Triangles

Triangles :

Triangles 1 copyright reserved by Vatsala Singh

Slide 2:

Presentation by Vatsala Singh B.Sc. , B.Ed. 2 copyright reserved by Vatsala Singh

SELF INSTRUCTION PROGRAMME:

SELF INSTRUCTION PROGRAMME UNIT 2: TRIANGLES BRANCH: GEOMETRY STANDARD: 9 TH 3 copyright reserved by Vatsala Singh

PLANNING AND ORGANISING OF INSTRUCTION MATERIAL BASED ON ICT AS PER THE UNIT AND OBJECTIVES:

PLANNING AND ORGANISING OF INSTRUCTION MATERIAL BASED ON ICT AS PER THE UNIT AND OBJECTIVES 4 copyright reserved by Vatsala Singh

OBJECTIVES.:

OBJECTIVES . To Help Student Understand The Concept Of Triangles. To Help Student Understand Types Of Triangles. To Help Student Understand The Properties Of Triangles. To Help Student To Solve The Problems Related To Properties Of Triangles. To Help Students To Understand The Similarity Of Triangles. To Help Student To Apply The Knowledge Of Triangles In Day To Day Life. 5 copyright reserved by Vatsala Singh

DEVELOPMENT OF INSTRUCTIONS.:

DEVELOPMENT OF INSTRUCTIONS . READ EACH SLIDE PROPERLY. GO STEP BY STEP. DO NOT EDIT ANY THING. TO GO TO NEXT SLIDE PRESS ENTER. ANSWER THE QUESTIONS AND THEN PROCEED TO NEXT SLIDE. FOR ANSWER PRESS TABS LOCK. ENJOY LEARNING. 6 copyright reserved by Vatsala Singh

CAN YOU IDENTIFY THESE SHAPES?:

CAN YOU IDENTIFY THESE SHAPES? 7 copyright reserved by Vatsala Singh

AND THESE SHAPES?:

AND THESE SHAPES? THESE ARE ALL TRIANGLES …. copyright reserved by Vatsala Singh 8

If A,B,C are three non Collinear points ,the figure made up by three line segments AB,BC and CA is called a triangle:

If A,B,C are three non Collinear points ,the figure made up by three line segments AB,BC and CA is called a triangle A B C 9 copyright reserved by Vatsala Singh

The Six elements of a triangle are-:

The Six elements of a triangle are- 1.Side AB 2. Side BC 3. Side AC 4.Angle ABC 5.Angle ACB 6. Angle BAC A B C 10 copyright reserved by Vatsala Singh

Interior & Exterior of a triangle:

Interior & Exterior of a triangle A B C P Q m 11 copyright reserved by Vatsala Singh

Types of triangles:

Types of triangles 12 copyright reserved by Vatsala Singh

Various types of triangles:

Various types of triangles Equilateral Triangle Has 3 EQUAL sides or 3 EQUAL angles (which will always be 60 degrees) copyright reserved by Vatsala Singh 13 6 cm 6 cm 6 cm

Isosceles triangle:

Isosceles triangle Has 2 EQUAL sides and 2 EQUAL angles copyright reserved by Vatsala Singh 14 6 cm 6 cm 4 cm

Scalene triangle:

Scalene triangle Has NO EQUAL sides and NO EQUAL angles copyright reserved by Vatsala Singh 15 5 cm 13 cm 6 cm

Acute triangle:

Acute triangle Each of the 3 angles are less than 90 degrees copyright reserved by Vatsala Singh 16 40 60 80

Right triangle:

Right triangle One angle is 90 degrees copyright reserved by Vatsala Singh 17 90

Obtuse Triangle:

Obtuse Triangle Has 1 angle greater than 90 degrees copyright reserved by Vatsala Singh 18 120

Perimeter of a Triangle:

Perimeter of a Triangle 5cm 5cm 4cm Perimeter: =5cm+5cm+4cm =14cm 19 copyright reserved by Vatsala Singh

