Examples: 1. Write the angles in order from 2. Write the sides in order from

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Examples: 1. Write the angles in order from 2. Write the sides in order from"

Transcription

1 Lesson 1 Triangle Inequalities 17. I can apply the triangle inequalities theorems When considering triangles, two basic questions arise: Can any three sides form a triangle? What is the relationship between the angles and sides of a triangle? NO show how sides of 4, 6, and 1 can t SHORTEST SIDE ACROSS FROM SMALLEST ANGLE, LONGEST SIDE ACROSS FROM LARGEST ANGLE Examples: 1. Write the angles in order from. Write the sides in order from smallest to largest shortest to longest F, H, G PQ, QR, PR 3. Determine whether a triangle can have the given side lengths. Show work or explain your reasoning. 7, 10, 19.3, 3.1, 4.6 8, 13, 1 NO YES NO ( = does not count!) 4. A triangle has side lengths 7 and 1. What is the range of possible side lengths for the other side? 1 7 5, x 19 Third side is between 5 and 19

2

3 Lesson Ratios and Proportions 1. I can express ratios in multiple formats. I can solve proportions A ratio compares two numbers by division. It can be written in three ways: 10 : to 1 A proportion is an equation stating two ratios are equal. a b c d Cross Products Property: In a proportion, a c a d b c b d Other Properties of Ratios The proportion a c b d a b is equivalent to: * * d c c d b a * b d a c Examples: 1. Solve the following proportions: 7 56 a. x 7 56x 504 x 9 b. x 3 x 5 4 5x 4(x3) 5x 8x1 3x 1 x 4 c. x x 4 d. x 3 8 x 3 ( x 4) 100 x 4 10 x 14 or x 6 ( x 3)( x 3) 16 x x x 5

4 . Given that 6 5 x y, then x y =? x y According to a recent study, 5 out of 6 high school students have a smart phone. If there are 900 students at Gull Lake High School, approximately how many of them have a smart phone? 5 x 6x 4500 x students Let x be the length of the model in centimeters. The rectangular model of the racing car is similar to the rectangular racing car, so the corresponding lengths are proportional. Find the length of the model to the nearest tenth of a centimeter. actual x 31.5 x 17.5 model 6.3 x Length of model is 17.5 cm 5. The ratio of the side lengths of a triangle is 4:7:5, and its perimeter is 96 cm. What is the length of the shortest side? 4x 7x 5x 96 16x 96 x 6 5(6) 30 Shortest side is 30 cm

5 Lesson 3 Similarity in Figures 3. I can state the properties of similarity 4. I can find the similarity ratio of similar triangles 6. I can find missing angles of similar triangles 10. I can write a similarity statement 11. I can verify that triangles are similar We have already learned what it means for two figures to be congruent. We are going to see what it means for two figures to be similar. Two figures are congruent if and only if they have the same shape and size. If two figures are congruent, then their corresponding angles are congruent and their corresponding side lengths are congruent. Congruence Transformations: Two figures are similar if and only if they have the same shape, but not necessarily the same size. If two figures are similar, then their corresponding angles are congruent and their corresponding side lengths are proportional. Similarity Transformations: The similarity ratio is ratio of lengths of corresponding sides. The similarity ratio of ABCD to EFGH is 1: The similarity ratio of EFGH to ABCD is :1 When figures are congruent to each other their similarity ratio is 1: 1

6 Examples: Determine if each pair of figures is similar to each other. If so, identify the corresponding sides and angles and determine the similarity ratio and similarity statement. If not, explain why they are not similar. M N L P J S Ratio: LM MJ JL PN NS SP Statement: JLM SPN A E B F C G D H Ratio: 3 Statement: ABCD BC CD DA AB FG GH HE EF EFGH NOT similar, angles are not congruent A J B G C H AB BC AC JG GH JH Ratio: 1 Statement: ABC JGH

7 Lesson 4 Triangle Similarity 5. I can find the missing side lengths of similar triangles 7. I can show triangles are similar using the AA Postulate 8. I can show triangles are similar using the SAS Theorem 9. I can show triangles are similar using the SSS Theorem 10. I can write a similarity statement 11. I can verify that triangles are similar Just like we learned that there are shortcuts to proving triangles congruent to each other, there are shortcuts to proving triangles similar to each other. They are: AA~ SSS~ SAS~ Examples: Prove or explain why the triangles are similar and write a similarity statement. A D (right 's) BCA ECD (vertical 's) ABC DEC by AA~ PR PQ QR SU ST TU PQR STU 3 by SSS~ TX UX VX WX TXU WXV (vertical 's) TUX VWX by SAS~ D H (given) DE FD 5.8 HJ KH DEF HJK by SAS~ B E (right 's) A D (both 43 ) ABC DEF by AA~

8 Refer to the diagram to the right. Explain why ABE ~ ACD. B C (right 's) A A (reflexive) ADC AEB by AA~ Now that you know the triangles are similar, determine CD. small big x 60 9x x 6.6 Refer to the diagram at the right. Explain why ABE ~ ACD. A A (reflexive) ABE ACD (corr. 's) ABE ACD by AA~ Now that you know the triangles are similar, determine BE and CD. small 3 x big 7.5 x 6 3x x x x 4 BE 4 CD 10

