high school geometry days 054067

202

STRUCTURED CURRICULUM LESSON PLAN

Day: 054 Subject: Geometry Grade Level: High School

Correlations (SG,CAS,CFS): 9A3

TAP:Identify, describe, compare, and classify

figures

ISAT:Understand and apply geometric concepts and

relationships

Unit Focus/Foci

Identifying Congruent Triangles

Instructional Focus/Foci

Classifying Triangles

Materials

Copies of Classifying Triangles Activity 1

Educational Strategies/Instructional Procedures

Have students sit in groups of four. Students should work with a partner in their group. Have astudent read the guidelines aloud. Answer any questions before they begin. Try to not answerany questions during the activity. Give the students as much time as you feel they need tocomplete the activity. Students should compare definitions with the members of their group andthen with the class. Bring the class together and agree on a final definition for each term.

Integration with Core Subject(s)

LA: Understand explicit, factual informationUnderstand the meaning of words in context

SC: Apply scientific method to solve problemsSS: Distinguish fact from opinion and relevant from irrelevant information

Connection(s)

Enrichment:

Fine Arts:

Home: Have parents sign assignment sheet.

203

Remediation:

Technology:

Assessment

Teacher observation

Homework

Assign from your text appropriate problems on classifying triangles.

Teacher Notes

Prepare copies of Classifying Triangles 2.

204

GUIDELINES FOR CLASSIFYING TRIANGLES ACTIVITY

1. Students should sit in groups of four.

2. Students should work with a partner within their group.

3. Each student should write a definition based on the given information.

4. Students will then, as a group, determine the best possible definition for the term.

5. Students should choose a recorder (who will write the group’s definition of each term).

6. Each group should be prepared to present their definitions.

205

CLASSIFYING TRIANGLES ACTIVITY

Write the best definition possible based on the information given.

1. Right triangle Not a right triangle

58°

88° 34° 50°

93° 37°

2. Acute triangle

35°

73° 72° 46°

45° 89°

Not an acute triangle

135° 46° 28 °

34° 17°

3. Obtuse triangle

28° 115°

122° 33° 32°

30°

Not an obtuse triangle

40° 35°

75° 70° 50°

26°

64°

47°

43°

55°

35°

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4. Scalene triangle

8

4

6 107

5

Not a scalene triangle

5 5

5

10 10

12

5. Isosceles triangle 15

15 10 10 19

7 7 7 14

Not an isosceles triangle

14.5 14 8 15

14 .6 12

6. Equilateral triangle

1 1

1

3 3 13 13 3

13

Not an equilateral triangle

7 12

6

7 7

6

207




TAP:Identify, describe, compare, and classify figures


relationships

Unit Focus/Foci




Materials

Copies of Classifying Triangles 2


Remind students that every triangle can be classified according to its angles or its sides. Havestudents note that a given triangle can fall into more than one category.

Triangle

Acute Right Obtuse

Scalene Isosceles Scalene Isosceles Scalene Isosceles

Equilateral

208

The parts of triangles have special names.

Isosceles RightVertex

Adjacent sides of a triangle – two sides that share a common vertex.Base of an isosceles triangle – side opposite the vertex angleBase angles of an isosceles triangle – angles opposite the legs with the base as one sideHypotenuse – the side opposite the right anglesLegs of the isosceles triangle – the congruent sidesLegs of a right triangle – the sides adjacent to the right angle.

Have the students complete Classifying Triangles 2.




Leg Leg

Base

Base Angles

Leg

Leg

Hypotenu

Side adjacent to ∠ ASide opposite to ∠ A

Side adjacent to ∠ AC

A

B

209

Connection(s)

Enrichment: Form 4 congruent triangles using 6 toothpicks. You cannot bend, break, oroverlay the toothpicks.

(Solution: Form a tetrahedron.)

Fine Arts:

Home: Have parents sign assignment sheet.

