# MATHS SYMBOLS - TRIANGLES - FIRST PROPERTIES

Ist. Superiore Marini-Gioia - Enzo ExposytoINGEGNERIA at Università degli Studi di Napoli 'Federico II'
1 of 58

## Recommended

t2 sine and cosine law inverse trig-functions by
t2 sine and cosine law inverse trig-functionsmath260
3.4K views77 slides
t4 sum and double half-angle formulas by
t4 sum and double half-angle formulasmath260
2.1K views65 slides
51 basic shapes and formulas by
51 basic shapes and formulasalg1testreview
621 views120 slides
6 trigonometric functions sohcahtoa-nat by
6 trigonometric functions sohcahtoa-natmath260
843 views81 slides
Problems in-plane-geometry-Sharygin by
Problems in-plane-geometry-SharyginArmando Cavero
14K views412 slides
Vistas Learning-Class-Maths by
Vistas Learning-Class-MathsPavithraT30
93 views29 slides

## What's hot

Activity 10: My True World! by
Activity 10: My True World!Sophia Marie Verdeflor
26.4K views9 slides
Module 3 geometric relations by
Module 3 geometric relationsdionesioable
3.7K views22 slides
Trigonometry for class xi by
Trigonometry for class xiindu psthakur
39.5K views63 slides
Module 4 geometry of shape and size by
Module 4 geometry of shape and sizedionesioable
5K views19 slides
Chapter 1 lines and angles ii [compatibility mode] by
Chapter 1 lines and angles ii [compatibility mode]Khusaini Majid
15K views26 slides
Module 1 geometric relations by
Module 1 geometric relationsdionesioable
3.8K views24 slides

### What's hot(17)

Module 3 geometric relations by dionesioable
Module 3 geometric relations
dionesioable3.7K views
Trigonometry for class xi by indu psthakur
Trigonometry for class xi
indu psthakur39.5K views
Module 4 geometry of shape and size by dionesioable
Module 4 geometry of shape and size
dionesioable5K views
Chapter 1 lines and angles ii [compatibility mode] by Khusaini Majid
Chapter 1 lines and angles ii [compatibility mode]
Khusaini Majid15K views
Module 1 geometric relations by dionesioable
Module 1 geometric relations
dionesioable3.8K views
Module 6 geometry of shape and size by dionesioable
Module 6 geometry of shape and size
dionesioable3.5K views
Haley 23.1K views
Module 2 geometric relations by dionesioable
Module 2 geometric relations
dionesioable2.7K views
Presentación1 by koalabites
Presentación1
koalabites482 views
SHARIGUIN_problems_in_plane_geometry_ by Armando Cavero
SHARIGUIN_problems_in_plane_geometry_
Armando Cavero1.5K views
C20 20.1 by BGEsp1
C20 20.1
BGEsp117 views
Assignment # 4 by Aya Chavez
Assignment # 4
Aya Chavez7.9K views
Module 5 geometry of shape and size by dionesioable
Module 5 geometry of shape and size
dionesioable2.6K views
53 pythagorean theorem and square roots by alg1testreview
53 pythagorean theorem and square roots
alg1testreview3.5K views

## Similar to MATHS SYMBOLS - TRIANGLES - FIRST PROPERTIES

MATHS SYMBOLS - GEOMETRY - FIRST ELEMENTS by
MATHS SYMBOLS - GEOMETRY - FIRST ELEMENTSIst. Superiore Marini-Gioia - Enzo Exposyto
760 views84 slides
14 right angle trigonometry by
14 right angle trigonometryKamarat Kumanukit
772 views12 slides
327759387-Trigonometry-Tipqc.ppt by
327759387-Trigonometry-Tipqc.pptSnCarbonel1
73 views62 slides
Triangles by
TrianglesFidelfo Moral
448 views12 slides
Solution kepler chap 1 by
Solution kepler chap 1Kamran Khursheed
265 views6 slides
Obj. 43 Laws of Sines and Cosines by
Obj. 43 Laws of Sines and Cosinessmiller5
2.6K views8 slides

### Similar to MATHS SYMBOLS - TRIANGLES - FIRST PROPERTIES(20)

327759387-Trigonometry-Tipqc.ppt by SnCarbonel1
327759387-Trigonometry-Tipqc.ppt
SnCarbonel173 views
Obj. 43 Laws of Sines and Cosines by smiller5
Obj. 43 Laws of Sines and Cosines
smiller52.6K views
Invention of the plane geometrical formulae - Part II by IOSR Journals
Invention of the plane geometrical formulae - Part II
IOSR Journals209 views
GCSE-TrigonometryOfRightAngledTriangles.pptx by Hasifa5
GCSE-TrigonometryOfRightAngledTriangles.pptx
Hasifa531 views
International Journal of Engineering Research and Development (IJERD) by IJERD Editor
International Journal of Engineering Research and Development (IJERD)
IJERD Editor709 views
International Journal of Engineering and Science Invention (IJESI) by inventionjournals
International Journal of Engineering and Science Invention (IJESI)
inventionjournals123 views
C2 st lecture 8 pythagoras and trigonometry handout by fatima d
C2 st lecture 8 pythagoras and trigonometry handout
fatima d1.8K views
Right triangle trigonometry by Ramesh Kumar
Right triangle trigonometry
Ramesh Kumar1.7K views

