Department of Mathematics : The University of Akron



Table of Contents

Part I: Unit Plan…………………………………………………..3-8

1. Unit Title

2. Unit Summary

3. Key Words

4. Background Knowledge

5. NCTM Standard(s) Addressed

a. State Standards, Benchmarks and

Grade Level Indicators

6. Learning Objectives

7. Materials

8. Suggested Procedures

a. Attention Getter

b. Suggested Grouping

9. Assessment(s)

Part II: Inquiry-based Activities………………………….9-57

Lesson One: The Geometric Mean w/homework

Lesson Two: Visual Proof of Pythagorean Theorem

w/homework

Lesson Three: Investigations into the Converse of

the Pythagorean Theorem w/homework

Lesson Four: Special right triangles w/homework

Lesson Five: Investigations of trig ratios

w/homework

Lesson Six: Investigations of Inverse Trig

Functions w/homework

Part III: Solutions ………………………………………….58-73

Unit Title

Investigations, explorations, and applications of right triangles and trigonometry

Lesson Summary

The inquiry-based activities in this lesson include formulating and testing ideas involving right triangles. Students will employ a variety of problem-solving techniques including using trigonometry, indirect measurements, constructions and the concept of geometric mean. Students will begin by exploring similar right triangles and the geometric mean. Students will then review the Pythagorean Theorem, formulate and test its converse, investigate Pythagorean triples, 45-45-90 and 30-60-90 special right triangles. Finally students will generalize formulas to solve right triangles and their real-world applications by correct selection and use of the tangent, sine and cosine ratios.

Key Words

Right triangles, acute triangles, obtuse triangles, trigonometry, geometric mean, special right triangles, Converse of the Pythagorean Theorem, leg, hypotenuse, adjacent segment, altitude, and trigonometric ratios.

Background Knowledge

Prior to this lesson students should have substantial knowledge and skills in the following areas:

• Properties of right triangles

• Applying the Pythagorean Theorem

• Finding converses given conditional statements

• Differentiating between < and >inequalities

• Ratios and Proportions

• Determining similarity of triangles

• Finding squares and square roots

• Definition of geometric mean

• Construction of a triangle given three sides using a compass and straightedge

• Geometer's Sketchpad or Cabri

• Recognizing complementary angles

• Using the trigonometric function keys on the calculator

Ohio Standards Addressed

Ohio Content Standards Addressed: Number, Number Sense and Operations, Measurement, Geometry and Spatial Sense, Patterns, Functions and Algebra

Benchmarks- Number Sense

8-10 D: Connect physical, verbal and symbolic representations of integers, rational numbers and irrational numbers.

8-10 E: Compare, order and determine equivalent forms of real numbers.

8-10 H: Find the square root of perfect squares, and approximate the square root of non-perfect squares

11-12 E: Represent and compute with complex numbers.

Benchmarks- Measurement

8-10 E: Estimate and compute various attributes, including length, angle measure, area, surface area and volume to a specified level of precision.

8-10 G: Use proportional reasoning and apply indirect measuring techniques, including right triangle trigonometry and properties of similar triangles to solve problems involving measurements and rates.

Benchmarks- Geometry and Spatial Sense

8-10 B: Describe and apply the properties of similar and

congruent figures; and justify conjectures involving similarity

and congruence.

8-10 E: Draw and construct representations of two-and three-

dimensional geometric objects using a variety of tools, such as

straight edge, compass and technology.

8-10 H: Establish the validity of conjectures about geometric

objects their properties and relationships by counterexamples,

inductive and deductive reasoning and critiquing arguments made

by others

8-10 I: Use right triangle trigonometric relationships to

determine length and angle measures.

11-12A: Use trigonometric relationships to verify and determine

solutions in problem situations.

Benchmarks-Patterns, Functions and Algebra

8-10 B: Identify and classify functions as linear or non-linear and contrast their properties using tables, graphs or equations.

8-10 D. Use algebraic representations such as tables, graphs, expressions, functions and inequalities to model and solve problem situations.

Grade level Indicators – Number Sense Standard

E 9-2: Compare, order and determine equivalent forms for rational and irrational numbers.

D 10-1: Connect physical, verbal and symbolic representations of irrational numbers; e.g., construct the square root of 2 as a hypotenuse or on a number line.

