CONSTRUCTIONS EXERCISE 11.1

Size: px
Start display at page:

Download "CONSTRUCTIONS EXERCISE 11.1"

Transcription

1 CONSTRUCTIONS EXERCISE 11.1 In each of the following, give the justification of the construction also: Q1. Draw a line segment of length 7.6cm and divides it in the 5:8. Measure the two parts. Step-I : Draw line segment =7.6cm and draw an acute angle with and produce the ray from A to X. Step-II : Locate 13=(5+8) points A 1, A 2, A 3 A 13 such that AA 1 = A 1 A 2 = A 2 A 3 = A 12 A 13 Step-III : Join BA 13. Step-IV: Through the pointa 5, draw a line parallel to BA 13 (by making an angle equal to AA 13 B) at A 5 intersecting at the point C. Then AC:CB= 5:8 AC=2.9 cm, BC= 4.7cm respectively. Since A 5 c is parallel to A 13 B By using basic proportionality a theorem AA 5 A 5 A 13 BC AA 5 A 5 A 13 = 5 8 Using equation and equation, we get

2 AA 5 = 5 A 5 A 13 BC 8 This justify the C divides in the ratio of 5:8. Q2. Construct a triangle sides 4cm, 5cm, 6cm and then a triangle similar it to whose sides are 3 the corresponding sides of the first triangle. 2 Step-I : Draw line segment =4, BC=5cm, AC=6cm and draw an acute angle with and produce the ray from A to X which is opposite to vertex C Step-II : Locate 3=(the greater of 2 and 3 in 3 2 ) points A 1, A 2, A 3 such that AA 1 = A 1 A 2 = A 2 A 3 Step-III : Join BA 3. Step-IV : Through the pointa 2, draw a line parallel to BA 3 to intersecting at the point B. C is the required triangle. To Proof: = 2 3 BC = 2 3 Bc Ac = 2 3 Ac B C ll BC C = C (Corresponding In C and A B C C = C B AC = BAC (Corresponding (Common C ~ A B C (AA similarity criterion) = B C BC AC

3 In AA 2 B and AA 3 B A 2 = A 3 (Corresponding AA 2 B = AA 3 B (Common AA 2 B ~ AA 3 B (AA similarity criterion) = AA 2 AA 3 = 2 3 By equation and equation = B C BC AC = 2 3 = 2 3 BC = 2 3 Bc Ac = 2 3 Ac Q3. Construct a triangle with sides 5cm, 6cm, 7cm and then another triangle whose sides are 7 of the corresponding sides of the first triangle. 5 Step-I : Draw line segment =5, BC=6cm, AC=7cm and draw an acute angle with and produce the ray from A to X which is opposite to vertex C Step-II : Locate 7=(the 7greater of between 7 and 5 in 7 5 ) points A 1, A 2, A 3,. A 7 such that AA 1 = A 1 A 2 = A 2 A 3 =..A 6 A 7 Step-III : Join BA 5. Step-IV :Through the pointa 7, draw a line parallel to BA 5 to intersecting extended line at the point B. Step-V : Draw a line through B parallel to BC intersecting the extended line AC at C. C is required triangle.

4 To Proof : = 7 5 B C = 7 5 BC AC = 7 5 AC B C ll BC C = C (Corresponding In C and A B C C = C BAC = B AC (Corresponding (Common C ~ C (AA similarity criterion) B C AC In AA 5 B and AA 7 B A 5 = A 7 (Common AA 5 B = AA 7 B (Corresponding AA 2 B ~ AA 3 B (AA similarity criterion) = AA 5 AA 7 = 5 7 By equation and equation = 5 B C AC 7 = 5 7 BC = 5 7 Bc Ac = 5 7 Ac Q4. Construct an isosceles triangle whose base is 8cm and the altitude 4cm and then another triangle whose sides are1 1 times the corresponding sides of 2 the isosceles triangle.

5 Step-I : Draw line segment =8cm, And draw the same line segment from point A and point B on the both side of the line segment and mark the both intersecting point O and O and OO intersect at D. Step-II : from D cut an arc 4cm on the extended line segment of OO at point C. An isosceles triangle C is formed, having CD(altitude) as 4cm and as 8 cm., A 2, A 3,. A 7 such that AA 1 = A 1 A 2 = A 2 A 3 =..A 6 A 7 Step-III : Draw a ray AX an acute angle with line segment on the opposite side on opposite side on vertex C. Step-IV :Locate 3 points (as 3 is greater between 3 and 2) A 1, A 2, A 3 on AX such that AA 1 =A 1 A 2 =A 2 A 3. Step-V : Join BA2 and draw a line through A3 parallel to BA2 to intersect extended line segment at point B. Step-VI : Draw a line through B parallel to BC intersecting the extended line segment AC at C. C is the required triangle. To Proof : = 3 2 B C = 3 2 BC AC = 3 2 AC In C and A B C C = C BAC = B AC (Corresponding (Common C ~ C (AA similarity criterion) B C AC In AA 2 B and AA 3 B A 2 = A 3 (Common

6 AA 2 B = AA 3 B (Corresponding AA 2 B ~ AA 3 B (AA similarity criterion) = AA 2 AA 3 = 2 3 By equation and equation = 2 B C AC 3 = 3 2 BC = 3 2 Bc Ac = 3 2 Ac Q5. Draw a triangle C with sides BC= 6cm, =5cm, and C=60. Then, construct a triangle whose sides are 3 times the corresponding sides of 4 the triangle C. StepI: Draw a C with BC=6cm, =5cm, C=60. StepII: Draw a ray BX making an actue angle with BC on the opposite side of vertex A. StepIII: Locate 4 points (as 4 is greater between 3 and 4),B 1, B 2, B 3, B 4 On the line segment BX. StepIV: Join B 4 C and draw a line through B 3, parallel to B 4 C intersecting BC at C. StepV: Draw a line through C parallel to AC intersecting at A. A B C is required triangle. To Proof :A B = 3 4 BC = 3 4 BC A C = 3 4 AC