Remediation: Teachers’ Half Dozen

Technology:

Assessment

Teacher observation

Homework

Assign from your text appropriate problems on classifying triangles

Teacher Notes

Solutions to Classifying Triangles 2:

1. ∆ABC 2. ∠ ADC

3. ∠ CAB 4. AC

5. AC and AB 6. AB and BC

7. AC and CD 8. Possible combinations of: ∠ ACD, ∠ CAD,∠ ACB, ∠ ABC, or ∠ CAB

9. ∠ ACB and ∠ ABC 10. CB

11. Scalene 12. Isosceles

210

GUIDELINES FOR CLASSIFYING TRIANGLES ACTIVITY

1. Students should sit in groups of four.

2. Students should work with a partner within their group.

3. Each student should write a definition based on the given information.

4. Students will then, as a group, determine the best possible definition for the term.

5. Students should choose a recorder (who will write the group’s definition of each term).

6. Each group should be prepared to present their definitions.

211

Classifying Triangles 2

A

B

D C

1. Name the isosceles triangle

2. Name the right angle

3. Name the vertex angle

4. Name the hypotenuse

5. Name the legs of the isosceles triangle

6. Name the sides adjacent to ∠ ABC

7. Name the sides adjacent to ∠∠∠∠ ACD

8. Name two acute angles

9. Name a pair of congruent angles

10. Name the base of the isosceles triangle

212

Graph the vertices of each triangle and determine if the triangle is scalene, isosceles, orequilateral.

11. 12.

A (0,3) R (0,4)B (3,3) S (4,0)C (-3,0) T (8,4)

213



Correlations (SG,CAS,CFS): 7A6; 9A5; 9A3

TAP:Understand geometric properties and

relationships; apply geometric concepts andformulas

Demonstrate understanding of measurementconcepts and apply measurement skills

Identify, compare, describe and classify figures


relationshipsDemonstrate an understanding of measurement

concepts and skills

Unit Focus/Foci



Applying the Angle Sum Theorem and the Exterior Angle Theorem

Materials

Copies of Exploring Angles in TrianglesChart paperMarkers


Review last night’s homework and answer students’ questions.

Explain the difference between a theorem and a corollary. A theorem is a statement that mustbe proved to be true. A corollary is a theorem that follows from a previously proven theorem.

Have the students work in groups of four. Students should work with a partner in their group tocomplete the Exploring Angles in Triangles activity.

214

EXPLORING ANGLES IN TRIANGLES

1. One group member will draw 2 large acute triangles and number the angles2. One group member will draw 2 large obtuse triangles and number triangles consecutively3. Measure angles of each triangle.4. Find the sum of the measures of the three angles in each triangle. Discuss the results as a

group.5. Each pair should cut out one triangle and tear off the angles.6. Arrange the angles so that their vertices meet at a point.

As a group make a conjecture about sum of the angles of a triangle.

7. Each pair should extend the side of their remaining triangle and label the angle formed 4.

8. Tear off angles 1 and 2 and place them on top of 4.

Discuss the results as a group and make a conjecture.

Randomly select two groups to share their observations with the class.

Discuss the Angle Sum Theorem (the sum of the measures of the angles of a triangle is 1800) andthe Exterior Angle Theorem (the measure of an exterior angle of a triangle is equal to the sum ofthe remote interior angles).

Relate to students that the Exploring Angles in Triangles activity was an informal proof of thesetheorems.

1

2

3

12

3

1

2

3 4

215

Using chart paper and markers each group will complete an formal or informal proof of the ThirdAngle Theorem: If two angles of one triangle are congruent to two angles of a second triangle,then the third angles are congruent.


LA: Understand explicit, factual information Understand the meaning of words in contextSC: Apply scientific method to solve problemsSS: Distinguish fact from opinion and relevant from irrelevant information

Connection(s)

Enrichment: Have the students construct a triangle whose angles measure 70°, 70°, and 40°,with sides 4 cm, 4 cm, 2.5 cm.

Fine Arts:.

Home: Have parents sign homework.

Remediation:

Technology: Have the students use a computer drawing program to explore the angles in atriangle.

Assessment

Evaluate the in-class assignment using the Structured Curriculum Scoring Rubric.

Homework

Assign from your text appropriate problems on exterior angles and angle sums.

Teacher Notes

216



Correlations (SG,CAS,CFS): 7A6; 9A3, 5

TAP:Identify, compare, describe, and classify

figures


relationships

Unit Focus/Foci



Naming and Labeling Corresponding Parts of Congruent Triangles

Materials


Have the students discuss in pairs the two corollaries and complete an informal proof.