E 11-7: Compute sums, differences, products and quotients of complex numbers.

Grade level Indicators – Measurement Standard

E 10-6: Estimate lengths of missing segments and measurements of missing angles using trigonometric charts and tables and interpolation to a specified number of significant digits.

G 9-4: Use scale drawings, properties of similar polygons, and right triangle trigonometry to solve problems that include unknown distances and angle measurements.

Grade level Indicators – Geometry and Spatial Sense Standard

H 10-3: Prove the Pythagorean Theorem.

I 9-1: Define the basic trigonometric ratios in right triangles: sine, cosine, and tangent.

I 9-2: Solve right triangle problems by correct selection and use of the tangent, sine and cosine ratios.

Grade level Indicators – Patterns, Functions and Algebra Standard

B 8-9: Solve and use linear inequalities to describe parameters of geometric figures.

D 8-8: Determine the lengths of two sides of special right triangles when the length of the third side is known.

Learning Objectives

Upon completion of this lesson, students will demonstrate knowledge and be able to:

• State and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle

• State and apply the Pythagorean Theorem, its converse, and related theorems about obtuse and acute triangles

• Recognize Pythagorean triples and their multiples in right triangle problems

• Determine the lengths of two sides of a 45-45-90 or a 30-60-90 triangle when the length of a third side is given

• Define the tangent, sine, and cosine ratios for both the acute angles in a right triangle

• State and apply the relationship that exists between the trig functions of an acute angle of a right triangle and those of its complement

• Recognize the secant, cosecant and cotangent functions as the inverse functions of sine, cosine and tangent

Materials

All handouts are included in this packet. Students will need pencil and eraser, a straightedge and a compass, a scientific or graphing calculator and access to dynamic geometric software. For the visual proof in Activity Two, they will also need scissors and tape/glue or sticky-back paper.

Suggested Procedures

a. See individual lesson plans for “attention getters”.

b. Unless otherwise noted in the activity, small groups are recommended (3-4 students per group).

Assessments

Both formal and informal assessments will be utilized:

• In-class handouts

• Contributions to group effort

• Peer evaluation

• Participation in class discussion

• Daily homework

• Quizzes

• Test

• Projects

Lesson One

Exploring the relationships created by the altitude of a right triangle.

Lesson summary: The students will explore the similar right triangles and geometric mean relationships created when the altitude from the right angle to the hypotenuse of a right triangle is drawn. First the students determine what angles of the three triangles (the original right triangle, and the two on each side of the altitude) are congruent. Then they use the congruent angles to determine similarity relationships between the three triangles. Next, they use the similar triangles to write proportion statements relating the sides of the triangle. Finally, the students will determine which proportion statements are geometric means.

Key words: Altitude of right triangle, and geometric mean.

Background knowledge: Definition of a geometric mean; Properties of similar triangles; Proving triangles similar; Parts of a right triangle; Definition of an altitude.

Standards: See overall project description.

Learning Objectives: The student will …

• Identify the similar triangles created when an altitude is drawn from the right angle of a right triangle to the hypotenuse.

• Realize the altitude of the right triangle is the geometric mean of the two segments of the hypotenuse created by the altitude.

• Realize that each leg of a right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse created by the altitude that is adjacent to the leg.

Materials: Handout, tape, chalk or other materials to create or draw a large right triangle and the altitude of that right triangle.

Suggested procedures:

“Attention getter”: Construct or draw a large right triangle (3 feet by 4 feet by 5 feet, for example) with altitude on the board, a wall, the floor or ceiling. First discuss why there appears to be only 1 altitude for this triangle (the other 2 altitudes are the legs of the triangle). Then have the students identify the legs, altitude, and hypotenuse of the right triangle. Then discuss the two segments of the hypotenuse that the altitude creates, and determine which leg is adjacent to each of these segments.

Groups: groups of 3-4 students are recommended. Random selection, heterogeneous skill level groups or groups that are previously established can be used.

Assessment: The worksheet for the activity should be collected. Homework problems are provided. Test/quiz questions are also recommended.

Name: _____________________

Lesson One: Exploring the relationships created by the altitude of a right triangle

Goal: Students will explore similar right triangles and geometric mean relationships created when the altitude from the right angle to the hypotenuse of a right triangle is drawn.