7 In A B C and C A C B = ACB (Corresponding A BC = C (Common A B C ~ C (AA similarity criterion) A B = A C BC AC In B 3 BC and BB 4 C B 3 BC = B 4 BC (Common BB 3 C = BB 4 C (Corresponding B 3 BC ~ BB 4 C (AA similarity criterion) BC BC = BB 3 BB 4 BC BC = 3 4 By equation and equation A B = 3 BC = 3 BC A C = 3 AC Q6.Draw a triangle C with sides BC= 7cm, B=45, A=105, Then construct a triangle whose sides are 4 times the corresponding sides of the 3 triangle C. A+ B+ c=180

8 c=180 C= 30 Step I: Draw a triangle C with side of BC=7cm, B= 45, C= 30. Step II: Draw a ray BX making an actue angle with BC on the opposite side of vertex A. Step III: Locate 4 points (as 4 is greater between 4 and 3) B 1, B 2, B 3, B 4 on BX. Step IV: Join B 3 C, Draw a line through B4 parallel to B 3 C intersecting extended BC at C. Step V: By C, draw a line parallel to AC intersecting extended line segment at C. A B C is the required triangle. To Proof :A B = 4 3 BC = 4 3 BC A C = 4 3 AC In A B C and C A C B = ACB A BC = C (Corresponding (Common A B C ~ C (AA similarity criterion) A B BC A C, In BB 3 C and BB 4 C B 3 BC = B 4 BC (Common

9 BB 3 C = BB 4 C^ (Corresponding BB 3 C ~ BB 4 C (AA similarity criterion) BC BC = BB 3 BB 4 BC BC = 3 4 By equation and equation A B BC A C = 3 4 A B = 4 3 BC = 4 3 BC A C = 4 3 AC Q7.Draw a right triangle in which the sides (other than hypotenuse) are of length 4cm and 3cm. Then construct another triangle whose sides are 5 3 times the corresponding sides of the triangle C. Step I: Draw a line segment =4cm, Draw a ray PA making 90 with it. Step II: Draw an arc of 3cm on the ray PA at C. Join BC triangle C is required. Step III: Draw a ray AX making an actue angle with, opposite to C. Step IV: Locate 5 points (as 5 is the greater between 5 and 3)A 1, A 2, A 3, A 4, A 5. Step V: Join A 3 B.Draw a line through A parallel toa 3 B intersecting extended line segment at B. Step VI: By B draw a line parallel BC intersecting extended line segment AC at C. Triangle C is the triangle. By constructio5 To Proof : = 5 3 B C = 5 3 BC AC = 4 3 AC

10 In C and C C = C B AC = BAC (Corresponding (Common C ~ C (AA similarity criterion) A B B C AC, In AA 3 B and AA 5 B A 3 = A 5 (Common AA 3 B = AA 5 B (Corresponding AA 3 B ~ AA 5 B (AA similarity criterion) = AA 3 AA 5 = 3 5 By equation and equation B C AC = 3 5 A B = 5 3 BC = 5 3 BC AC = 5 3 AC

CONSTRUCTIONS Introduction Division of a Line Segment

CONSTRUCTIONS Introduction Division of a Line Segment 216 MATHEMATICS CONSTRUCTIONS 11 111 Introduction In Class IX, you have done certain constructions using a straight edge (ruler) and a compass, eg, bisecting an angle, drawing the perpendicular bisector

More information

not to be republished NCERT CONSTRUCTIONS CHAPTER 10 (A) Main Concepts and Results (B) Multiple Choice Questions

not to be republished NCERT CONSTRUCTIONS CHAPTER 10 (A) Main Concepts and Results (B) Multiple Choice Questions CONSTRUCTIONS CHAPTER 10 (A) Main Concepts and Results Division of a line segment internally in a given ratio. Construction of a triangle similar to a given triangle as per given scale factor which may

More information

Class IX Chapter 11 Constructions Maths

Class IX Chapter 11 Constructions Maths 1 Class IX Chapter 11 Constructions Maths 1: Exercise 11.1 Question Construct an angle of 90 at the initial point of a given ray and justify the construction. Answer: The below given steps will be followed

More information

Chapter 6.1 Medians. Geometry

Chapter 6.1 Medians. Geometry Chapter 6.1 Medians Identify medians of triangles Find the midpoint of a line using a compass. A median is a segment that joins a vertex of the triangle and the midpoint of the opposite side. Median AD

More information

Mth 97 Winter 2013 Sections 4.3 and 4.4

Mth 97 Winter 2013 Sections 4.3 and 4.4 Section 4.3 Problem Solving Using Triangle Congruence Isosceles Triangles Theorem 4.5 In an isosceles triangle, the angles opposite the congruent sides are congruent. A Given: ABC with AB AC Prove: B C

More information

For all questions, E. NOTA means none of the above answers is correct. Diagrams are NOT drawn to scale.