A) the acute angles of a right triangle are complementaryB) there can be at most one right or obtuse angle in a triangle

Randomly select three pairs to present their findings to the class.

Define congruent segments: two segments that have the same length.

Define congruent angles: two angles that have the same measure. Introduce the symbol used forcongruency ( ≅ ).

Discuss the concept of corresponding parts of triangles:

Y T

W H N O

∆ ∆WHY NOT≅

217

Corresponds to:

WH NO W N

OT HY H O

WY NT Y T

→ ∠ → ∠

→ ∠ → ∠

→ ∠ → ∠

Discuss the definition of congruent triangles (triangles that have their corresponding partscongruent):

Place the following examples on the chalkboard or overhead and have the students completethem in their groups.

Example 1: Y N

W X O M

If WYX ≅ DEF, name the corresponding congruent parts of the triangle. B

Example 2: A C

D

Solutions:∠ Y ! ∠ N WY ! NM∠ X ! ∠ O YX ! NO∠ W ! ∠ M WX ! MO

Name the corresponding congruent parts of the triangles.

Solutions:

∠ ABD ! ∠ CBD AB ! BC∠ BAD ! ∠ BCD AD ! CD∠ BDA ! ∠ BDC BD ! BD

Discuss the solutions with the students and remind them of the reflexive property.

218




Connection(s)

Enrichment: In isosceles ∆ ABC, AB ≅ BC. If AB=5 × +10, BC=4 × +40, and AC= 3 × +30.Find the perimeter of ∆ ABC.

Solution: 275 units

Fine Arts: Have the students make a mobile using all triangles with their properties displayedon each.


Remediation: Teachers’Half Dozen

Technology: Have the students use a computer drawing program to practice identifyingcorresponding parts of congruent figures.

Assessment

Teacher observation

Homework

Assign appropriate problems from your text.

Teacher Notes

Remind students that a quiz will be given tomorrow. Prepare copies of Exploring TrianglesQuiz.

219


Day: 058-059 Subject: Geometry Grade Level: High School



figures


relationships

Unit Focus/Foci



Constructing Congruent Triangles

Materials

Copies of Exploring Triangles QuizCopies of Constructing Congruent Triangles Activities 1 and 2StraightedgesProtractorsCompassesScissors


Review last night’s homework.

Administer the Exploring Triangles Quiz.

Have students complete the following:

Name the corresponding sides and angles.

1. 2.

W Y

X

K P

OM

Z

N

220

Solution:

∠ ≅ ∠X XYZWZ WX ≅ YZ ∠ ≅ ∠MNK ONP MN ≅ NP∠ ≅ ∠X W XZYZ WZ ≅ YZ ∠ ≅ ∠KMN OPN KN ≅ NO∠ ≅ ∠ZXW ZXY XZ ≅ XZ ∠ ≅ ∠NKM PON MK ≅ OP

Have the students work along as you demonstrate how to duplicate a triangle according to thefollowing:

a. the included angle is formed by two sides of a triangleb. the included side forms a side of two given angles

Have the students complete constructing Congruent Triangles Activities 1 and 2. Discuss theresults with the class using a compass and straightedge.


LA: Understand explicit, factual informationSC: Apply scientific method to solve problems

Connection(s)

Enrichment: Have the students construct an isosceles triangle with a 6 cm base and 500 vertexangle.

Fine Arts:


Remediation: Teachers’ Half-Dozen

Technology:

Assessment

Evaluate the Exploring Triangles Quiz using the Structured Curriculum Scoring Rubric.

221

Homework

Assign from your text appropriate problems.

Teacher Notes

This activity may take longer than one class period. If so, continue the next day.