Given a right triangle, when the altitude is

drawn from the vertex of the right angle to

the hypotenuse, two new triangles are formed.

Below, these three triangles are drawn side

by side. The sides a, b, and c from the original

triangle are labeled, along with the two

segments of the hypotenuse (d and e) that are

created by the altitude (f).

1. List the corresponding congruent angles between [pic] and [pic].

2. Now consider [pic] and [pic]. List their congruent corresponding angles.

3. Finally, write the corresponding angle congruence statements for [pic] and [pic].

4. If two or more angles of a triangle are congruent to two or more angles in a second triangle, then the triangles are similar. List all sets of similar triangles above.

5. Since all three triangles are similar to each other, we can write proportions relating sets of corresponding sides. Write a proportion that contains side f twice (the altitude from the right angle of [pic]).

6. By Definition, x is the geometric mean of m and n if [pic] or[pic].

Are any of the segments in the above proportion a geometric mean? Write a definition for the geometric mean you discovered in question 5, using the names of the segments related to the original triangle (leg, hypotenuse, altitude, segments of the hypotenuse).

7. Write a proportion that contains side b twice (one of the legs of[pic]).

8. Write a definition for the geometric mean you discovered in question 5, using the names of the segments related to the original triangle (leg, hypotenuse, altitude, segment of the hypotenuse adjacent to the leg, segment of the hypotenuse not adjacent to the leg).

9. Write a proportion that contains side a twice (the other leg of[pic])

10. Can you use the same definition for the geometric mean relationship from 8 to describe the geometric mean relationship related to side a from question 9?

Name:__________________

Lesson One : Homework

Solve for all unknowns. Show work for any credit.

1.

2.

Extension: Find the value of X.

Lesson Plan

Lesson Two – Revisiting the Pythagorean Theorem – A Visual Proof

Lesson Summary This lesson leads students through a visual proof of the Pythagorean Theorem using a provided pattern. In addition, students are asked to derive both a verbal and algebraic statement of this extensively used theorem.

Background knowledge Prior to this lesson students should have substantial knowledge and skills in the following areas:

• Properties of right triangles

• Applying the Pythagorean Theorem

• Finding converses given conditional statements

Ohio Standards/Benchmarks/Grade Level Indicators Addressed

Benchmarks- Geometry and Spatial Sense - 8-10 H: Establish the validity of

conjectures about geometric objects their properties and relationships by

counterexamples, inductive and deductive reasoning and critiquing arguments made

by others. Grade Level Indicator H 10-3: Prove the Pythagorean Theorem.

Learning Objectives Students will be able to verbally state, algebraically

represent and visually prove the Pythagorean Theorem.

Materials Needed

- paper, pencil, eraser, straightedge, scissors, glue/tape

- Individual copies of the handout

- Individual copy of "net" copied onto sticky back paper

- Individual copy of "net" with 2-column proof of the Pythagorean Theorem on the reverse side (This one will be saved and put into the student's notes)

- 2 overhead transparencies (incomplete and completed 2-column proof)

- Overhead projector or S-video or Smartboard and screen

Suggested Procedures

The following require preparation in advance: 1) Individual student copies of the packet and the "net," and two overhead transparencies. 2) Predetermination of preferred grouping (2 to 4 suggested). The remaining procedures are suggested:

← "Attention Grabber"-Place the transparency of the incomplete 2-column proof of the Pythagorean Theorem (provided) on the overhead projector. Have it projected onto the screen as students enter the room. Meet them at the door and excitedly tell them that TODAY is the day they are going to PROVE the Pythagorean Theorem. (Listen to them groan!)

← Transition from Activity One (Geometric Mean) by asking students if they recognize the diagram used for this formal proof.

← Review yesterday's activity through several quick questions.

← Refocus the students' attention to the Proof. Elicit ideas on how to start and finalize the proof.

← Give in reluctantly and place the completed proof on the overhead. You might have one or two students come to the front and read the statements and reasons of the proof to the class.

← Interrupt the students long enough to distribute the packet so that the students can follow along.

← Direct the students to underline all the algebraic equations in the proof.

← When the proof has been presented in its entirety, tell the students, "Now it's your turn!" Divide the students into their predetermined groups' point out where the activity starts and then walk around the class assisting as needed.