9 Lesson 5 Properties of Similar Triangles 1. I can apply the Triangle Proportionality Theorem 13. I can apply the Converse of the Triangle Proportionality Theorem 14. I can apply the Two-Transversal Proportionality Corollary 15. I can apply the Triangle Angle Bisector Theorem Using the theorems involving proportional relationships, you can complete the following examples. Examples: In each example, find what s asked for and NAME the theorem you used. Find PN Find SU SU 4 10 SU 56 SU 5.6 Side-Splitter Theorem 3 PN 5 PN 15 PN 7.5 Side-Splitter Theorem Given that AC 36, what value of BC would make DE AB? Verify that DE BC Conv. Side-Split Thm BE 0 BE 40 BE 1 BC 7

10 LM 4.5 LM 1.74 LM MN 4.5 MN 0.09 MN 4.5 Find LM and MN Find PS and SR x 3 x x 80 3x 160 8x 40 x 30 PS 8 SR 35 Find AC and CD y y 4y 4.5y 9.5x 9 x 18 AC 16 DC 9

11 Lesson 6 Indirect Measurement 16. I can solve real-world problems using similar triangles Indirect measurement is any method that uses formulas, similar figures, and/or proportions to measure an object. The following examples use indirect measurement to find a missing measure. For each of the following, draw a sketch of the situation and use similar triangles to determine the missing length. Using an object s shadow Follow along: Tyler wants to find the height of a telephone pole. He measured the pole s shadow and his own shadow and then made a diagram. What is the height h of the pole? height 5'9" H 69 H 9H H '9" shadow 7'8" 38' 4" You Try: A student who is 5 ft 6 in. tall measured shadows to find the height LM of a flagpole. What is LM? height 5' 6" H 66 H 60H 110 H '7" shadow 5' 14' " Using a scale drawing Follow along: On a Wisconsin road map, Kristin measured a distance of 11 in. from Madison to Wausau. The scale of this map is 1inch: 13 miles. What is the actual distance between Madison and Wausau to the nearest mile? map actual 1 11 x 143 miles 13 x You Try: The rectangular central chamber of the Lincoln Memorial is 74 ft long and 60 ft wide. Make a scale drawing of the floor of the chamber using a scale of 1 in.:0 ft. drawing 1 L 0L inches long actual 0 74 drawing 1 L 0L 60 3 inches long actual 0 60

12 What is the relationship between Similar Figures, Perimeter, and Area? The following figures are similar squares, similar triangles, and similar rectangles. Find the similarity ratio, and perimeter and area of each figure. A. Similar Squares B. Similar Triangles C. Similar Rectangles P: 8 P: 4 P: 1 P: 4 P: 30 P: 18 A: 4 A: 36 A: 6 A: 4 A: 50 A: 18 Similarity Ratio: 3:1 :1 5:3 Perimeter Ratio: 3:1 :1 5:3 Area Ratio: 9:1 4:1 5:9 Follow along: Given that LMN QRT, find the perimeter P and area A of QRS. a 9.1 b Perimeter: cm Area: cm You try: ABC ~ DEF, BC = 4 mm, and EF = 1 mm. If P = 4 mm and A = 96 mm for DEF, find the perimeter and area of ABC. a 1 b 3 1 Perimeter: 4 14 cm 3 1 Area: cm

6.1 Ratios, Proportions, and the Geometric Mean

6.1 Ratios, Proportions, and the Geometric Mean 6.1 Ratios, Proportions, and the Geometric Mean Obj.: Solve problems by writing and solving proportions. Key Vocabulary Ratio - If a and b are two numbers or quantities and b 0, then the ratio of a to

More information

7-1 Ratio and Proportion

7-1 Ratio and Proportion 7-1 Ratio and Proportion Ratio 1) Find the slope of line m provided that points and lie on m. 2) The ratio of the angle measures in a triangle is 1:6:13. What is the measure of each angle? Proportion Cross

More information

Unit 8: Congruent and Similar Triangles Lesson 8.1 Apply Congruence and Triangles Lesson 4.2 from textbook

Unit 8: Congruent and Similar Triangles Lesson 8.1 Apply Congruence and Triangles Lesson 4.2 from textbook Unit 8: Congruent and Similar Triangles Lesson 8.1 Apply Congruence and Triangles Lesson 4.2 from textbook Objectives Identify congruent figures and corresponding parts of closed plane figures. Prove that

More information

The common ratio in (ii) is called the scaled-factor. An example of two similar triangles is shown in Figure 47.1. Figure 47.1

The common ratio in (ii) is called the scaled-factor. An example of two similar triangles is shown in Figure 47.1. Figure 47.1 47 Similar Triangles An overhead projector forms an image on the screen which has the same shape as the image on the transparency but with the size altered. Two figures that have the same shape but not