Solutions to Exploring Triangles Quiz:

1. See student work

2. x = 100°, y = 80°

3. x = 60°

4. x = 242

3°

5. x = 48°

6. a. See student workb. Right triangle

222

CONSTRUCTING CONGRUENT TRIANGLES

ACTIVITY 1

1. Use a straightedge to draw any line l and choose a point on the line; call it D.

2. Use a compass to construct DE on l such that DE ≅ AB .

3. Use a compass to construct an angle congruent to ∠ A using GE as a side of the angle andpoint D as a vertex.

4. Use a compass to construct G on the new side of the angle such that GF AC≅ .

5. Compare the triangle to ∆ABC.

ACTIVITY 2

1. Use a straightedge to draw a line n.2. On line n choose a point G.

3. Use a compass to draw GH such that GH ≅ AB .

4. Use a compass to construct an angle congruent to ∠ A using GH as a side of the angle andpoint G as the vertex.

5. Use a compass to construct an angle congruent to ∠ B using HG as a side of the angle andpoint H as the vertex.

6. Label the point where the new sides meet I.7. How does ∆GHI compare to ∆ABC? (Use a ruler to measure the sides and a protractor to

measure the angles.)

A

C

B

223

Exploring Triangles Quiz

1. Draw a diagram of the two congruent triangles and identify the corresponding sides andangles. ∆ MCH ≅ ∆ RJD

Find the missing angle measure:

2. 3.

Find the value of X.

4. 5.

6. a. Construct ∆ ABC with AB = 2 cm, ∠ C = 600, and BC = 1 cm.b. Classify ∆ ABC according to its sides and angles.

(3x-5)0

(x+20)0 (2x+17)0

x0 620

480

450

350 x0y0

400

x01000

224



Correlations (SG,CAS,CFS): 6C1, 2; 7A6; 9A3, B2

TAP:Understand geometric properties and


Analyze and interpret data presented in charts,graphs, tables, and other displays



relationshipsUse mathematical skills to estimate,

approximate, and predict outcomes and tojudge reasonableness of results

Demonstrate an understanding of measurementconcepts and skills

Unit Focus/Foci



Using S.S.S., S.A.S., and A.S.A. Postulates to Test for Congruency of Triangles

Materials

Copies of Exploring CongruenceStraws/Pipe cleanersScissorsProtractorsRulers


Students will work in groups of 2 or 4. Complete Exploring Congruence to explore the trianglecongruence postulates SSS, SAS, and ASA. (15 min.)

Discuss the results with the class.




225

Connection(s)

Enrichment: Have the students create a set of flash cards with the triangle congruencepostulates on one side and a diagram on the other.

Fine Arts: See Enrichment.


Remediation:

Technology: Have students use the pieces of information to draw and explore triangles using acomputer drawing program.

Assessment

Evaluate the in-class assignment.

Homework

Assign from your text applications of the congruence postulates.

Teacher Notes

Review the distance formula with the students.

226

Exploring Congruence

1. Construct and label the following triangles:

a. ABC: AB = 7 m∠ B = 45° BC = 7b. DEF: DE = 8 m∠ E = 52° EF = 4c. GHJ: GH = 12 m∠ H = 60° HJ = 14

2. Compare your constructions with triangles created by other classmates. What do you noticewhen you compare constructions?

3. Are the constructions with the same names congruent?

4. What would be the reasoning behind their being congruent?

5. Can you draw two non-congruent triangles that have the same set of measures?

6. If this is possible, what type of result would this provide?

7. Complete the following statement based upon your observations:

“If the ___________________ of one triangle are congruent to the _____________________ ofa second triangle, the triangles are congruent.”

Repeat /complete this same exercise using new constructions:

∆ MNO m ∠ M = 30°, MN = 10, m ∠ N = 60°∆ PQR: m ∠ P = 45°, PQ = 6, m ∠ Q = 45°∆ STV: m ∠ S = 35°, ST = 8, m ∠ T = 50°

Repeat this same exercise using new constructions:

∆ JKL: J K = 5 in., K L = 4 in., J L = 3 in.∆ UVW: UV = 3 in., VW = 6 in., UV = 7 in.∆ XYZ: XY = 2 in., Y Z = 2 in., XZ = 2 in.

227



Correlations (SG,CAS,CFS): 6C1, 2; 7A6; 9A2, 3


figures


relationships

Unit Focus/Foci



Using the Angle Angle Side Theorem to Test Congruent Triangles

Materials

Angle Angle Side (A.A.S.) ActivityPlain paperStraight edges


Review last night’s homework and answer questions.

Review the guidelines with the students for the activity: Proving AAS.This activity may take more than 25 minutes. Allow students as much time as they may need. Atthe end of the activity, allow one or two groups to present their results to the class.