← At the end of class bring the students back to their "whole class" seating, take a quick oral survey of how they did, and distribute and explain the homework and the plan for the next day.

Assessments

Informal: observation during group work, questioning, participation in review

Formal: Check answers to the activity, homework, quiz after Activity 3, and Test at end of unit.

[pic]

[pic]

Lesson Two: "Now You See It …Don't Say You Don't."

Name: ___________________________ Per ____ Date ________

Steps for proof:

- Cut out the smaller square (number 5) and parts 1-4 of the middle square in the attached net.

- Arrange these pieces to exactly cover the larger square ABCD.

- When you are absolutely sure that you have accomplished this:

a) lay the pieces out on the second (uncut) sheet,

b) trace the outline of these pieces onto the largest square

c) peel back the sticky tab and permanently place the pieces on the second (uncut) sheet.

*** When you have successfully completed the above steps, you have demonstrated the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the two legs of your triangle.

1) Compare your results with the results of other groups near you.

2) If the lengths of the two legs of a right triangle are named [pic]and b, then the areas of the

squares on the legs would be ________ and ___________. (Refer to the diagram on the uncut

sheet).

3) If the length of the hypotenuse is c, then the area of the square of the hypotenuse is ________.

4) Using the visual diagram, combine the results from steps 4 and 5 into an equation and write it

on the line below.

_____________________________________

5) What have you just discovered?

6) Rewrite the concept you have just proven as a conditional statement in "if…then" form.

7) How does this visual help to make sense out of the famous theorem?

Do not discard this activity and especially your square. We will use this later!!!

[pic]

Name: ____________________

Homework for Pythagorean Theorem

1) The classical ladder problem:

There is a building with a 12 ft high window. You want to use a ladder to go up to the window, and you decide to keep the ladder 5 ft away from the building to have a good slant. How long should the ladder be?

2) Baseball diamond:

On a baseball diamond the bases are 90 ft apart. What is the distance from home plate to second base in a straight line?

[pic]

3) An algebraic problem:

Find the length of both of the missing sides on the following triangle:

[pic]

4) An iterative problem:

Look at the following figure. Start by finding the value for X1, then for X2, then X3, and so on until you get the value for X6. Write the lengths as square roots, as that makes it simpler.

[pic]

What is the value of X6?

Extension:

5) Equilateral Triangle:

An equilateral triangle has vertices (0,0) and (6,0) in a coordinate plane. What are the coordinates of the third vertex? You may want to sketch it out.

Note: the sides of an equilateral triangle are equal in length.

LESSON PLAN THREE

Lesson Three – Exploring the Converse of the Pythagorean Theorem and its Corollaries

Lesson Summary: This lesson is combined with lesson four to allow students to engage in a two-tiered activity in order to derive and apply the Converse of the Pythagorean Theorem. Students will construct triangles, and measure, calculate, reason, conjecture and finally justify the conclusion that triangles whose sides satisfy the equation a2 + b2 = c2 are in fact right triangles. They will also conjecture about the types of triangles formed when a2 + b2 > c2 and will work with Pythagorean Triples.

Background Knowledge: Students will need to be familiar with: Properties of right triangles, applying the Pythagorean Theorem, finding converses given conditional statements, finding squares and square roots, the geometric mean, construction of a triangle given three sides using a compass and straightedge, Geometer's Sketchpad or Cabri.

Ohio Standards Addressed: 8-10 G: Use proportional reasoning and apply indirect measuring techniques, including right triangle trigonometry and properties of similar triangles to solve problems involving measurements and rates.

Learning Objectives: Upon completion of this lesson, students will demonstrate knowledge and be able to:

• State and apply the Pythagorean Theorem, its converse, and related theorems about obtuse and acute triangles

• Recognize Pythagorean triples and their multiples in right triangle problems

Materials Needed: Pencil, paper, eraser, straightedge, ruler, compass, and handout.

Technology required: Access to student computers loaded with a dynamic Geometry software such as Geometer's Sketchpad or Cabri. Optional: S-video, Internet

Suggested Procedures:

← Requires advanced preparation: Installation of software, scheduled use of student computers, individual student copies of packet for Activity Three, determination of grouping for students (2 to 4 per group, 1 to 2 per computer).