For all questions, E. NOTA means none of the above answers is correct. Diagrams are NOT drawn to scale. For all questions, means none of the above answers is correct. Diagrams are NOT drawn to scale.. In the diagram, given m = 57, m = (x+ ), m = (4x 5). Find the degree measure of the smallest angle. 5. The

More information

Geometry Cheat Sheet

Geometry Cheat Sheet Geometry Cheat Sheet Chapter 1 Postulate 1-6 Segment Addition Postulate - If three points A, B, and C are collinear and B is between A and C, then AB + BC = AC. Postulate 1-7 Angle Addition Postulate -

More information

GEOMETRY R Unit 2: Angles and Parallel Lines

GEOMETRY R Unit 2: Angles and Parallel Lines GEOMETRY R Unit 2: Angles and Parallel Lines Day Classwork Homework Friday 9/15 Unit 1 Test Monday 9/18 Tuesday 9/19 Angle Relationships HW 2.1 Angle Relationships with Transversals HW 2.2 Wednesday 9/20

More information

Drill Exercise - 1. Drill Exercise - 2. Drill Exercise - 3

Drill Exercise - 1. Drill Exercise - 2. Drill Exercise - 3 Drill Exercise -. Find the distance between the pair of points, (a sin, b cos ) and ( a cos, b sin ).. Prove that the points (a, 4a) (a, 6a) and (a + 3 a, 5a) are the vertices of an equilateral triangle.

More information

If three points A (h, 0), P (a, b) and B (0, k) lie on a line, show that: a b 1.

If three points A (h, 0), P (a, b) and B (0, k) lie on a line, show that: a b 1. ASSIGNMENT ON STRAIGHT LINES LEVEL 1 (CBSE/NCERT/STATE BOARDS) 1 Find the angle between the lines joining the points (0, 0), (2, 3) and the points (2, 2), (3, 5). 2 What is the value of y so that the line

More information

1 www.gradestack.com/ssc Dear readers, ADVANCE MATHS - GEOMETRY DIGEST Geometry is a very important topic in numerical ability section of SSC Exams. You can expect 14-15 questions from Geometry in SSC

More information

UNIT 5 SIMILARITY AND CONGRUENCE

UNIT 5 SIMILARITY AND CONGRUENCE UNIT 5 SIMILARITY AND CONGRUENCE M2 Ch. 2, 3, 4, 6 and M1 Ch. 13 5.1 Parallel Lines Objective When parallel lines are cut by a transversal, I will be able to identify angle relationships, determine whether

More information

Solutions to the Test. Problem 1. 1) Who is the author of the first comprehensive text on geometry? When and where was it written?

Solutions to the Test. Problem 1. 1) Who is the author of the first comprehensive text on geometry? When and where was it written? Solutions to the Test Problem 1. 1) Who is the author of the first comprehensive text on geometry? When and where was it written? Answer: The first comprehensive text on geometry is called The Elements

More information

Chapter. Triangles. Copyright Cengage Learning. All rights reserved.

Chapter. Triangles. Copyright Cengage Learning. All rights reserved. Chapter 3 Triangles Copyright Cengage Learning. All rights reserved. 3.3 Isosceles Triangles Copyright Cengage Learning. All rights reserved. In an isosceles triangle, the two sides of equal length are

More information

SECONDARY MATH Area of a Triangle and Law of Sines

SECONDARY MATH Area of a Triangle and Law of Sines SECONDARY MATH 3 7-1 Area of a Triangle and Law of Sines Goal: Be the first team to find (r j h g f)(x). WARM UP COMPOSITION OF FUNCTIONS Person #1 f(x) = x 2 7x + 6 Person #2 g(x) = 2 +10 4 Person #3

More information

CBSE X Mathematics 2012 Solution (SET 1) Section C

CBSE X Mathematics 2012 Solution (SET 1) Section C CBSE X Mathematics 01 Solution (SET 1) Q19. Solve for x : 4x 4ax + (a b ) = 0 Section C The given quadratic equation is x ax a b 4x 4ax a b 0 4x 4ax a b a b 0 4 4 0. 4 x [ a a b b] x ( a b)( a b) 0 4x

More information

Lesson 23: Base Angles of Isosceles Triangles Day 1

Lesson 23: Base Angles of Isosceles Triangles Day 1 Lesson 23: Base Angles of Isosceles Triangles Day 1 Learning Targets I can examine two different proof techniques via a familiar theorem. I can complete proofs involving properties of an isosceles triangle.

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

Proof: Given ABC XYZ, with A X, B Y, and Our strategy is to show C Z and apply ASA. So, WLOG, we assume for contradiction that m C > m Z.

Proof: Given ABC XYZ, with A X, B Y, and Our strategy is to show C Z and apply ASA. So, WLOG, we assume for contradiction that m C > m Z. Theorem: AAS Congruence. If under some correspondence, two angles and a side opposite one of the angles of one triangle are congruent, respectively, to the corresponding two angles and side of a second

More information

Lab Area of Other Quadrilaterals

Lab Area of Other Quadrilaterals Name: Date: Period: Area of a Trapezoid: Compass Lab Area of Other Quadrilaterals Part 1: Constructing the Trapezoid Isosceles a. Using your straightedge, construct 2 intersecting lines in the space provided

More information

If B is the If two angles are

If B is the If two angles are If If B is between A and C, then 1 2 If P is in the interior of RST, then If B is the If two angles are midpoint of AC, vertical, then then 3 4 If angles are adjacent, then If angles are a linear pair,

More information

DE to a line parallel to Therefore

DE to a line parallel to Therefore Some Proofs 1. In the figure below segment DE cuts across triangle ABC, and CD/CA = CE/CB. Prove that DE is parallel to AB. Consider the dilation with center C and scaling factor CA/CD. This dilation fixes