Have the students answer the following questions and discuss them:

1. How well did your group work together?2. Did any one person dominate?3. Did each member of your group participate?4. Did you encourage everyone to work together?


LA: Understand explicit, factual informationSC: Apply scientific method to solve problemsSS: Distinguish fact from opinion and relevant from irrelevant information

228

Connection(s)

Enrichment: The measures of the angles ∆ RST.

∠ ∠ ∠R = 4x - 6, S = 2x = 12, and T = 5

4x. Show that ∆ RST is a right triangle.

Fine Arts: Have students create an abstract design using triangles and shading.


Remediation: Teachers’ Half Dozen

Technology:

Assessment

Teacher observation

Homework


Teacher Notes

Tell the students a test will be given in the next two or three days.

229

ANGLE - ANGLE - SIDE(A.A.S.) ACTIVITY

Materials: Plain paper Straightedge

" Draw a triangle given two angles and a nonincluded side.

" Draw a triangle on a piece of paper. Label the vertices X, Y, and Z

" Construct XY, Y and Z∠ ∠ on another piece of paper.

" Cut out the side and two angles.

" Place them together to form a triangle in which the side is not theincluded side of the angles.

" Trace the triangle onto another piece of paper.

" Place this triangle on top of ∆ XYZ. How do the triangles compare?

" Try again with another set of angles and a nonincluded side.

" Write a conjecture about the two angles and the nonincluded side of onetriangle compared to the two angles and the nonincluded side of theother congruent triangle.

230



Correlations (SG,CAS,CFS): 6C1, 2; 7A6, 9A3; 9B2


figures


relationships

Unit Focus/Foci



Using Properties of Isosceles Triangles

Materials

Copies of Congruent Triangle QuizActivity: Analyzing Isosceles Triangles ActivityPaperRulersScissorsProtractors


Administer the Congruent Triangle Quiz.

Review last night’s homework; answer any questions.

Review the guidelines for the activity with the students. (Let a student read the guidelinesaloud.) Students should be in groups of four for today’s activity: Analyzing Isosceles Triangles.Students should discuss and compare results of the activity with the members of their group.Bring the class together and discuss the results as a group.



231

Connection(s)

Enrichment: ∆ABC ≅ ∆EFG

If ∠ A = x+10, ∠ E = y+20, ∠ B = 2x, ∠ F = x+3y:Find m ∠ A and m ∠ E

Solution:

x = 15 y = 5m ∠ A = 250 m ∠ B = 300

Fine Arts:

Home: Have parents sign homework sheet.

Remediation: Explain and allow the students to ask/answer questions.

Technology:

Assessment

Evaluate the Congruent Triangle Quiz using the Structured Curriculum Scoring Rubric.(Appendix F)

Homework


Teacher Notes

Place cutup proofs in envelopes for day 064-065.Remind students to study for the upcoming assessment.

Solutions to Congruent Triangle Quiz:

1. Corresponding sides Corresponding AnglesAB EF

BC FG

CD HG

AD EH

≅≅≅≅

∠ ≅ ∠∠ ≅ ∠∠ ≅ ∠∠ ≅ ∠

B F

C G

A E

D H2. S.A.S. 3. A.S.A. 4. Not congruent 5. A.S.A.6. a. AB ≅ DE b. ∠ C ≅ ∠ F c. ∠ B ≅ ∠ E

232

CONGRUENT TRIANGLE QUIZ

1. Polygon ABCD ≅ Polygon EFGH; list the corresponding parts.

Write a congruent statement for the triangles shown. If it is not possible, write “not congruent.”2.

3.

4.

B

A

C

D

F

E

G

H

BA

C

F

E

D

R

S U

T

V

NM

O P

RQ

233

5.

6. List the corresponding parts that need to be congruent to prove ∆ABC ≅ ∆DEF by:

a. S.A.S.b. A.S.A.c. A.A.S.

CA

B

F

E

D

R S

UT

234

ANALYZING ISOSCELES TRIANGLES ACTIVITY

• Draw an isosceles triangle on a sheet of paper. Then cut the triangle and fold itso that the two legs match up. What can you say about the base angles of thetriangle? Write a conjecture in if-then form.