More information

Study Guide and Review

Study Guide and Review Choose the letter of the word or phrase that best completes each statement. a. ratio b. proportion c. means d. extremes e. similar f. scale factor g. AA Similarity Post h. SSS Similarity Theorem i. SAS

More information

Testing for Congruent Triangles Examples

Testing for Congruent Triangles Examples Testing for Congruent Triangles Examples 1. Why is congruency important? In 1913, Henry Ford began producing automobiles using an assembly line. When products are mass-produced, each piece must be interchangeable,

More information

5-1 Perpendicular and Angle Bisectors

5-1 Perpendicular and Angle Bisectors 5-1 Perpendicular and Angle Bisectors 5-1 Perpendicular and Angle Bisectors Warm Up Lesson Presentation Lesson Quiz Holt 5-1 Perpendicular and Angle Bisectors Warm Up Construct each of the following. 1.

More information

Geometry Regents Review

Geometry Regents Review Name: Class: Date: Geometry Regents Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. If MNP VWX and PM is the shortest side of MNP, what is the shortest

More information

Similar Polygons. Copy both triangles onto tracing paper. Measure and record the sides of each triangle. Cut out both triangles.

Similar Polygons. Copy both triangles onto tracing paper. Measure and record the sides of each triangle. Cut out both triangles. -7 Similar Polygons MAIN IDEA Identify similar polygons and find missing measures of similar polygons. New Vocabulary polygon similar corresponding parts congruent scale factor Math Online glencoe.com

More information

NCERT. not to be republished TRIANGLES UNIT 6. (A) Main Concepts and Results

NCERT. not to be republished TRIANGLES UNIT 6. (A) Main Concepts and Results UNIT 6 TRIANGLES (A) Main Concepts and Results The six elements of a triangle are its three angles and the three sides. The line segment joining a vertex of a triangle to the mid point of its opposite

More information

Algebra III. Lesson 33. Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids

Algebra III. Lesson 33. Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids Algebra III Lesson 33 Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids Quadrilaterals What is a quadrilateral? Quad means? 4 Lateral means?

More information

4.1 Apply Triangle Sum Properties

4.1 Apply Triangle Sum Properties 4.1 Apply Triangle Sum Properties Obj.: Classify triangles and find measures of their angles. Key Vocabulary Triangle - A triangle is a polygon w it h three sid es. A t r ian gle w it h ver t ices A, B,

More information

End-of-Year Test Modules 1 23

End-of-Year Test Modules 1 23 Name Date Class For 1 2, use the graph. 7. Use the graph. 1. Which segment is congruent to EF? 2. What is the midpoint of GH? Write the vector (transformation) that maps RST to RST. _ Use the figure for

More information

10-4 Inscribed Angles. Find each measure. 1.

10-4 Inscribed Angles. Find each measure. 1. Find each measure. 1. 3. 2. intercepted arc. 30 Here, is a semi-circle. So, intercepted arc. So, 66 4. SCIENCE The diagram shows how light bends in a raindrop to make the colors of the rainbow. If, what

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name: School Name:

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name: School Name: GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, June 16, 2009 9:15 a.m. to 12:15 p.m., only Student Name: School Name: Print your name and the name of

More information

Test to see if ΔFEG is a right triangle.

Test to see if ΔFEG is a right triangle. 1. Copy the figure shown, and draw the common tangents. If no common tangent exists, state no common tangent. Every tangent drawn to the small circle will intersect the larger circle in two points. Every

More information

Quadrilateral Geometry. Varignon s Theorem I. Proof 10/21/2011 S C. MA 341 Topics in Geometry Lecture 19

Quadrilateral Geometry. Varignon s Theorem I. Proof 10/21/2011 S C. MA 341 Topics in Geometry Lecture 19 Quadrilateral Geometry MA 341 Topics in Geometry Lecture 19 Varignon s Theorem I The quadrilateral formed by joining the midpoints of consecutive sides of any quadrilateral is a parallelogram. PQRS is

More information

Geometry Chapter 7. Ratios & Proportions Properties of Proportions Similar Polygons Similarity Proofs Triangle Angle Bisector Theorem

Geometry Chapter 7. Ratios & Proportions Properties of Proportions Similar Polygons Similarity Proofs Triangle Angle Bisector Theorem Geometry Chapter 7 Ratios & Proportions Properties of Proportions Similar Polygons Similarity Proofs Triangle Angle Bisector Theorem Name: Geometry Assignments Chapter 7 Date Due Similar Polygons Section

More information

Geometry FSA Mathematics Practice Test Answer Key

Geometry FSA Mathematics Practice Test Answer Key Geometry FSA Mathematics Practice Test Answer Key The purpose of these practice test materials is to orient teachers and students to the types of questions on paper-based FSA tests. By using these materials,

More information

Chapter 4 Study guide

Chapter 4 Study guide Name: Class: Date: ID: A Chapter 4 Study guide Numeric Response 1. An isosceles triangle has a perimeter of 50 in. The congruent sides measure (2x + 3) cm. The length of the third side is 4x cm. What is

