationships using exact values and approximations.

3.2.a Find the angle measure in degrees when given the trigonometric ratio.

3.2.b Find the trigonometric ratio given the angle measure in degrees, using a calculator.

3.2.c Find unknown measures of right triangles using sine, cosine, and tangent functions and inverse trigonometric functions.

4.2.c Solve problems involving trigonometric ratios.

Investigation. Trig ratios.

Trigonometry-

Trigonometric ratio-

*related to the acute angles of a right triangle (NOT the right angle).

Diagram:

Sine

Cosine

Tangent

SOH-CAH-TOA

Practice Problems.

Find sinL sinN

cosL cosN

tanL tanN

*we’ll find a fraction and decimal for each.

If we know the degree of our angles, we can find the ratio of sides on our calculator.

*we can find the trig ratios!

Practice.

Find the sin67. Find tan56.

Find cos89. Find tan11.

Using trig ratios to solve problems.

Practice. p.568 #5

p.575#9

Solving for an angle measure, knowing the ratio of sides.

We set up trig equations just as in the previous example. The missing piece is now the angle!

We use the inverse of sine, cosine, and tangent to help us solve these equations.

sinA = x to find angle A A = sin-1(x) “A is the inverse sine of x”

cosA= x to find angle A A = cos-1(x) “A is the inverse cosine of x”

tanA= x to find angle A A = tan-1 (x) “A is the inverse tangent of x”

On the calculator

ex. to find angle L

Ex. to find angle N

Angle of Elevation-

Diagram

Angle of Depression-

Diagram

Practice p. 568 #6

p.575 #10

HW: p.568 #9-19 odds, 20 p.576 #13-27 odds, 31, 36

CDAS

Utah State Standards

1.4 Analyze characteristics and properties of polygons.

1.4.c Derive, justify, and use formulas for the perimeter and lines of symmetry of regular polygons.

Polygon-

*sides that have a common endpoint are _________________.

*each side intersects exactly two other sides, but only at their ______________.

Naming a polygon

Sketches of some polygon examples and names

Concave vs. Convex polygons

Classifying polygons by the number of sides.

|Number of |Polygon Name |

|Sides | |

|3 | |

|4 | |

|5 | |

|6 | |

|7 | |

|8 | |

|9 | |

|10 | |

|12 | |

|N | |

Regular polygon- A convex polygon with all __________ congruent and all ___________ congruent.

Diagram

Practice. Listen and Talk worksheet

Practice p. 404 #2,4-7,9,10

Diagonals of a polygon

Perimeter of a polygon-

Practice p.405 #8

Symmetry

Reflection symmetry

Need a mirror line

Practice. Determine how many lines of symmetry a square has.

Determine how many lines of symmetry a regular pentagon has.

p.436 #8,9

Rotation symmetry

Can a figure have both reflection and rotation symmetry? Draw an example.

HW: p.405 #13-33 p.437 #13-17 odds

CDAS

Utah State Standards

1.6.a Identify and classify prisms, pyramids, cylinders and cones based on the shape of their base(s).

1.6.b Identify three-dimensional objects from different perspectives using nets, cross-sections, and two-dimensional views.

1.6.c Describe the symmetries of three-dimensional figures.

1.6.d Describe the relationships between the faces, edges, and vertices of polyhedra.

Orthogonal drawings

[pic]

Ohio Dept. of Education

Practice drawings with cube designs.

Build 3D Figures from nets.

Polyhedron

Face

Edge

Vertex

Examples:

Prism

Naming a prism

Examples

Regular Prism

Pyramid

Naming a pyramid

Examples

Euler’s Theorem:

Regular polyhedron

Platonic Solids

Names and descriptions

Solid Non-polyhedrons

Cylinder

Sketch

Cone

Sketch

Sphere

Sketch

Cross-sections of Solids

The circle on the right is a cross-section of the cylinder on the left.

[pic]

The triangle on the right is a cross-section of the cube on the left.

[pic]

Homework. p. 499 #15,17,21-29 odds, 32,33

CDAS

Utah State Standards

1.4.c Derive, justify and use formulas for the number of diagonals, lines of symmetry, angle measures of regular polygons.

Investigation. Interior angles of polygons.

Interior Angle Sum Theorem-

Each interior angle

Ex. Finding the sum of the interior angles of a polygon.

In a pentagon, what is the sum of the interior angles?

Ex. Finding the measure of EACH interior angle of a polygon.