• Draw a triangle with two angles of the triangle congruent. Cut out the triangleand fold it so that the two angles match up. What can you say about the twolegs of the triangle? Write a conjecture in if-then form.

• Look at the folds in the two triangles. What can you say about the folds in eachtriangle? Give as much information as you can.

• After folding the triangle, compare your conjectures and the information yougained with the other members of your group.

235



Correlations (SG,CAS,CFS): 6C1, 2; 9A3, B2


figures


relationships

Unit Focus/Foci

Understanding and Applying Geometric Concepts and Relationships



Materials

Activity: Putting It Together


Have the students sit in groups of three or four. Have a student read the guidelines aloud for theactivity. Try not to answer any questions during the activity. Give the students as much time asyou feel they need to complete this activity.

Putting It Together

Given: BD ⊥ AC, AD ≅ DCProve: ∆ BDA ≅ ∆ BDC

Statements Reasons1. BD ⊥ AC 1. Given2. ∆ BDA and ∆ BDC are rightangles

2. ⊥ lines intersect to form right angles

3. ∆ BDA ≅ ∆ BDC 3. All right angles are congruent.4. AD ≅ DC 4. Given5. BD ≅ BD 5. Reflexive Property6. ∆ BDA ≅ ∆ BDC 6. S.A.S.

A

B

D C

236

Teacher should make as many copies as needed of the proof. Cutup the proofs, includingincorrect statements and reasons, and place them in an envelope. Give each group a proof. Thefirst group to complete the proof should be allowed to present their proof (even if it is wrong). Ifit is wrong, allow another group to try to present the correct proof. After about three tries, workwith the class as a whole to see if they can come up with the correct answer.

A blank copy of Putting It Together is here for your use.(You may want to use a proof of your own.)



Connection(s)

Enrichment:

PO ≅ OW ≅ EW ≅ RE ≅ RP∠ R ≅ ∠ O

Fine Arts:

Home: Have family members quiz students on material from the text.

Remediation: Teachers’ Half-Dozen

Technology:

R

E W

P

O

237

Assessment

Teacher observation and student activity

Homework


Teacher Notes

As a time saver, make a transparency of the given proof and any additional proofs you might liketo use, cut up the statements and reasons, and have them available for the students to put togetheron the overhead.

238

ACTIVITY

Guidelines

1. Students should work in groups of four. (Groups should represent a wide range of abilitylevels.)

2. Each group should be given a packet containing all the parts of a proof.3. After your group has finished their proof, then pass it on to another group until each group

has worked all the proofs.4. You should take all parts of the proof and rearrange them to complete the proof.

B

A D C

Given: BD⊥ AC, AD ≅ DC Prove: ∆ BDA ≅ ∆ BDC

Statements Reasons

BD⊥ AC Given

∠ BDA and ∠ BDC are rightangles.

⊥ lines intersect to form right angles.

∠ BDA ≅ ∠ BDC All right angles are congruent.

AD ≅ DC Given

BD ≅ BD Reflexive Property

∆ BDA ≅ ∆ BDC S.A.S.

∠ ABD ≅ ∠ CBD Definition of congruent angles

BD ≅ BD Given

AD ≅ DC Definition of midpoint

239



Correlations (SG,CAS,CFS): 6C1, 2; 7A6; 9A3; 9B2

TAP:Perform arithmetic operations involving

integers, fractions, decimals and percents,explicitly stated or within context

Understand number systemsUnderstand geometric properties and


Apply a variety of estimation strategies:standard rounding, order of magnitude,front-ending, compatible numbers, andcompensation

Use variables, number sentences, and equationsto represent solutions and solve problems


ISAT:Solve problems requiring computations with

whole numbers, fractions, decimals, ratios,percent, and proportions

Understand and apply geometric concepts andrelationships

Use mathematical skills to estimate,approximate, and predict outcomes and tojudge reasonableness of results

Identify, analyze, and solve problems usingequations, inequalities, functions, and theirgraphs


Unit Focus/Foci



Reviewing Congruent Triangles

Materials


Review the homework with students.

Review Unit Four with the students using a review format from Appendix D.

240




Connection(s)

Enrichment:

Fine Arts:


Remediation:

Technology:

Assessment

Teacher observation

Homework

Study for the Unit Four Assessment.