More about triangles::

More about triangles: Giovanni Ceva known for Ceva’s theoram in elementary geometry: 20 copyright reserved by Vatsala Singh

CEVIAN:

CEVIAN A cevian is a line segment which joins a vertex of a triangle with a point on the opposite side . Seg CF,AD And BE are Cevians. 21 copyright reserved by Vatsala Singh

2.2 Properties of Triangles :

2.2 Properties of Triangles 22 copyright reserved by Vatsala Singh

Angle sum property of a Triangle:

Angle sum property of a Triangle The sum of the angles of a triangle is 180˚. A B C B C A 23 copyright reserved by Vatsala Singh

Angle sum property of a Triangle:

Angle sum property of a Triangle 1 2 3 4 5 Angle 4 + Angle 3 + Angle 5=180º Angle 1 + Angle 3 + Angle 2=180º 24 copyright reserved by Vatsala Singh

The sum of the measures of angles of a triangle is 180o :

The sum of the measures of angles of a triangle is 180 o Given: ABC is a triangle. To prove: ABC+CAB+BCA = 180 0 Construction: draw a line XY through the point b such that line XY ll side AC. 25 copyright reserved by Vatsala Singh

Proof::

Proof: Line XY ll side AC and BA is transversal since, XBA=CBA Line XY ll side OC and BC is transversal since, YBC= BCA (alternate angles) XBA+YBC= BAC+BCA+ABC Adding ABC to both sides: XBA+YBC+ABC=BAC+BCA+ABC 180 degrees=BAC+BCA+ABC (angles in linear pair) Hence, ABC+BAC+BCA=180 degree. 26 copyright reserved by Vatsala Singh

Solve this….:

Solve this…. Find x in each case. ANSWER 27 copyright reserved by Vatsala Singh

Q.The degree measures of the angles of a triangle are in the ratio 4 : 5 : 9. What is the degree measure of the SMALLEST angle of the triangle? :

Q.The degree measures of the angles of a triangle are in the ratio 4 : 5 : 9. What is the degree measure of the SMALLEST angle of the triangle? ANSWER copyright reserved by Vatsala Singh 28

In triangle ABC below, angle A = 40 degrees and angle B = 60 degrees. What is the measure of angle C? :

In triangle ABC below, angle A = 40 degrees and angle B = 60 degrees. What is the measure of angle C? Answer copyright reserved by Vatsala Singh 29

Exterior & Interior Opposite Angles:

Exterior & Interior Opposite Angles A B C D 30 copyright reserved by Vatsala Singh

Slide 31:

The angle forming a linear pair with the interior angle of a triangle is called an exterior angle. 31 copyright reserved by Vatsala Singh

Exterior angle property of a Triangle:

Exterior angle property of a Triangle A B D C Angle A + Angle B = Angle BCD 32 copyright reserved by Vatsala Singh

Exterior angle property of a Triangle:

Exterior angle property of a Triangle A B D C Angle A + Angle B + Angle BCA=180º Angle BCA + Angle BCD=180º Angle A + Angle B +Angle BCA = Angle BCA+ Angle BCD

Triangle Inequality Property:

Triangle Inequality Property The sum of any two sides of a triangle is greater than the third side. A B C AB+BC >AC AB+AC>BC AC+BC>AB 34 copyright reserved by Vatsala Singh

Triangle Inequality Property:

Triangle Inequality Property A B C AB+BC >AC 35 copyright reserved by Vatsala Singh

Triangle Inequality Property:

Triangle Inequality Property A B C AB+AC>BC 36 copyright reserved by Vatsala Singh

Triangle Inequality Property:

Triangle Inequality Property A B C AC+BC>AB 37 copyright reserved by Vatsala Singh

Lets solve for the value of x in this case: :

Lets solve for the value of x in this case: Answer copyright reserved by Vatsala Singh 38