More information

5-1 Perpendicular and Angle Bisectors

5-1 Perpendicular and Angle Bisectors 5-1 Perpendicular and Angle Bisectors Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up Construct each of the following. 1. A perpendicular bisector. 2. An angle bisector. 3. Find the midpoint and

More information

Unit 1: Similarity, Congruence, and Proofs

Unit 1: Similarity, Congruence, and Proofs Unit 1: Similarity, Congruence, and Proofs This unit introduces the concepts of similarity and congruence. The definition of similarity is explored through dilation transformations. The concept of scale

More information

Geometry EOC Practice Test #4

Geometry EOC Practice Test #4 Class: Date: Geometry EOC Practice Test #4 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. In the diagram below, which expression represents x, the degree

More information

CHAPTER 8 QUADRILATERALS. 8.1 Introduction

CHAPTER 8 QUADRILATERALS. 8.1 Introduction CHAPTER 8 QUADRILATERALS 8.1 Introduction You have studied many properties of a triangle in Chapters 6 and 7 and you know that on joining three non-collinear points in pairs, the figure so obtained is

More information

Similar Polygons. Similar Polygons

Similar Polygons. Similar Polygons Similar Polygons In this unit, we will define similar polygons, investigate ways to show two polygons are similar, and apply similarity postulates and theorems in problems and proofs. Similar Polygons

More information

http://www.castlelearning.com/review/teacher/assignmentprinting.aspx 5. 2 6. 2 1. 10 3. 70 2. 55 4. 180 7. 2 8. 4

http://www.castlelearning.com/review/teacher/assignmentprinting.aspx 5. 2 6. 2 1. 10 3. 70 2. 55 4. 180 7. 2 8. 4 of 9 1/28/2013 8:32 PM Teacher: Mr. Sime Name: 2 What is the slope of the graph of the equation y = 2x? 5. 2 If the ratio of the measures of corresponding sides of two similar triangles is 4:9, then the

More information

Geometry FSA Mathematics Practice Test Questions

Geometry FSA Mathematics Practice Test Questions Geometry FSA Mathematics Practice Test Questions The purpose of these practice test materials is to orient teachers and students to the types of questions on paper-based FSA tests. By using these materials,

More information

A. 3y = -2x + 1. y = x + 3. y = x - 3. D. 2y = 3x + 3

A. 3y = -2x + 1. y = x + 3. y = x - 3. D. 2y = 3x + 3 Name: Geometry Regents Prep Spring 2010 Assignment 1. Which is an equation of the line that passes through the point (1, 4) and has a slope of 3? A. y = 3x + 4 B. y = x + 4 C. y = 3x - 1 D. y = 3x + 1

More information

Coordinate Algebra 1- Common Core Test -1. Diagnostic. Test. Revised 12/5/13 1:19 pm

Coordinate Algebra 1- Common Core Test -1. Diagnostic. Test. Revised 12/5/13 1:19 pm Coordinate Algebra 1- Common Core Test -1 Diagnostic Test Revised 12/5/13 1:19 pm 1. A B C is a dilation of triangle ABC by a scale factor of ½. The dilation is centered at the point ( 5, 5). Which statement

More information

Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition.

Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. 1. measures less than By the Exterior Angle Inequality Theorem, the exterior angle ( ) is larger than

More information

The mid-segment of a triangle is a segment joining the of two sides of a triangle.

The mid-segment of a triangle is a segment joining the of two sides of a triangle. 5.1 and 5.4 Perpendicular and Angle Bisectors & Midsegment Theorem THEOREMS: 1) If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment.

More information

0810ge. Geometry Regents Exam 0810

0810ge. Geometry Regents Exam 0810 0810ge 1 In the diagram below, ABC XYZ. 3 In the diagram below, the vertices of DEF are the midpoints of the sides of equilateral triangle ABC, and the perimeter of ABC is 36 cm. Which two statements identify

More information

2nd Semester Geometry Final Exam Review

2nd Semester Geometry Final Exam Review Class: Date: 2nd Semester Geometry Final Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The owner of an amusement park created a circular

More information

8-1 Geometric Mean. or Find the geometric mean between each pair of numbers and 20. similar triangles in the figure.

8-1 Geometric Mean. or Find the geometric mean between each pair of numbers and 20. similar triangles in the figure. 8-1 Geometric Mean or 24.5 Find the geometric mean between each pair of numbers. 1. 5 and 20 4. Write a similarity statement identifying the three similar triangles in the figure. numbers a and b is given

More information

Unit 7 - Test. Name: Class: Date: 1. If BCDE is congruent to OPQR, then DE is congruent to?. A. PQ B. OR C. OP D. QR 2. BAC?