In a regular hexagon, what is the measure of each interior angle?

Ex. Finding the number of sides, given the sum of the measure.

Given that the sum of the measure of interior angles of a polygon is 900, how many sides does the polygon have?

Exterior angles of a polygon-

Diagram

Relationship between an exterior and interior angle

Investigation. Exterior angles of a polygon.

Exterior Angle Sum Theorem-

Diagram

Ex. Find the measure of each exterior angle of a regular nonagon.

Ex. Find the measure of an interior angle and an exterior angle of a regular 18-gon.

HW: p.411 #9-17 odds, 18

Utah State Standards

1.4.a Use examples and counterexamples to classify subsets of quadrilaterals.

1.4.b Prove properties of quadrilaterals using triangle congruence relationships, postulates, and theorems.

Quadrilateral

Consecutive sides

Non-consecutive sides

Parallelogram-

Diagram

Symbol

Investigation. What Does It Take to Make a Parallelogram?

Properties of Parallelograms

Theorem- Opposite sides of a parallelogram are __________________.

Theorem- Opposite angles of a parallelogram are __________________.

Theorem- Consecutive angles in a parallelogram are __________________.

Theorem- If a parallelogram has 1 right angle, then _________________________.

Theorem- The diagonals of a parallelogram _________________________________.

Practice. p.319 #2,4-8

The ONLY ways to prove a quadrilateral is a parallelogram.

Definition- Both pairs of opposite sides are _________________.

Theorem- Both pairs of opposite sides are __________________.

Theorem- Both pairs of opposite angles are __________________.

Theorem- The diagonals __________________________________.

Theorem- One pair of sides is both _______________ and ________________.

Practice. p.325 #3,4

HW: p.320 #11-23 odds, 28,29 p.325 #7-12, 14,15,16

Utah State Standards

1.4.a Use examples and counterexamples to classify subsets of quadrilaterals

1.4.b Prove properties of quadrilaterals using triangle congruence relationships, postulates, and theorems.

Rectangle-

Diagram

Investigation. What Does It Take to Make a Rectangle?

Theorem- If a parallelogram is a rectangle, then the diagonals are __________________.

*It also has all the same properties of a parallelogram!

Ex. Quadrilateral RSTU is a rectangle. If RT = 6x + 4 and SU = 7x – 4, find x.

Ex. Quadrilateral LMNP is a rectangle. Find x and y.

Theorem- If the diagonals of a parallelogram are congruent, then the parallelogram is _______________________.

Rhombus-

Plural “rhombi”

Diagram

Investigation. What Does It Take to Make a Rhombus?

*All properties of a parallelogram still apply!

Theorem- The diagonals of a rhombus are _________________

Theorem- If the diagonals of parallelogram are perpendicular, then that parallelogram is a ___________________.

Theorem- Each diagonal of a rhombus bisects a pair of _________________________.

Ex. Use rhombus LMNP and the given information to find each value.

1. Find [pic]if [pic].

Square-

Diagram

*Has all properties of a parallelogram, rectangle, and rhombus.

Kite-

Diagram

HW: p.330 #16-21, 23-37 odds, 39-44

Utah State Standards

1.4.a Use examples and counterexamples to classify subsets of quadrilaterals

2.1.a Verify the classifications of geometric figures using coordinate geometry to find lengths and slopes.

Trapezoid-

Diagram

Bases

Base angles

Legs

Isosceles Trapezoid-

Diagram

Theorem- Both pairs of base angles are __________________ in an isosceles trapezoid.

Theorem- The diagonals of an isosceles trapezoid are __________________.

Median of a trapezoid-

Sometimes called midsegment

Diagram

Theorem- The median of any trapezoid is ______________ to the bases, and its measure is equal to _________________ the __________ of the measures of the bases.

Formula

Ex. DEFG is an isosceles trapezoid with median [pic].

1. Find DG if EF = 20 and MN = 30.

2. Find [pic] if [pic].

HW: p.337 #10,13-21 odds, 30,32,33,34

Utah State Standards

1.4.d Define radius, diameter, chord, secant, arc, sector, central angle, inscribed angle, and tangent of a circle, and solve problems using their properties

Circle-

Center

Radius

Radii

Naming a circle-

Chord

Diameter

*made of collinear radii

Diameter =

Practice Problems.

Concentric circles-

What is π?

Circumference

Formula

Practice Problems. Leave π in your answers. Then use a calculator for a decimal approximation..