← "Attention Grabber" Write on the board or overhead for the students to complete in their journals: "If [Mrs. Millin] had a dollar for ever Geometry student who hated proofs, then……………"

← Answer questions on the previous night's homework. Have students put 2 or 3 of the problems on the board. Have students check their work for accuracy. Collect.

← Generate a discussion of conditional statements and their converses with voluntary student conditionals. Ask: "Is this conditional statement true? Is its converse true? "Is there a general rule about conditionals and their converses?"

← Ask about the conditional statement the students wrote on Question # 6 of yesterday's class activity. "Who is pretty sure he wrote the converse of the Pythagorean Theorem correctly? Be careful: It's tricky!"

← Have a student write the converse on the board being sure to use the correct wording. Instruct students to get out their compasses, straightedges and rulers while you distribute the packet.

← Transition to today's activity through a summary of the concepts explored yesterday.

← Explain to the students that this activity will be combined with a second activity that will take two days and the second day they will be using computers. Dismiss them to their predetermined groups.

← Walk around the classroom assisting students as needed. Observe students level of involvement and cooperation within their groups for assessment purposes.

← Stop three or four minutes before the bell. Give students their homework

Assessments

Informal: Observation during group work, questioning, participation in review.

Formal: Check answers to the activity, homework, quiz after Activity 3, and

Test at end of unit.

Name: __________________

Lesson Three: If a triangle's not right, then it's…

Lesson Goals: Through measurement, constructions omit and calculations, and using their knowledge of the Pythagorean Theorem, the students will identify right, acute, and obtuse triangles based on the length of their sides,

RECALL the converse of the Pythagorean Theorem from today's discussion and write it out in the space below:

How to construct a triangle:

[pic]

Example for you to trace over and practice on: Example: Construct a triangle with side lengths of (1,[pic], 2) in inches ([pic][pic] 1.73)

[pic]

1. DRAW the following triangles with sides as given. (Use cms for your unit of measurement)

[pic]CAT (3,4,5) [pic] SIT(9,12,15) [pic]DOG (6,8,10) [pic]RUN (5,12,13)

2. For each of the triangles in Exercise 1 compare the sum of the squares of the lengths of the two shorter sides with the square of the length of the longest side (using or =).

Example: [pic]HOW (15, 20, 25) [pic] 152 + 202 ? 252 [pic] 225 + 400 = 625

[pic]CAT (3, 4, 5) [pic]DOG (6, 8,10)

(__)2 + (__)2 _?_ (__)2 (__)2 + (__)2 _?_ (__)2

____ + ____ ___ ____ ____ + ____ ___ ____

[pic] SIT (9, 12, 15) [pic] RUN (5, 12, 13)

(__)2 + (__)2 _?_ (__)2 (__)2 + (__)2 _?_ (__)2

____ + ____ ___ ____ ____ + ____ ___ ____

3. Have a teammate check the accuracy of your calculations. _______What do your

Initials

calculations in exercise 2 suggest about all of the triangles you constructed?

4. Verify your answer by using your protractor to measure (to the nearest degree) the angle opposite the longest side length in each of the triangles in Exercise 1. Write your angle measurements inside the angles.

Have a teammate check the accuracy of your measurements. ______

Initials

5. Do you think your findings in Exercises 2 and 4 are true …for all triangles? (Yes, NO) … or for only certain triangles? (Yes, NO). Explain your reasoning.

6. Using only your straightedge and a compass, construct [pic]s with the given side lengths (in inches or centimeters).

[pic]CBS (2,3,3) [pic]NOT (1,2,3) [pic]ESP (2,5,6) [pic]MTV (1,1, [pic])

7. Now for each triangle in Exercise 6, compare the sum of the squares of the two shorter lengths with the square of the longest length as you did in Exercise 2, only this time put the square of the longest length on the left of your equation.

[pic]CBS (2,3,3) [pic]NOT (1,2,3)

(__)2 ? (__)2 + (__)2 (__)2 ? (__)2 + (__)2

___ ___ ____ +____ ___ ___ ____ +____

[pic]ESP (2,5,6) [pic]MTV (1,1,[pic])

(__)2 ? (__)2 + (__)2 (__)2 ? (__)2 + (__)2

___ ___ ____ +____ ___ ___ ____ +____

8. Did all the sets of sides in Exercise 6 form triangles? (Yes No) How can you be sure that three given sides will combine to make a triangle?