Unit 7 - Test. Name: Class: Date: 1. If BCDE is congruent to OPQR, then DE is congruent to?. A. PQ B. OR C. OP D. QR 2. BAC? Class: Date: Unit 7 - Test 1. If BCDE is congruent to OPQR, then DE is congruent to?. A. PQ B. OR C. OP D. QR 2. BAC? A. PNM B. NPM C. NMP D. MNP 3. Given QRS TUV, QS = 3v + 2, and TV = 7v 6, find the

More information

Geometry, Final Review Packet

Geometry, Final Review Packet Name: Geometry, Final Review Packet I. Vocabulary match each word on the left to its definition on the right. Word Letter Definition Acute angle A. Meeting at a point Angle bisector B. An angle with a

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

More information

The Pythagorean Packet Everything Pythagorean Theorem

The Pythagorean Packet Everything Pythagorean Theorem Name Date The Pythagorean Packet Everything Pythagorean Theorem Directions: Fill in each blank for the right triangle by using the words in the Vocab Bo. A Right Triangle These sides are called the of

More information

Review for Final - Geometry B

Review for Final - Geometry B Review for Final - Geometry B Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A model is made of a car. The car is 4 meters long and the model is 7 centimeters

More information

Formal Geometry S1 (#2215)

Formal Geometry S1 (#2215) Instructional Materials for WCSD Math Common Finals The Instructional Materials are for student and teacher use and are aligned to the Course Guides for the following course: Formal Geometry S1 (#2215)

More information

DEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.

DEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle. DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent

More information

Warm Up. Use A ( 2, 3) and B (1, 0) 1. Find the slope of AB. 2. Find the midpoint of AB. 3. Find the distance of AB. 4. Simplify.

Warm Up. Use A ( 2, 3) and B (1, 0) 1. Find the slope of AB. 2. Find the midpoint of AB. 3. Find the distance of AB. 4. Simplify. Use A ( 2, 3) and B (1, 0) 1. Find the slope of AB. 2. Find the midpoint of AB. 3. Find the distance of AB. Warm Up 4. Simplify. 5. Draw an example of vertical angles. GOALS Develop and apply the

More information

Geometry EOC Practice Test #2

Geometry EOC Practice Test #2 Class: Date: Geometry EOC Practice Test #2 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Rebecca is loading medical supply boxes into a crate. Each supply

More information

Law of Cosines. If the included angle is a right angle then the Law of Cosines is the same as the Pythagorean Theorem.

Law of Cosines. If the included angle is a right angle then the Law of Cosines is the same as the Pythagorean Theorem. Law of Cosines In the previous section, we learned how the Law of Sines could be used to solve oblique triangles in three different situations () where a side and two angles (SAA) were known, () where

More information

THE DISTANCE FORMULA

THE DISTANCE FORMULA THE DISTANCE FORMULA In this activity, you will develop a formula for calculating the distance between any two points in a coordinate plane. Part 1: Distance Along a Horizontal or Vertical Line To find

More information

Review of Ratio and Proportion Ratio- a comparison of two quantities. the ratio of p and q is p q

Review of Ratio and Proportion Ratio- a comparison of two quantities. the ratio of p and q is p q Review of Ratio and Proportion Ratio- a comparison of two quantities. the ratio of p and q is p q or p:q or p to q The ratio of to is EX1/ Find the ratio of shaded boxes to unshaded. EX2/ Find the ratio

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name: GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, June 17, 2010 1:15 to 4:15 p.m., only Student Name: School Name: Print your name and the name of your

More information

Geometry Sample Problems

Geometry Sample Problems Geometry Sample Problems Sample Proofs Below are examples of some typical proofs covered in Jesuit Geometry classes. Shown first are blank proofs that can be used as sample problems, with the solutions

More information

Semester Exam Review. Multiple Choice Identify the choice that best completes the statement or answers the question.

Semester Exam Review. Multiple Choice Identify the choice that best completes the statement or answers the question. Semester Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Are O, N, and P collinear? If so, name the line on which they lie. O N M P a. No,

More information

Investigating Relationships of Area and Perimeter in Similar Polygons

Investigating Relationships of Area and Perimeter in Similar Polygons Investigating Relationships of Area and Perimeter in Similar Polygons Lesson Summary: This lesson investigates the relationships between the area and perimeter of similar polygons using geometry software.

More information

Name Period Right Triangles and Trigonometry Section 9.1 Similar right Triangles

Name Period Right Triangles and Trigonometry Section 9.1 Similar right Triangles Name Period CHAPTER 9 Right Triangles and Trigonometry Section 9.1 Similar right Triangles Objectives: Solve problems involving similar right triangles. Use a geometric mean to solve problems. Ex. 1 Use

More information

1. An isosceles trapezoid does not have perpendicular diagonals, and a rectangle and a rhombus are both parallelograms.

1. An isosceles trapezoid does not have perpendicular diagonals, and a rectangle and a rhombus are both parallelograms. Quadrilaterals - Answers 1. A 2. C 3. A 4. C 5. C 6. B 7. B 8. B 9. B 10. C 11. D 12. B 13. A 14. C 15. D Quadrilaterals - Explanations 1. An isosceles trapezoid does not have perpendicular diagonals,

More information

How do changes in dimensions of similar geometric figures affect the perimeters and the areas of the figures? ACTIVITY: Creating Similar Figures

How do changes in dimensions of similar geometric figures affect the perimeters and the areas of the figures? ACTIVITY: Creating Similar Figures .6 Perimeters and Areas of Similar Figures How do changes in dimensions of similar geometric figures affect the perimeters and the areas of the figures? ACTIVITY: Creating Similar Figures Work with a partner.