More information

Sample Question Paper

Sample Question Paper Time : 3hrs. MM : 90 Sample Question Paper Term - II General Instructions: (i) (ii) All questions are compulsory. The question paper consists of 34 questions divided into 4 sections. A, B, C and D. Section

More information

LINES AND ANGLES CHAPTER 6. (A) Main Concepts and Results. (B) Multiple Choice Questions

LINES AND ANGLES CHAPTER 6. (A) Main Concepts and Results. (B) Multiple Choice Questions CHAPTER 6 LINES AND ANGLES (A) Main Concepts and Results Complementary angles, Supplementary angles, Adjacent angles, Linear pair, Vertically opposite angles. If a ray stands on a line, then the adjacent

More information

Chapter 1-2 Points, Lines, and Planes

Chapter 1-2 Points, Lines, and Planes Chapter 1-2 Points, Lines, and Planes Undefined Terms: A point has no size but is often represented by a dot and usually named by a capital letter.. A A line extends in two directions without ending. Lines

More information

Moore Catholic High School Math Department

Moore Catholic High School Math Department Moore Catholic High School Math Department Geometry Vocabulary The following is a list of terms and properties which are necessary for success in a Geometry class. You will be tested on these terms during

More information

Geometry Review for Test 3 January 13, 2016

Geometry Review for Test 3 January 13, 2016 Homework #7 Due Thursday, 14 January Ch 7 Review, pp. 292 295 #1 53 Test #3 Thurs, 14 Jan Emphasis on Ch 7 except Midsegment Theorem, plus review Betweenness of Rays Theorem Whole is Greater than Part

More information

Lesson 3.6 Overlapping Triangles

Lesson 3.6 Overlapping Triangles Lesson 3.6 Overlapping Triangles Getting Ready: Each division in the given triangle is 1 unit long. Hence, the side of the largest triangle is 4- unit long. Figure 3.6.1. Something to think about How many

More information

November 21, Angles of Triangles

November 21, Angles of Triangles Geometry Essential Question How are the angle measures of a triangle related? Goals Day 1 Classify triangles by their sides Classify triangles by their angles Identify parts of triangles. Find angle measures

More information

SOME IMPORTANT PROPERTIES/CONCEPTS OF GEOMETRY (Compiled by Ronnie Bansal)

SOME IMPORTANT PROPERTIES/CONCEPTS OF GEOMETRY (Compiled by Ronnie Bansal) 1 SOME IMPORTANT PROPERTIES/CONCEPTS OF GEOMETRY (Compiled by Ronnie Bansal) 1. Basic Terms and Definitions: a) Line-segment: A part of a line with two end points is called a line-segment. b) Ray: A part

More information

Moore Catholic High School Math Department

Moore Catholic High School Math Department Moore Catholic High School Math Department Geometry Vocabulary The following is a list of terms and properties which are necessary for success in a Geometry class. You will be tested on these terms during

More information

9.2 SECANT AND TANGENT

9.2 SECANT AND TANGENT TOPICS PAGES. Circles -5. Constructions 6-. Trigonometry -0 4. Heights and Distances -6 5. Mensuration 6-9 6. Statistics 40-54 7. Probability 55-58 CIRCLES 9. CIRCLE A circle is the locus of a points which

More information

5.4 Medians and Altitudes in Triangles

5.4 Medians and Altitudes in Triangles 5.4. Medians and Altitudes in Triangles www.ck12.org 5.4 Medians and Altitudes in Triangles Learning Objectives Define median and find their point of concurrency in a triangle. Apply medians to the coordinate

More information

CHAPTER TWO. . Therefore the oblong number n(n + 1) is double the triangular number T n. , and the summands are the triangular numbers T n 1 and T n.

CHAPTER TWO. . Therefore the oblong number n(n + 1) is double the triangular number T n. , and the summands are the triangular numbers T n 1 and T n. CHAPTER TWO 1. Since AB BC; since the two angles at B are equal; and since the angles at A and C are both right angles, it follows by the angle-side-angle theorem that EBC is congruent to SBA and therefore

More information

Problem 3.1 (Building up geometry). For an axiomatically built-up geometry, six groups of axioms needed:

Problem 3.1 (Building up geometry). For an axiomatically built-up geometry, six groups of axioms needed: Math 3181 Dr. Franz Rothe September 29, 2016 All3181\3181_fall16h3.tex Names: Homework has to be turned in this handout. For extra space, use the back pages, or put blank pages between. The homework can

More information

Chapter 9: Surface Area and Volume CHAPTER 9: ANGLES AND PYTHAGOREAN THEOREM. Date: Lesson: Learning Log Title:

Chapter 9: Surface Area and Volume CHAPTER 9: ANGLES AND PYTHAGOREAN THEOREM. Date: Lesson: Learning Log Title: CHAPTER 9: ANGLES AND PYTHAGOREAN THEOREM Date: Lesson: Learning Log Title: Date: Lesson: Learning Log Title: Date: Lesson: Learning Log Title: Date: Lesson: Learning Log Title: MATH NOTES ANGLE VOCABULARY

More information

Complementary, Supplementary, & Vertical Angles

Complementary, Supplementary, & Vertical Angles Unit 4: Lesson 1: Complementary and Supplementary Angles Date: Complementary, Supplementary, & Vertical Angles Type of Angles Definition/Description Complementary Angles Diagram Supplementary Angles Vertical

More information

Chapter 2 Similarity and Congruence

Chapter 2 Similarity and Congruence Chapter 2 Similarity and Congruence Definitions Definition AB = CD if and only if AB = CD Remember, mab = AB. Definitions Definition AB = CD if and only if AB = CD Remember, mab = AB. Definition ABC =

More information

Unit No: F3HW 11. Unit Title: Maths Craft 2. 4 Trigonometry Sine and Cosine Rules. Engineering and Construction

Unit No: F3HW 11. Unit Title: Maths Craft 2. 4 Trigonometry Sine and Cosine Rules. Engineering and Construction Unit No: F3HW 11 Unit Title: Maths Craft 4 Trigonometry Sine and Cosine Rules SINE AND COSINE RULES TRIGONOMETRIC RATIOS Remember: The word SOH CAH TOA is a helpful reminder. In any right-angled triangle,

More information

Geometry Practice Questions Semester 1

Geometry Practice Questions Semester 1 Geometry Practice Questions Semester 1 MAFS.912.G-CO.1.1 - Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line,

More information

Instructional Focus: Use dilations to show figures similar.