Teacher Notes

Prepare copies of the Unit Four Assessment.

241



Correlations (SG,CAS,CFS): 6C1, 2; 7A7; 9A3; 9B2

TAP:Perform arithmetic operations involving

integers, fractions, decimals and percents,explicitly stated or within context

Understand number systemsUnderstand geometric properties and


Apply a variety of estimation strategies:standard rounding, order of magnitude,front-ending, compatible numbers, andcompensation

Use variables, number sentences, and equationsto represent solutions and solve problems


ISAT:Solve problems requiring computations with

whole numbers, fractions, decimals, ratios,percent, and proportions

Understand and apply geometric concepts andrelationships

Use mathematical skills to estimate,approximate, and predict outcomes and tojudge reasonableness of results

Identify, analyze, and solve problems usingequations, inequalities, functions, and theirgraphs


Unit Focus/Foci



Assessing Unit Four

Materials

Copies of the Unit Four Assessment


Administer the Unit Four Assessment.

242




Connection(s)

Enrichment:

Fine Arts:

Home:

Remediation:

Technology:

Assessment

Evaluate the Unit Four Assessment.

Homework

Relax.

Teacher Notes

Solutions to Unit Four Assessment:

1. See student drawing 2. Isosceles triangle3. Isoscles 4. Equilateral (Equiangular)5. Scalene N

8 9.4

O 5 M

6. Scalene

7. m∠ P = 521

3°

8. m∠ P = -4°

243

9.

No SSA

10.

Yes by ASA N

11. Given:

NO NP

MO PQ

NM NQ

≅

≅

≅Prove: ∆ NMO ≅ ∆ MQP M O P Q

Statements Reasons

1. NO NP≅ 145

272 5= . !

2. MO PQ≅3. NM MQ≅4. ∆ NMO ≅ ∆ MQP

1. Given2. Given3. Given4. AAS Theorem

B

12. Given: ∆ ABC is an isosceles triangle with base TN∠ 2 ≅ ∠ 3∠ A ≅ ∠ C

Prove: ∆ ABE ≅ ∆ CBD 1 2 3 4 A D E C

Statements Reasons

1. ∆ ABC is an isosceles triangle with base TN

2. AB BC≅3. ∠ 2 ≅ ∠ 34. ∠ A ≅ ∠ C5. ∆ ABE ≅ ∆ CBD

1. Given2. Definition of isosceles triangle3. Given4. Given5. AAS Theorem

13. x = 72.5° y = 72.5°

14. 180 – 150° = 30° x° = 30°

A E

F

DB

C

1 2

244

UNIT FOUR ASSESSMENT

Show all work for complete credit:

1. Draw and label the following triangles:a. acute triangleb. obtuse trianglec. right triangle

Fill in the blanks for problems 2 – 4.

2. A triangle with two congruent sides is an ____________________________triangle.

3. If at least two sides of a triangle are congruent, then the triangle is ______________.

4. If all sides of a triangle are congruent, then the triangle is ____________________.

Use the distance formula to classify the triangles by the measure of their sides.

5. ∆ MNO with vertices M(5, 0), N(0, 8), and O(0, 0).

6. ∆ QRS with vertices Q(2, 4), R(-5, 1) and S(0, 0).

245

Find m∠ P for each triangle.

7. 8.

Determine whether the following are congruent or not. Explain your answer.

9. 10.

Given:

AB ED

Is AD EB

=∠ ≅ ∠

≅

1 2

Given: ∠ ≅ ∠Y V W is the midpoint of VY .

Is WZ WX≅

11. Given:

NO NP≅ MO PQ≅ NM NQ≅Prove: ∆ ∆NMO NQP≅

Statements Reasons

N

M O P Q

A E

F

DB

C

1 2

246

Write an informal proof for problem #12.

12. Given: ∆ABC is an isosceles triangle

with base TN ∠ 2 ≅ ∠ 3 ∠ A ≅ ∠ C

Prove: ∆ABE ≅ ∆CBD

Find the missing measures in the following problems.

13. 14.

75°

75°

x°

high school geometry days 054067

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isosceles triangle

scalene triangle

right triangle

obtuse triangle

acute triangle

given triangle

equilateral triangle

groups definition