More information

Identifying Triangles 5.5

Identifying Triangles 5.5 Identifying Triangles 5.5 Name Date Directions: Identify the name of each triangle below. If the triangle has more than one name, use all names. 1. 5. 2. 6. 3. 7. 4. 8. 47 Answer Key Pages 19 and 20 Name

More information

Find the unknown number in the given proportion. Round your answer to the nearest hundredth, if necessary. x 1) 34 = 7 17

Find the unknown number in the given proportion. Round your answer to the nearest hundredth, if necessary. x 1) 34 = 7 17 Cy-Fair College PRACTICE: Departmental Review for Math 0306 Exit Assessment (Calculator Portion) Name Find the unknown number in the given proportion. Round your answer to the nearest hundredth, if necessary.

More information

5.1 Midsegment Theorem and Coordinate Proof

5.1 Midsegment Theorem and Coordinate Proof 5.1 Midsegment Theorem and Coordinate Proof Obj.: Use properties of midsegments and write coordinate proofs. Key Vocabulary Midsegment of a triangle - A midsegment of a triangle is a segment that connects

More information

GEOMETRY (Common Core)

GEOMETRY (Common Core) GEOMETRY (COMMON CORE) The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY (Common Core) Tuesday, June 2, 2015 1:15 to 4:15 p.m., only Student Name: School Name: The possession

More information

CHAPTER 38 INTRODUCTION TO TRIGONOMETRY

CHAPTER 38 INTRODUCTION TO TRIGONOMETRY CHAPTER 38 INTRODUCTION TO TRIGONOMETRY EXERCISE 58 Page 47. Find the length of side x. By Pythagoras s theorem, 4 = x + 40 from which, x = 4 40 and x = 4 40 = 9 cm. Find the length of side x. By Pythagoras

More information

VOLUME of Rectangular Prisms Volume is the measure of occupied by a solid region.

VOLUME of Rectangular Prisms Volume is the measure of occupied by a solid region. Math 6 NOTES 7.5 Name VOLUME of Rectangular Prisms Volume is the measure of occupied by a solid region. **The formula for the volume of a rectangular prism is:** l = length w = width h = height Study Tip:

More information

Pythagorean Theorem & Trigonometric Ratios

Pythagorean Theorem & Trigonometric Ratios Algebra 2012-2013 Pythagorean Theorem & Trigonometric Ratios Name: Teacher: Pd: Table of Contents DAY 1: SWBAT: Calculate the length of a side a right triangle using the Pythagorean Theorem Pgs: 1-4 HW:

More information

Unit 7: Right Triangles and Trigonometry Lesson 7.1 Use Inequalities in a Triangle Lesson 5.5 from textbook

Unit 7: Right Triangles and Trigonometry Lesson 7.1 Use Inequalities in a Triangle Lesson 5.5 from textbook Unit 7: Right Triangles and Trigonometry Lesson 7.1 Use Inequalities in a Triangle Lesson 5.5 from textbook Objectives Use the triangle measurements to decide which side is longest and which angle is largest.

More information

Geometry Second Semester Final Exam Review

Geometry Second Semester Final Exam Review Name: Class: Date: ID: A Geometry Second Semester Final Exam Review 1. Mr. Jones has taken a survey of college students and found that 1 out of 6 students are liberal arts majors. If a college has 7000

More information

Unit 3: Triangle Bisectors and Quadrilaterals

Unit 3: Triangle Bisectors and Quadrilaterals Unit 3: Triangle Bisectors and Quadrilaterals Unit Objectives Identify triangle bisectors Compare measurements of a triangle Utilize the triangle inequality theorem Classify Polygons Apply the properties

More information

9-1 Similar Right Triangles (Day 1) 1. Review:

9-1 Similar Right Triangles (Day 1) 1. Review: 9-1 Similar Right Triangles (Day 1) 1. Review: Given: ACB is right and AB CD Prove: ΔADC ~ ΔACB ~ ΔCDB. Statement Reason 2. In the diagram in #1, suppose AD = 27 and BD = 3. Find CD. (You may find it helps

More information

Chapter 11. Areas of Plane Figures You MUST draw diagrams and show formulas for every applicable homework problem!

Chapter 11. Areas of Plane Figures You MUST draw diagrams and show formulas for every applicable homework problem! Chapter 11 Areas of Plane Figures You MUST draw diagrams and show formulas for every applicable homework problem! Objectives A. Use the terms defined in the chapter correctly. B. Properly use and interpret

More information

Pythagorean Theorem: 9. x 2 2

Pythagorean Theorem: 9. x 2 2 Geometry Chapter 8 - Right Triangles.7 Notes on Right s Given: any 3 sides of a Prove: the is acute, obtuse, or right (hint: use the converse of Pythagorean Theorem) If the (longest side) 2 > (side) 2