Instructional Focus: Use dilations to show figures similar. Instructional Focus: Use dilations to show figures. Properties of Dilations (G.SRT.1) Suppose we apply a dilation by a factor of 2, centered at the point P to the figure. a. In the picture, locate the

More information

1 I am given. (Label the triangle with letters and mark congruent parts based on definitions.)

1 I am given. (Label the triangle with letters and mark congruent parts based on definitions.) Name (print first and last) Per Date: 12/13 due 12/15 5.2 Congruence Geometry Regents 2013-2014 Ms. Lomac SLO: I can use SAS to prove the isosceles triangle theorem. (1) Prove: If a triangle is isosceles

More information

How to Construct a Perpendicular to a Line (Cont.)

How to Construct a Perpendicular to a Line (Cont.) Geometric Constructions How to Construct a Perpendicular to a Line (Cont.) Construct a perpendicular line to each side of this triangle. Find the intersection of the three perpendicular lines. This point

More information

CHAPTER - 10 STRAIGHT LINES Slope or gradient of a line is defined as m = tan, ( 90 ), where is angle which the line makes with positive direction of x-axis measured in anticlockwise direction, 0 < 180

More information

Chapter 4 UNIT - 1 AXIOMS, POSTULATES AND THEOREMS I. Choose the correct answers: 1. In the figure a pair of alternate angles are

Chapter 4 UNIT - 1 AXIOMS, POSTULATES AND THEOREMS I. Choose the correct answers: 1. In the figure a pair of alternate angles are STD-VIII ST. CLARET SCHOOL Subject : MATHEMATICS Chapter 4 UNIT - 1 AXIOMS, POSTULATES AND THEOREMS I. Choose the correct answers: 1. In the figure a pair of alternate angles are a) and b) and c) and d)

More information

Sample Question Paper. Time : 3hrs. MM : 90. Time allowed: 3 hours Maximum Marks: 90

Sample Question Paper. Time : 3hrs. MM : 90. Time allowed: 3 hours Maximum Marks: 90 Sample Question Paper SOLVED SAMPLE Term QUESTION - II PAPER Time : 3hrs. MM : 90 General Instructions: (i) (ii) (iii) All questions are compulsory. The question paper consists of 34 questions divided

More information

Proving Theorems about Lines and Angles

Proving Theorems about Lines and Angles Proving Theorems about Lines and Angles Angle Vocabulary Complementary- two angles whose sum is 90 degrees. Supplementary- two angles whose sum is 180 degrees. Congruent angles- two or more angles with

More information

7 CONGRUENCE OF TRIANGLES

7 CONGRUENCE OF TRIANGLES 7 CONGRUENCE OF TRIANGLES Exercise 7.1 Q.1. Complete the following statements : (a) Two line segments are congruent if. (b) Among two congruent angles, one has a measure 70 ; the measure of the other angle

More information

Angles. An angle is: the union of two rays having a common vertex.

Angles. An angle is: the union of two rays having a common vertex. Angles An angle is: the union of two rays having a common vertex. Angles can be measured in both degrees and radians. A circle of 360 in radian measure is equal to 2π radians. If you draw a circle with

More information

Unit 2 Triangles Part 1

Unit 2 Triangles Part 1 Graded Learning Targets LT 2.1 I can Unit 2 Triangles Part 1 Supporting Learning Targets I can justify, using a formal proof, that the three angles in a triangle add up to 180. I can justify whether or

More information

CBSE SAMPLE PAPERS SUMMATIVE ASSESSMENT-II (MATHS) CLASS 10

CBSE SAMPLE PAPERS SUMMATIVE ASSESSMENT-II (MATHS) CLASS 10 CBSE SAMPLE PAPERS SUMMATIVE ASSESSMENT-II (MATHS) CLASS 10 Time: 3 Hrs Max Marks: 90 General Instructions: A) All questions are compulsory. B) The question paper consists of 34 questions divided into

More information

no triangle can have more than one right angle or obtuse angle.

no triangle can have more than one right angle or obtuse angle. Congruence Theorems in Action Isosceles Triangle Theorems.3 Learning Goals In this lesson, you will: Prove the Isosceles Triangle Base Theorem. Prove the Isosceles Triangle Vertex Angle Theorem. Prove

More information

Type of Triangle Definition Drawing. Name the triangles below, and list the # of congruent sides and angles:

Type of Triangle Definition Drawing. Name the triangles below, and list the # of congruent sides and angles: Name: Triangles Test Type of Triangle Definition Drawing Right Obtuse Acute Scalene Isosceles Equilateral Number of congruent angles = Congruent sides are of the congruent angles Name the triangles below,

More information

Mth 97 Fall 2013 Chapter 4

Mth 97 Fall 2013 Chapter 4 4.1 Reasoning and Proof in Geometry Direct reasoning or reasoning is used to draw a conclusion from a series of statements. Conditional statements, if p, then q, play a central role in deductive reasoning.