1. Find C if r = 7. 2. Find C if d = 12.5

3. Find d and r to the nearest hundredth if C=136.9

HW: p.457 #13-27,29,30 p.481 #11,13,17,19,22,26

Utah State Standards

1.4.d Define radius, diameter, chord, secant, arc, sector, central angle, inscribed angle, and tangent of a circle, and solve problems using their properties

1.4.e Show the relationship between intercepted arcs and inscribed or central angles, and find their measures.

Central angle

Diagram

Sum of central angles (with no interior points in common)

Practice Problem

1. Find x.

2. Find [pic].

Arc

*Measure of an arc =

Different than the length of an arc.

Minor Arc

Major Arc

Semicircle

Arc addition postulate

*What is the sum of all non-overlapping arcs on a circle?

Practice Problems.

1. Find [pic]

2. Find mCBE

3. Find ACE

Investigation. Intercepted Arcs.

Inscribed Angles

Diagram

Intercepted arc

Measure of inscribed angle =

Practice Problems.

Inscribed angle intercepting a semicircle-

HW: p.466 #13-23 odds, 31,41 p.590 #9,11,12-14,19,21,27

Utah State Standards

1.4.d Define radius, diameter, chord, secant, arc, sector, central angle, inscribed angle, and tangent of a circle, and solve problems using their properties

1.4.e Show the relationship between intercepted arcs and inscribed or central angles, and find their measures.

Tangent

Point of tangency

Diagram

If a line is tangent to a circle, then it is _______________ to the radius drawn to the point of tangency.

Diagram

If two segments from the same exterior point are tangent to the same circle, then they are _____________________.

Practice. p.595 #6

Secant

Diagram

Measures of angles formed by secants and tangents

2 secants/chords intersecting in the interior of the circle

Diagram

Secant and a tangent

Diagram

2 secants

Diagram

2 tangents

Diagram

Practice. p.603 #4,5

Practice.

HW: p.596 #9-14, 19,27 p.604 #9-17 odds

Utah State Standards

1.4.d Define radius, diameter, chord, secant, arc, sector, central angle, inscribed angle, and tangent of a circle, and solve problems using their properties

1.4.e Show the relationship between intercepted arcs and inscribed or central angles, and find their measures.

Segments in a circle

Two chords intersecting in the circle

Diagram

Two secant segments

Diagram

Secant and a tangent segment

Diagram

Practice.

HW: p.616 #8-13

Utah State Standards

2.2.a Graph a circle given the equation in the form [pic].

Equation of a circle

Center

Radius

Practice Problems.

1. Write an equation of a circle with center at (-2,4) and a radius of 2.

2. Write an equation of a circle with center at the origin and diameter of 6.

3. Graph the circle with equation (x+2)2 +(y-3)2=16.

HW: p.621 #15-25 odds,26,27,34,36

CDAS

Utah State Standards

1.4.c Derive, justify, and use formulas for the area of regular polygons.

4.1.d Calculate or estimate the area of an irregular region.

What is area?

Practice. p.416 #4,5,7

Area of a rectangle

Ex. What is the area of rectangle ABCD?

Investigation. Area of a Polygon

Area of a parallelogram

Ex. Find the area of each parallelogram.

1.

2.

Area of a triangle

Ex. Find the area of the triangle.

Area of a trapezoid

Ex. Find the area of the trapezoid.

Area of a rhombus

Ex. Find the area of the rhombus.

Area of a square

*a square is a rectangle, parallelogram, and a rhombus.

Ex. Find the area of the square.

What happens when you have a very strange shape like…?

HW: p.416 #9,12,13,15,21 p.423 #11,12,14,15

Utah State Standards

1.4.c Derive, justify, and use formulas for the area of regular polygons.

4.1.d Calculate or estimate the area of an irregular region.

Area of a regular polygon

Apothem

Central angle

Ex. Find the area of each regular polygon.

1. A pentagon with side length of 10 and apothem 6.9.

2. A hexagon with perimeter of 72 and apothem [pic].

Area of a circle

*derived from the polygon formula

Ex. Find the area for each problem.

1. A circle with radius of 8.

2. A caterer has a 48-inch diameter table that is 34 inches tall. She wants a tablecloth that will touch the floor. Find the area of the tablecloth in square inches.

Irregular Figures

Ex.

HW: p.429 #7,8,9,11,12,15 p.486 #9,11,13,17

Utah State Standards

4.1.e Find the length of an arc and the area of a sector when given the angle measure and radius.