9. Now use your protractor to measure the angles in each of the triangles you constructed in Exercise 6. Classify your triangles as acute, right or obtuse.

Have a teammate check the accuracy of your measurements. ______

Initials

[pic]CBS (2,3,3) [pic]NOT (1,2,3) [pic]ESP (2,5,6) [pic]MTV (1,1,[pic])

___________ ____________ ___________ _____________

10. Compare your findings in Exercises 3 and 9. Summarize this lesson by writing 4 possible conclusions you can make from today's activities.

1. If______________________________________________then___________________

________________________________________________________________________

2. If______________________________________________then___________________

________________________________________________________________________

3. If______________________________________________then___________________

________________________________________________________________________

4. If______________________________________________then___________________

________________________________________________________________________

Extension: The right triangle side lengths in Exercise 1 are all positive integers.

A set of three positive integers that satisfy the equation c2 = a2 + b2 is called a Pythagorean Triple. List all the Pythagorean Triples found in this lesson.

Bonus: How many more Pythagorean Triples can you come up with?

Homework Lesson Three--Converse of the Pythagorean Theorem

___________________________________________ ______ __________________

Name Period Date

Directions: Graph points P, Q, and R. Connect the points to form [pic]PQR. Use the distance formula and the converse of the Pythagorean Theorem to show whether [pic]PQR is right, acute or obtuse.

Distance formula = [pic]

1. [pic]

[pic]

2. [pic] [pic]

LESSON PLAN

Lesson Four – "They're SPECIAL!"

Lesson Summary: This lesson is the second part of a two-tiered activity designed to help students learn right triangle Geometry. Students will continue their study of right triangles as they use Geometer's Sketchpad to explore 45-45-90 and 30-60-90 triangles of various lengths. They will then conjecture about the relationships between the legs and the sides of each. They will chart their findings and then generalize them for each of the special triangles. Finally, they will complete a chart using only the variables and no specific number values.

Background Knowledge: Students will need to be familiar with: Properties of right triangles, applying the Pythagorean Theorem, finding squares and square roots in both decimal and radical form. They also will need a working knowledge of the measuring tools in Geometer's Sketchpad.

Ohio Standards Addressed:

Benchmarks- Number Sense 8-10 H: Find the square root of perfect squares, and approximate the square root of non-perfect squares.

Benchmarks- Geometry and Spatial Sense 8-10 E: Draw and construct representations of two-and three-dimensional geometric objects using a variety of tools, such as a straight edge, compass and technology.

Benchmarks-Patterns, Functions and Algebra 8-10 D: Use algebraic representations such as tables, graphs, expressions, functions and inequalities to model and solve problem situations.

Learning Objectives: Upon completion of this lesson, students will demonstrate knowledge and be able to:

• State and apply the Pythagorean Theorem, its converse

• Recognize Pythagorean triples and their multiples in right triangle problems

• Determine the lengths of two sides of a 45-45-90 or a 30-60-90 triangle when the length of a third side is given.

Materials Needed: Pencil, paper, eraser, and handout.

Technology Needed: Individual computers with Geometer's Sketchpad software.

Suggested Procedures:

← Requires advanced preparation: Individual student copies of packet for Activity Four, determination of grouping for students (1 to 2 per computer).

← "Attention Grabber": Choose your tool QUIZ. As students enter the room ask them to choose between three different tools laid out on a table; e.g. Protractor, Compass, Ruler.

Each tool signifies a different problem to solve for a quiz grade.

← Transition: refer to the baseball diamond from last night's homework. Ask the students to guess at the measure of the angles formed when a diagonal is drawn from second base to homeplate. Have them verify their guesses. Why do they think it is 45? Are the angle measurements the same for the angles formed by third and second? Verify. Why is this so?

← GO TO THE COMPUTER LAB (If you have not taken the students to the lab before, make sure to explain the rules of the lab and consequences for non adherence.

Name: __________________

Date: __________________

Period: __________________

Lesson Four: They're Special

Directions: Follow the step by step instructions on this page. Place your answers to all questions directly on this activity sheet.

1) Double click on the Geometer's Sketchpad icon on your desktop. Open the file imspecial.gsp.

2) Measure all the acute angles of triangles ONE, DAY, GEO, and LAB and identify them in the space provided. Ex: m ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download