More information

Proving Triangles and Quadrilaterals Satisfy Transformational Definitions

Proving Triangles and Quadrilaterals Satisfy Transformational Definitions Proving Triangles and Quadrilaterals Satisfy Transformational Definitions 1. Definition of Isosceles Triangle: A triangle with one line of symmetry. a. If a triangle has two equal sides, it is isosceles.

More information

Lesson 12: Angles Associated with Parallel Lines

Lesson 12: Angles Associated with Parallel Lines Lesson 12 Lesson 12: Angles Associated with Parallel Lines Classwork Exploratory Challenge 1 In the figure below, LL 1 is not parallel to LL 2, and mm is a transversal. Use a protractor to measure angles

More information

Lesson 3: Rectangles Inscribed in Circles

Lesson 3: Rectangles Inscribed in Circles Classwork Opening Exercise Using only a compass and straightedge, find the location of the center of the circle below. Follow the steps provided. Draw chord. AAAA Construct a chord perpendicular to AAAA

More information

MST Topics in the History of Mathematics

MST Topics in the History of Mathematics MST Topics in the History of Mathematics Paul Yiu Department of Mathematics Florida Atlantic University Summer 2011 3B: Euclid s Book VI (Similar triangles) Euclide VI.1,2 VI.1. Triangles and parallelograms

More information

Geometry Vocabulary Word Wall Cards

Geometry Vocabulary Word Wall Cards Geometry Vocabulary Word Wall Cards Mathematics vocabulary word wall cards provide a display of mathematics content words and associated visual cues to assist in vocabulary development. The cards should

More information

Indicate whether the statement is true or false.

Indicate whether the statement is true or false. Math 121 Fall 2017 - Practice Exam - Chapters 5 & 6 Indicate whether the statement is true or false. 1. The simplified form of the ratio 6 inches to 1 foot is 6:1. 2. The triple (20,21,29) is a Pythagorean

More information

Geometry: Traditional Pathway

Geometry: Traditional Pathway GEOMETRY: CONGRUENCE G.CO Prove geometric theorems. Focus on validity of underlying reasoning while using variety of ways of writing proofs. G.CO.11 Prove theorems about parallelograms. Theorems include:

More information

Geometry (H) Worksheet: 1st Semester Review:True/False, Always/Sometimes/Never

Geometry (H) Worksheet: 1st Semester Review:True/False, Always/Sometimes/Never 1stSemesterReviewTrueFalse.nb 1 Geometry (H) Worksheet: 1st Semester Review:True/False, Always/Sometimes/Never Classify each statement as TRUE or FALSE. 1. Three given points are always coplanar. 2. A

More information

*Chapter 1.1 Points Lines Planes. Use the figure to name each of the following:

*Chapter 1.1 Points Lines Planes. Use the figure to name each of the following: Name: Period Date Pre- AP Geometry Fall 2015 Semester Exam REVIEW *Chapter 1.1 Points Lines Planes Use the figure to name each of the following: 1) three non-collinear points 2) one line in three different

More information

Visualizing Triangle Centers Using Geogebra

Visualizing Triangle Centers Using Geogebra Visualizing Triangle Centers Using Geogebra Sanjay Gulati Shri Shankaracharya Vidyalaya, Hudco, Bhilai (Chhattisgarh) India http://mathematicsbhilai.blogspot.com/ sanjaybhil@gmail.com ABSTRACT. In this

More information

m 6 + m 3 = 180⁰ m 1 m 4 m 2 m 5 = 180⁰ m 6 m 2 1. In the figure below, p q. Which of the statements is NOT true?

m 6 + m 3 = 180⁰ m 1 m 4 m 2 m 5 = 180⁰ m 6 m 2 1. In the figure below, p q. Which of the statements is NOT true? 1. In the figure below, p q. Which of the statements is NOT true? m 1 m 4 m 6 m 2 m 6 + m 3 = 180⁰ m 2 m 5 = 180⁰ 2. Look at parallelogram ABCD below. How could you prove that ABCD is a rhombus? Show that

More information

Shortcuts, Formulas & Tips

Shortcuts, Formulas & Tips & present Shortcuts, Formulas & Tips For MBA, Banking, Civil Services & Other Entrance Examinations Vol. 3: Geometry Lines and Angles Sum of the angles in a straight line is 180 Vertically opposite angles

More information

An Approach to Geometry (stolen in part from Moise and Downs: Geometry)

An Approach to Geometry (stolen in part from Moise and Downs: Geometry) An Approach to Geometry (stolen in part from Moise and Downs: Geometry) Undefined terms: point, line, plane The rules, axioms, theorems, etc. of elementary algebra are assumed as prior knowledge, and apply

More information

Postulates, Theorems, and Corollaries. Chapter 1

Postulates, Theorems, and Corollaries. Chapter 1 Chapter 1 Post. 1-1-1 Through any two points there is exactly one line. Post. 1-1-2 Through any three noncollinear points there is exactly one plane containing them. Post. 1-1-3 If two points lie in a

More information

Student Instruction Sheet: Unit 4, Lesson 2. Ratios of Sides of Right-Angle Triangles

Student Instruction Sheet: Unit 4, Lesson 2. Ratios of Sides of Right-Angle Triangles Student Instruction Sheet: Unit 4, Lesson 2 Ratios of Sides of Right-Angle s Suggested Time: 75 minutes What s important in this lesson: In this lesson, you will learn through investigation, the relationship

More information

Teacher: Mr. Samuels. Name: 1. 2

Teacher: Mr. Samuels. Name: 1. 2 Teacher: Mr. Samuels Name: 1. 2 As shown in the diagram below of ΔABC, a compass is used to find points D and E, equidistant from point A. Next, the compass is used to find point F, equidistant from points

More information

Review Test 1 Chapters 1 & 2 and Appendix L

Review Test 1 Chapters 1 & 2 and Appendix L ath 61 pring 2007 Review Test 1 hapters 1 & 2 and Appendix L 1 www.timetodare.com To prepare for the test, learn all definitions, be familiar with all theorems and postulates and study the following problems.