## More from Ist. Superiore Marini-Gioia - Enzo Exposyto

Gli Infiniti Valori Derivanti dalla Frazione 1 su 6 - Cinque Formule - Molte ... by
Gli Infiniti Valori Derivanti dalla Frazione 1 su 6 - Cinque Formule - Molte ...Ist. Superiore Marini-Gioia - Enzo Exposyto
380 views40 slides
Gli Infiniti Valori Derivanti dalla Frazione 1 su 9 - Quattro Formule - Numer... by
Gli Infiniti Valori Derivanti dalla Frazione 1 su 9 - Quattro Formule - Numer...Ist. Superiore Marini-Gioia - Enzo Exposyto
756 views33 slides
Gli Infiniti Valori Derivanti dalla Frazione 5 su 3 - Due Formule con Sette D... by
Gli Infiniti Valori Derivanti dalla Frazione 5 su 3 - Due Formule con Sette D...Ist. Superiore Marini-Gioia - Enzo Exposyto
341 views27 slides
Valori della Frazione 4 su 3 - Due Formule con Tante Dimostrazioni e Molti E... by
Valori della Frazione 4 su 3 - Due Formule con Tante Dimostrazioni e Molti E...Ist. Superiore Marini-Gioia - Enzo Exposyto
241 views25 slides
ICDL/ECDL FULL STANDARD - IT SECURITY - CONCETTI di SICUREZZA - SICUREZZA dei... by
ICDL/ECDL FULL STANDARD - IT SECURITY - CONCETTI di SICUREZZA - SICUREZZA dei...Ist. Superiore Marini-Gioia - Enzo Exposyto
396 views58 slides
ICDL/ECDL FULL STANDARD - IT SECURITY - CONCETTI di SICUREZZA - SICUREZZA dei... by
ICDL/ECDL FULL STANDARD - IT SECURITY - CONCETTI di SICUREZZA - SICUREZZA dei...Ist. Superiore Marini-Gioia - Enzo Exposyto
208 views67 slides

### More from Ist. Superiore Marini-Gioia - Enzo Exposyto(20)

ICS3211_lecture 08_2023.pdf by
ICS3211_lecture 08_2023.pdfVanessa Camilleri
149 views30 slides
Pharmaceutical Inorganic Chemistry Unit IVMiscellaneous compounds Expectorant... by
Pharmaceutical Inorganic Chemistry Unit IVMiscellaneous compounds Expectorant...Ms. Pooja Bhandare
93 views45 slides
EIT-Digital_Spohrer_AI_Intro 20231128 v1.pptx by
EIT-Digital_Spohrer_AI_Intro 20231128 v1.pptxISSIP
369 views50 slides
AI Tools for Business and Startups by
AI Tools for Business and StartupsSvetlin Nakov
107 views39 slides
AUDIENCE - BANDURA.pptx by
AUDIENCE - BANDURA.pptxiammrhaywood
84 views44 slides
REPRESENTATION - GAUNTLET.pptx by
REPRESENTATION - GAUNTLET.pptxiammrhaywood
100 views26 slides

Pharmaceutical Inorganic Chemistry Unit IVMiscellaneous compounds Expectorant... by Ms. Pooja Bhandare
Pharmaceutical Inorganic Chemistry Unit IVMiscellaneous compounds Expectorant...
EIT-Digital_Spohrer_AI_Intro 20231128 v1.pptx by ISSIP
EIT-Digital_Spohrer_AI_Intro 20231128 v1.pptx
ISSIP369 views
AI Tools for Business and Startups by Svetlin Nakov
AI Tools for Business and Startups
Svetlin Nakov107 views
AUDIENCE - BANDURA.pptx by iammrhaywood
AUDIENCE - BANDURA.pptx
iammrhaywood84 views
REPRESENTATION - GAUNTLET.pptx by iammrhaywood
REPRESENTATION - GAUNTLET.pptx
iammrhaywood100 views
Classification of crude drugs.pptx by GayatriPatra14
Classification of crude drugs.pptx
GayatriPatra1486 views
Ch. 7 Political Participation and Elections.pptx by Rommel Regala
Ch. 7 Political Participation and Elections.pptx
Rommel Regala97 views
CUNY IT Picciano.pptx by apicciano
CUNY IT Picciano.pptx
apicciano54 views
Narration lesson plan by TARIQ KHAN
Narration lesson plan
TARIQ KHAN58 views
Create a Structure in VBNet.pptx by Breach_P
Create a Structure in VBNet.pptx
Breach_P75 views
Pharmaceutical Inorganic chemistry UNIT-V Radiopharmaceutical.pptx by Ms. Pooja Bhandare