4.2.d Solve problems involving geometric probability.

Sector

Area of a Sector

Practice Problem. Find the area of the sector with central angle of 80 degrees in a circle with radius 6.

Length of an Arc

*different than the measure of an arc.

Practice Problem. Find the length of an arc with central angle of 80 degrees in a circle with radius 10.

Geometric Probability and Area

Comparing the area of part with the area of the whole figure.

P =

Practice Problems.

1. Find the probability that a point chosen at random lies in a sector of a circle with central angle measure of 60 degrees. The radius of the circle is 9.

2. Find the probability that a point chosen at random lies in the shaded region.

Investigation. Geometric Probability.

HW: p.487 #19-21, 24,25,27,29 p.482 #27

Utah State Standards

1.6.b Identify three-dimensional objects from different perspectives using nets, cross-sections, and two-dimensional views.

4.1.b Develop surface area and volume formulas for polyhedra, cones, and cylinders.

4.1.c Determine the perimeter, area, surface area, lateral area, and volume for prisms, cylinders, pyramids, cones, and spheres when given the formulas.

Surface Area

Lateral Area

Prisms

*Named by their ________________.

Lateral Area of Prisms

Sketch

Shape of lateral faces of prism

Surface Area of Prism = Lateral Area + 2(Area of 1 Base)

Practice. p.508 #3,4,6

HW: p.508 #7,9,11,16,18,19

Utah State Standards

4.1.b Develop surface area and volume formulas for polyhedra, cones, and cylinders.

4.1.c Determine the perimeter, area, surface area, lateral area, and volume for prisms, cylinders, pyramids, cones, and spheres when given the formulas.

Lateral Area of a Cylinder

Sketch

Base Area of a Cylinder

Surface Area of a Cylinder = Lateral Area + 2(Area of 2 base)

Practice. Find the surface area of a cylinder with a height of 18 feet and a base radius of 14 feet.

Practice. Find the radius of the base of a right cylinder if the surface area is [pic] square feet and the height is 10 feet.

HW: p.509 #13-15,17

Lateral Area of Regular Pyramids

*Named by their _______________.

Sketch

Slant height l

Shape of lateral faces of a pyramid

Surface Area of Regular Pyramids = Lateral Area + Area of 1 Base

Practice. Find the surface area of the regular pyramid to the nearest tenth.

Lateral Area of a Cone

Sketch

Surface Area of a Cone = Lateral Area + Area of 1 Base (Circle)

Practice. Find the surface area of a cone with height of 3.2cm and base radius of 1.4 cm. Draw a sketch.

Surface Area of a Sphere

Practice. Find the surface area of a sphere with radius of 41 cm.

Practice. Find the surface area of a ball with a circumference of 24 inches to determine how much leather is need to make the ball.

HW: p.520 #7-14 all p.531 #7-13 (surface area only)

Utah State Standards

4.1.b Develop surface area and volume formulas for polyhedra, cones, and cylinders.

4.1.c Determine the perimeter, area, surface area, lateral area, and volume for prisms, cylinders, pyramids, cones, and spheres when given the formulas.

Volume

Introduce project. There’s No Place Like Home

Volume of a Prism

Height

Area of the Base

Practice. p. 513 #8

Practice. The weight of water is 0.036 pounds times the volume of water in cubic inches. How many pounds of water would fit into a rectangular child’s pool that is 12 inches deep, 3 feet wide, and 4 feet long?

Volume of a Cylinder

Practice. Find the volume of a cylinder with height of 1.8cm and base radius of 2.2cm, to the nearest tenth.

Investigation. Pyramids and Prisms. Cones and Cylinders.

Volume of a Pyramid

Ex. A student has a solid clock that is in the shape of a square pyramid. The clock has a base side of 3 inches and a height of 7 inches. Find the volume of the clock.

Sketch

Volume of a Cone

Ex. Find the volume of a cone with a height of 12 feet and a base radius of 5 feet.

Volume of a Sphere

Practice. Find the volume of a sphere with radius of 15 cm.

Practice. Find the volume of a sphere with a Circumference of 25 cm.

Practice. Compare the volumes of a sphere and a cylinder with the same radius and height as the radius of the sphere.

Sketch

HW: p.513 #12,14,15,18,19,20 p.525 #9-17 odds, 22 p.531 #8,11,16,21

CDAS

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