More information

3. SOLUTION: Since point T is on the x-axis, the y-coordinate of the point will be 0. On the triangle it is indicated that.

3. SOLUTION: Since point T is on the x-axis, the y-coordinate of the point will be 0. On the triangle it is indicated that. Position and label each triangle on the coordinate plane. 1. right with legs and so that is 2a units long and leg is 2b units long Since this is a right triangle, two sides can be located on axis. Place

More information

Unit 1: Fundamentals of Geometry

Unit 1: Fundamentals of Geometry Name: 1 2 Unit 1: Fundamentals of Geometry Vocabulary Slope: m y x 2 2 Formulas- MUST KNOW THESE! y x 1 1 *Used to determine if lines are PARALLEL, PERPENDICULAR, OR NEITHER! Parallel Lines: SAME slopes

More information

PROPORTION PLANAR GEOMETRIC TRANSFORMATIONES AND GEOMETRIC RELATIONSHIP

PROPORTION PLANAR GEOMETRIC TRANSFORMATIONES AND GEOMETRIC RELATIONSHIP PROPORTION PLANAR GEOMETRIC TRANSFORMATIONES AND GEOMETRIC RELATIONSHIP The relationship regarding dimensions between two or more figures, or between two parts or a part and the whole is called PROPORTION.

More information

Geometry - Concepts 9-12 Congruent Triangles and Special Segments

Geometry - Concepts 9-12 Congruent Triangles and Special Segments Geometry - Concepts 9-12 Congruent Triangles and Special Segments Concept 9 Parallel Lines and Triangles (Section 3.5) ANGLE Classifications Acute: Obtuse: Right: SIDE Classifications Scalene: Isosceles:

More information

Chapter 4 Triangles: Congruency & Similarity

Chapter 4 Triangles: Congruency & Similarity 1 Chapter 4 Triangles: Congruency & Similarity Concepts & Skills Quilting is a great American pastime especially in the heartland of the United States. Quilts can be simple in nature or as in the photo

More information

Triangle Congruence Packet #3

Triangle Congruence Packet #3 Triangle Congruence Packet #3 Name Teacher 1 Warm-Up Day 1: Identifying Congruent Triangles Five Ways to Prove Triangles Congruent In previous lessons, you learned that congruent triangles have all corresponding

More information

Ch. 2 Trigonometry Notes

Ch. 2 Trigonometry Notes First Name: Last Name: Block: Ch. Trigonometry Notes.0 PRE-REQUISITES: SOLVING RIGHT TRIANGLES.1 ANGLES IN STANDARD POSITION 6 Ch..1 HW: p. 83 #1,, 4, 5, 7, 9, 10, 8. - TRIGONOMETRIC FUNCTIONS OF AN ANGLE

More information

You MUST know the big 3 formulas!

You MUST know the big 3 formulas! Name: Geometry Pd. Unit 3 Lines & Angles Review Midterm Review 3-1 Writing equations of lines. Determining slope and y intercept given an equation Writing the equation of a line given a graph. Graphing

More information

Geometry. Geometry is one of the most important topics of Quantitative Aptitude section.

Geometry. Geometry is one of the most important topics of Quantitative Aptitude section. Geometry Geometry is one of the most important topics of Quantitative Aptitude section. Lines and Angles Sum of the angles in a straight line is 180 Vertically opposite angles are always equal. If any

More information

CHAPTER 2 REVIEW COORDINATE GEOMETRY MATH Warm-Up: See Solved Homework questions. 2.2 Cartesian coordinate system

CHAPTER 2 REVIEW COORDINATE GEOMETRY MATH Warm-Up: See Solved Homework questions. 2.2 Cartesian coordinate system CHAPTER 2 REVIEW COORDINATE GEOMETRY MATH6 2.1 Warm-Up: See Solved Homework questions 2.2 Cartesian coordinate system Coordinate axes: Two perpendicular lines that intersect at the origin O on each line.

More information

INSIDE the circle. The angle is MADE BY. The angle EQUALS

INSIDE the circle. The angle is MADE BY. The angle EQUALS ANGLES IN A CIRCLE The VERTEX is located At the CENTER of the circle. ON the circle. INSIDE the circle. OUTSIDE the circle. The angle is MADE BY Two Radii Two Chords, or A Chord and a Tangent, or A Chord

More information

describes a ray whose endpoint is point A. g. A plane has no thickness. h. Symbols XY and YX describe the same line. i. Symbols AB

describes a ray whose endpoint is point A. g. A plane has no thickness. h. Symbols XY and YX describe the same line. i. Symbols AB RVIW FOR TH GOMTRY MITRM XM. 1. True or False? e prepared to explain your answer. a. efinitions and theorems are very important in mathematics but every mathematical system must contain some undefined

More information

Geometry- Unit 6 Notes. Simplifying Radicals

Geometry- Unit 6 Notes. Simplifying Radicals Geometry- Unit 6 Notes Name: Review: Evaluate the following WITHOUT a calculator. a) 2 2 b) 3 2 c) 4 2 d) 5 2 e) 6 2 f) 7 2 g) 8 2 h) 9 2 i) 10 2 j) 2 2 k) ( 2) 2 l) 2 0 Simplifying Radicals n r Example

More information

Analytic Spherical Geometry:

Analytic Spherical Geometry: Analytic Spherical Geometry: Begin with a sphere of radius R, with center at the origin O. Measuring the length of a segment (arc) on a sphere. Let A and B be any two points on the sphere. We know that

More information

Construction: Draw a ray with its endpoint on the left. Label this point B.