Sub-Unit 2.3:

Sub-Unit 2.3 copyright reserved by Vatsala Singh 39 (AA, SSS, SAS) Proving Triangles Similar

AA Similarity (Angle-Angle):

AA Similarity (Angle-Angle) copyright reserved by Vatsala Singh 40 If 2 angles of one triangle are congruent to 2 angles of another triangle, then the triangles are similar. Conclusion: and Given:

SSS Similarity (Side-Side-Side):

SSS Similarity (Side-Side-Side) copyright reserved by Vatsala Singh 41 If the measures of the corresponding sides of 2 triangles are proportional, then the triangles are similar. Given: Conclusion: 5 11 22 8 16 10

SAS Similarity (Side-Angle-Side):

SAS Similarity (Side-Angle-Side) copyright reserved by Vatsala Singh 42 If the measures of 2 sides of a triangle are proportional to the measures of 2 corresponding sides of another triangle and the angles between them are congruent, then the triangles are similar. Given: Conclusion: 5 11 22 10

Similarity is reflexive, symmetric, and transitive.:

Similarity is reflexive, symmetric, and transitive. copyright reserved by Vatsala Singh 43 1. Mark the Given. 2. Mark … Shared Angles or Vertical Angles 3. Choose a Method. ( AA, SSS , SAS) Think about what you need for the chosen method and be sure to include those parts in the proof. Steps for proving triangles similar: Proving Triangles Similar

Slide 44:

copyright reserved by Vatsala Singh 44 Problem #1 C D E G F Step 1: Mark the given … and what it implies Step 2: Mark the vertical angles Step 3: Choose a method: (AA,SSS,SAS) Step 4: List the Parts in the order of the method with reasons Step 5: Is there more? Statements Reasons Given Alternate Interior <s AA Similarity Alternate Interior <s AA

Slide 45:

copyright reserved by Vatsala Singh 45 Problem #2 Step 1: Mark the given … and what it implies Step 2: Choose a method: (AA,SSS,SAS) Step 4: List the Parts in the order of the method with reasons Step 5: Is there more? Statements Reasons Given Division Property SSS Similarity Substitution SSS 1. IJ = 3LN ; JK = 3NP ; IK = 3LP

Slide 46:

copyright reserved by Vatsala Singh 46 Problem #3 Step 1: Mark the given … and what it implies Step 3: Choose a method: (AA,SSS,SAS) Step 4: List the Parts in the order of the method with reasons Next Slide…………. Step 5: Is there more? SAS Step 2: Mark the reflexive angles

Slide 47:

Statements Reasons G is the Midpoint of H is the Midpoint of Given 2. EG = DG and EH = HF Def. of Midpoint 3. ED = EG + GD and EF = EH + HF Segment Addition Post. 4. ED = 2 EG and EF = 2 EH Substitution Division Property Substitution Reflexive Property SAS Postulate copyright reserved by Vatsala Singh 47

Lets quickly summarize what we learnt today::

Lets quickly summarize what we learnt today: A triangle is a polygon with three sides. It has six elements viz 3 sides and 3 angles. Types of triangles: Equilateral, Isosceles and Scalene Trangles. Theoram: The sum of measures of the angles of triangles is 180 0. Remote interior angles: the measures of an exterior angle of a triangle is equal to the sum of measures of its remote interior angles. Exterior angles theoram: the measure of an exterior angle is always greater than the measure of each of its remote interior angles. Tests of Similarity:SSS tests, AAA tests, SAS tests. 48 copyright reserved by Vatsala Singh

Homework:

Homework 1.Find the value of x in the given figure: 49 copyright reserved by Vatsala Singh

Slide 50:

2. Find the value of x in the given figure: 50 copyright reserved by Vatsala Singh

Hope u enjoyed learning about Triangles……:

Hope u enjoyed learning about Triangles…… 51 copyright reserved by Vatsala Singh