More information

RATIO, PROPORTION, AND SIMILARITY

RATIO, PROPORTION, AND SIMILARITY HPTER 474 12 HPTER TLE OF ONTENTS 12-1 Ratio and Proportion 12-2 Proportions Involving Line Segments 12-3 Similar Polygons 12-4 Proving Triangles Similar 12-5 ilations 12-6 Proportional Relations mong

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, January 26, 2016 1:15 to 4:15 p.m., only.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, January 26, 2016 1:15 to 4:15 p.m., only. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, January 26, 2016 1:15 to 4:15 p.m., only Student Name: School Name: The possession or use of any communications

More information

as a fraction and as a decimal to the nearest hundredth.

as a fraction and as a decimal to the nearest hundredth. Express each ratio as a fraction and as a decimal to the nearest hundredth. 1. sin A The sine of an angle is defined as the ratio of the opposite side to the hypotenuse. So, 2. tan C The tangent of an

More information

MATH 139 FINAL EXAM REVIEW PROBLEMS

MATH 139 FINAL EXAM REVIEW PROBLEMS MTH 139 FINL EXM REVIEW PROLEMS ring a protractor, compass and ruler. Note: This is NOT a practice exam. It is a collection of problems to help you review some of the material for the exam and to practice

More information

6-5 Rhombi and Squares. ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure.

6-5 Rhombi and Squares. ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure. ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure. 1. If, find. A rhombus is a parallelogram with all four sides congruent. So, Then, is an isosceles triangle. Therefore, If a parallelogram

More information

PROPERTIES OF TRIANGLES AND QUADRILATERALS

PROPERTIES OF TRIANGLES AND QUADRILATERALS Mathematics Revision Guides Properties of Triangles, Quadrilaterals and Polygons Page 1 of 21 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier PROPERTIES OF TRIANGLES AND QUADRILATERALS

More information

Tips for doing well on the final exam

Tips for doing well on the final exam Name Date Block The final exam for Geometry will take place on May 31 and June 1. The following study guide will help you prepare for the exam. Everything we have covered is fair game. As a reminder, topics

More information

Math 311 Test III, Spring 2013 (with solutions)

Math 311 Test III, Spring 2013 (with solutions) Math 311 Test III, Spring 2013 (with solutions) Dr Holmes April 25, 2013 It is extremely likely that there are mistakes in the solutions given! Please call them to my attention if you find them. This exam

More information

Math 531, Exam 1 Information.

Math 531, Exam 1 Information. Math 531, Exam 1 Information. 9/21/11, LC 310, 9:05-9:55. Exam 1 will be based on: Sections 1A - 1F. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/531fa11/531.html)

More information

POTENTIAL REASONS: Definition of Congruence:

POTENTIAL REASONS: Definition of Congruence: Sec 6 CC Geometry Triangle Pros Name: POTENTIAL REASONS: Definition Congruence: Having the exact same size and shape and there by having the exact same measures. Definition Midpoint: The point that divides

More information

Geo, Chap 4 Practice Test, EV Ver 1

Geo, Chap 4 Practice Test, EV Ver 1 Class: Date: Geo, Chap 4 Practice Test, EV Ver 1 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. (4-3) In each pair of triangles, parts are congruent as

More information

Lesson 28: Properties of Parallelograms

Lesson 28: Properties of Parallelograms Student Outcomes Students complete proofs that incorporate properties of parallelograms. Lesson Notes Throughout this module, we have seen the theme of building new facts with the use of established ones.

More information

Geometry Notes Chapter 12. Name: Period:

Geometry Notes Chapter 12. Name: Period: Geometry Notes Chapter 1 Name: Period: Vocabulary Match each term on the left with a definition on the right. 1. image A. a mapping of a figure from its original position to a new position. preimage B.

More information

Therefore, x is 3. To find BC. find 2x or 2(3) = 6. Thus, BC is Linear Measure

Therefore, x is 3. To find BC. find 2x or 2(3) = 6. Thus, BC is Linear Measure Find the length of each line segment or object. 1. Refer to Page 18. The ruler is marked in centimeters. The tail of the fish starts at the zero mark of the ruler and the mouth appears to end 7 tenth marks

More information

11-4 Areas of Regular Polygons and Composite Figures

11-4 Areas of Regular Polygons and Composite Figures 1. In the figure, square ABDC is inscribed in F. Identify the center, a radius, an apothem, and a central angle of the polygon. Then find the measure of a central angle. Center: point F, radius:, apothem:,

More information

6-5 Rhombi and Squares. ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure.