Construction: Draw a ray with its endpoint on the left. Label this point B. Name: Ms. Ayinde Date: Geometry CC 1.13: Constructing Angles Objective: To copy angles and construct angle bisectors using a compass and straightedge. To construct an equilateral triangle. Copy an Angle:

More information

theorems & postulates & stuff (mr. ko)

theorems & postulates & stuff (mr. ko) theorems & postulates & stuff (mr. ko) postulates 1 ruler postulate The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of

More information

Section Congruence Through Constructions

Section Congruence Through Constructions Section 10.1 - Congruence Through Constructions Definitions: Similar ( ) objects have the same shape but not necessarily the same size. Congruent ( =) objects have the same size as well as the same shape.

More information

Eureka Math. Grade 7, Module 6. Student File_A. Contains copy-ready classwork and homework

Eureka Math. Grade 7, Module 6. Student File_A. Contains copy-ready classwork and homework A Story of Ratios Eureka Math Grade 7, Module 6 Student File_A Contains copy-ready classwork and homework Published by the non-profit Great Minds. Copyright 2015 Great Minds. No part of this work may be

More information

Geometry Definitions, Postulates, and Theorems. Chapter 3: Parallel and Perpendicular Lines. Section 3.1: Identify Pairs of Lines and Angles.

Geometry Definitions, Postulates, and Theorems. Chapter 3: Parallel and Perpendicular Lines. Section 3.1: Identify Pairs of Lines and Angles. Geometry Definitions, Postulates, and Theorems Chapter : Parallel and Perpendicular Lines Section.1: Identify Pairs of Lines and Angles Standards: Prepare for 7.0 Students prove and use theorems involving

More information

H Geo Final Review Packet Multiple Choice Identify the choice that best completes the statement or answers the question.

H Geo Final Review Packet Multiple Choice Identify the choice that best completes the statement or answers the question. H Geo Final Review Packet Multiple Choice Identif the choice that best completes the statement or answers the question. 1. Which angle measures approximatel 7?.. In the figure below, what is the name of

More information

Carnegie Learning High School Math Series: Geometry Indiana Standards Worktext Correlations

Carnegie Learning High School Math Series: Geometry Indiana Standards Worktext Correlations Carnegie Learning High School Math Series: Logic and Proofs G.LP.1 Understand and describe the structure of and relationships within an axiomatic system (undefined terms, definitions, axioms and postulates,

More information

Geometry Ch 7 Quadrilaterals January 06, 2016

Geometry Ch 7 Quadrilaterals January 06, 2016 Theorem 17: Equal corresponding angles mean that lines are parallel. Corollary 1: Equal alternate interior angles mean that lines are parallel. Corollary 2: Supplementary interior angles on the same side

More information

Geometry. Oklahoma Math Day INSTRUCTIONS:

Geometry. Oklahoma Math Day INSTRUCTIONS: Oklahoma Math Day November 16, 016 Geometry INSTRUCTIONS: 1. Do not begin the test until told to do so.. Calculators are not permitted. 3. Be sure to enter your name and high school code on the answer

More information

A Correlation of. To the. New York State Next Generation Mathematics Learning Standards Geometry

A Correlation of. To the. New York State Next Generation Mathematics Learning Standards Geometry A Correlation of 2018 To the New York State Next Generation Mathematics Learning Standards Table of Contents Standards for Mathematical Practice... 1... 2 Copyright 2018 Pearson Education, Inc. or its

More information

Videos, Constructions, Definitions, Postulates, Theorems, and Properties

Videos, Constructions, Definitions, Postulates, Theorems, and Properties Videos, Constructions, Definitions, Postulates, Theorems, and Properties Videos Proof Overview: http://tinyurl.com/riehlproof Modules 9 and 10: http://tinyurl.com/riehlproof2 Module 9 Review: http://tinyurl.com/module9livelesson-recording

More information

Geometry Review for Semester 1 Final Exam

Geometry Review for Semester 1 Final Exam Name Class Test Date POINTS, LINES & PLANES: Geometry Review for Semester 1 Final Exam Use the diagram at the right for Exercises 1 3. Note that in this diagram ST plane at T. The point S is not contained

More information

H.Geometry Chapter 4 Definition Sheet

H.Geometry Chapter 4 Definition Sheet Section 4.1 Triangle Sum Theorem The sum of the measure of the angles in a triangle is Conclusions Justification Third Angle Theorem If two angles in one triangle are to two angles in another triangle,

More information

added to equal quantities, their sum is equal. Same holds for congruence.

added to equal quantities, their sum is equal. Same holds for congruence. Mr. Cheung s Geometry Cheat Sheet Theorem List Version 6.0 Updated 3/14/14 (The following is to be used as a guideline. The rest you need to look up on your own, but hopefully this will help. The original

More information

5 The Pythagorean theorem revisited

5 The Pythagorean theorem revisited 230 Chapter 5. AREAS 5 The Pythagorean theorem revisited 259. Theorem. The areas of squares constructed on the legs of a right triangle add up to the area of the square constructed on its hypotenuse. This

More information