6-5 Rhombi and Squares. ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure. ALGEBRA Quadrilateral ABCD is a rhombus. Find each value or measure. 3. PROOF Write a two-column proof to prove that if ABCD is a rhombus with diagonal. 1. If, find. A rhombus is a parallelogram with all

More information

Chapter 4: Congruent Triangles

Chapter 4: Congruent Triangles Name: Chapter 4: Congruent Triangles Guided Notes Geometry Fall Semester 4.1 Apply Triangle Sum Properties CH. 4 Guided Notes, page 2 Term Definition Example triangle polygon sides vertices Classifying

More information

Finding the Measure of Segments Examples

Finding the Measure of Segments Examples Finding the Measure of Segments Examples 1. In geometry, the distance between two points is used to define the measure of a segment. Segments can be defined by using the idea of betweenness. In the figure

More information

Triangle Congruence using SSS

Triangle Congruence using SSS Triangle Congruence using SSS CK12 Editor Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, August 13, 2013 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

More information

Section 7.1 Solving Right Triangles

Section 7.1 Solving Right Triangles Section 7.1 Solving Right Triangles Note that a calculator will be needed for most of the problems we will do in class. Test problems will involve angles for which no calculator is needed (e.g., 30, 45,

More information

Geometry Review. Here are some formulas and concepts that you will need to review before working on the practice exam.

Geometry Review. Here are some formulas and concepts that you will need to review before working on the practice exam. Geometry Review Here are some formulas and concepts that you will need to review before working on the practice eam. Triangles o Perimeter or the distance around the triangle is found by adding all of

More information

MHR Principles of Mathematics 10 Solutions 1

MHR Principles of Mathematics 10 Solutions 1 Chapter 5 Quadratic Expressions Chapter 5 Get Ready Chapter 5 Get Ready Question 1 Page 08 a 3y has one term. It is a monomial. 3 b 5+ 6a has two terms. It is a binomial. c 6 x + x 1 has three terms. It

More information

Perimeter and area formulas for common geometric figures:

Perimeter and area formulas for common geometric figures: Lesson 10.1 10.: Perimeter and Area of Common Geometric Figures Focused Learning Target: I will be able to Solve problems involving perimeter and area of common geometric figures. Compute areas of rectangles,

More information

SLOPES AND EQUATIONS OF LINES CHAPTER

SLOPES AND EQUATIONS OF LINES CHAPTER CHAPTER 90 8 CHAPTER TABLE OF CONTENTS 8- The Slope of a Line 8- The Equation of a Line 8-3 Midpoint of a Line Segment 8-4 The Slopes of Perpendicular Lines 8-5 Coordinate Proof 8-6 Concurrence of the

More information

8.1 Find Angle Measures in Polygons

8.1 Find Angle Measures in Polygons 8.1 Find Angle Measures in Polygons Obj.: To find angle measures in polygons. Key Vocabulary Diagonal - A diagonal of a polygon is a segment that joins two nonconsecutive vertices. Polygon ABCDE has two

More information

Congruence of Triangles

Congruence of Triangles Congruence of Triangles You've probably heard about identical twins, but do you know there's such a thing as mirror image twins? One mirror image twin is right-handed while the other is left-handed. And

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name: GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, August 18, 2010 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of

More information

3. If AC = 12, CD = 9 and BE = 3, find the area of trapezoid BCDE. (Mathcounts Handbooks)

3. If AC = 12, CD = 9 and BE = 3, find the area of trapezoid BCDE. (Mathcounts Handbooks) EXERCISES: Triangles 1 1. The perimeter of an equilateral triangle is units. How many units are in the length 27 of one side? (Mathcounts Competitions) 2. In the figure shown, AC = 4, CE = 5, DE = 3, and

More information

Winter 2016 Math 213 Final Exam. Points Possible. Subtotal 100. Total 100

Winter 2016 Math 213 Final Exam. Points Possible. Subtotal 100. Total 100 Winter 2016 Math 213 Final Exam Name Instructions: Show ALL work. Simplify wherever possible. Clearly indicate your final answer. Problem Number Points Possible Score 1 25 2 25 3 25 4 25 Subtotal 100 Extra

More information

Geometry Chapter 2: Geometric Reasoning Lesson 1: Using Inductive Reasoning to Make Conjectures Inductive Reasoning:

Geometry Chapter 2: Geometric Reasoning Lesson 1: Using Inductive Reasoning to Make Conjectures Inductive Reasoning: Geometry Chapter 2: Geometric Reasoning Lesson 1: Using Inductive Reasoning to Make Conjectures Inductive Reasoning: Conjecture: Advantages: can draw conclusions from limited information helps us to organize

More information

Chapter 8. Right Triangles

Chapter 8. Right Triangles Chapter 8 Right Triangles Objectives A. Use the terms defined in the chapter correctly. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the

More information

(a) 5 square units. (b) 12 square units. (c) 5 3 square units. 3 square units. (d) 6. (e) 16 square units

(a) 5 square units. (b) 12 square units. (c) 5 3 square units. 3 square units. (d) 6. (e) 16 square units 1. Find the area of parallelogram ACD shown below if the measures of segments A, C, and DE are 6 units, 2 units, and 1 unit respectively and AED is a right angle. (a) 5 square units (b) 12 square units

More information

Geometry CC Assignment #1 Naming Angles. 2. Complementary angles are angles that add up to degrees.

Geometry CC Assignment #1 Naming Angles. 2. Complementary angles are angles that add up to degrees. Geometry CC Assignment #1 Naming Angles 1. Name the given angle in 3 ways. A. Complementary angles are angles that add up to degrees. 3. Supplementary angles are angles that add up to degrees. #1 4. Given

More information