### MATHS SYMBOLS - TRIANGLES - FIRST PROPERTIES

• 1. ENZO EXPOSYTO MATHS SYMBOLS TRIANGLES - FIRST PROPERTIES  Enzo Exposyto 1
• 4. 1 - Deﬁnition 5 2 - Measures of the Sides 13 3 - Types of Triangles 23 4 - Sum of the Interior Angles 33 5 - Exterior Angles Theorem 41 6 - Sum of the Exterior Angles 53 7 - SitoGraphy 57 Enzo Exposyto 4
• 6. Polygonal Chains - Examples a simple open polygonal chain a simple closed polygonal chain Enzo Exposyto 6
• 7. Polygonal Chains - 2 polygonal it is a ﬁnite sequence chain of connected line-segments, called sides. This sides are connected by consecutive points, called vertices. For example: an angle has a simple open polygonal chain; a triangle, a square, … have a simple closed polygonal chain   Enzo Exposyto 7
• 8. Polygonal Chains - 3 polygonal More precisely, chain a closed polygonal chain is one in which the ﬁrst vertex coincides with the last one, or, alternatively, the ﬁrst and the last vertices are connected by a line segment. A simple closed polygonal chain in the plane is the boundary of a simple polygon. Enzo Exposyto 8
• 9. Polygons - 1 polygon is a planar ﬁgure that is bounded by a ﬁnite sequence of straight line-segments which form a closed polygonal chain. Often the term "polygon" is used in the meaning of "closed polygonal chain", but, in some cases, it's important to do a clear distinction between a polygonal area and a polygonal chain. Enzo Exposyto 9
• 10. Polygons - 2 polygon Since two line-segments (triangle) of an angle always form a simple open polygonal chain, are needed, at least, three line-segments to have a simple closed polygonal chain and, then, a polygon. This type of polygon, with 3 line-segments, is called triangle.   Enzo Exposyto 10
• 11. Deﬁnition - 1 A triangle is a polygon with THREE SIDES (a, b, c) and THREE VERTICES (A, B, C). It is one of the basic shapes in Geometry. The symbol ΔABC represents a triangle with vertices A, B, C.   Enzo Exposyto 11
• 12. Deﬁnition - 2 Euclid was a great greek scholar, author of “The elements”, written 3 centuries b. C.. In Euclidean Geometry, any three points, when non-collinear, determine an unique triangle and an unique plane, i. e. an Euclidean Plane.  Enzo Exposyto 12
• 14. Measures of the Sides - 1 a, b, c We can form a triangle if, and only if, every sum of the measures of two sides is greater than the measure of the third side. In other words, for every three line-segments which form a triangle and whose measures are a, b, c, it's a + b > c a + c > b b + c > a  Enzo Exposyto 14
• 15. Measures of the Sides - Example 1 a, b, c Since a = 4.31 b = 4.55 c = 10.15 and a + b = 4.31 + 4.55 = 8.86 then a + b < c because 8.86 < 10.15 It's impossible to form a triangle by line-segments with a, b, c lengths  Enzo Exposyto 15 b a c
• 16. Measures of the Sides - Example 2 a, b, c Since a = 5.17 b = 4.98 c = 10.15 and a + b = 5.17 + 4.98 = 10.15 then a + b = c because 10.15 = 10.15 It's impossible to form a triangle by line-segments with a, b, c lengths  Enzo Exposyto 16 b a c
• 17. Measures of the Sides - 3 a, b, c Besides, we can form a triangle if, and only if, every side has a measure greater than the absolute value of the diﬀerence of the measures of the other two sides. In other words, for every three line-segments which form a triangle and whose measures are a, b, c, it's a > |b - c| or a > |c - b| b > |a - c| or b > |c - a| c > |a - b| or c > |b - a|   Enzo Exposyto 17
• 18. Measures of the Sides - 4 a > |b - c| Let's start from (page 14) (proof) a + c > b We can subtract c from both sides: a + c - c > b - c and we get a > b - c If b < c then b - c will be a negative number, which hasn't geometric meaning. So, we must write a > |b - c| Enzo Exposyto 18
• 19. Measures of the Sides - 5 a > |c - b| If we consider (page 14) (proof) a + b > c we can subtract b from both sides: a + b - b > c - b and we have a > c - b If c < b then c - b will be a negative number, which hasn't geometric meaning. Thus, we must write a > |c - b| Enzo Exposyto 19
• 20. Measures of the Sides - 6 a > |b - c| Therefore, a > |c - b| a > |b - c| and a > |c - b| are both true. In a similar way, it's possible to proof that b > |a - c| or b > |c - a| c > |a - b| or c > |b - a| Enzo Exposyto 20
• 21. Measures of the Sides - Example 3 a, b, c Since a = 4.31 b = 4.55 c = 10.15 and b - c = 4.55 - 10.15 = - 5.60 then a < |b - c| because 4.31 < |-5.60| It's impossible to form a triangle by line-segments with a, b, c lengths  Enzo Exposyto 21 b a c
• 22. Measures of the Sides - Example 4 a, b, c Since a = 5.17 b = 4.98 c = 10.15 and b - c = 4.98 - 10.15 = -5.17 then a = |b - c| because 5.17 = |-5.17| It's impossible to form a triangle by line-segments with a, b, c lengths  Enzo Exposyto 22 c ab
• 24. Types of Angles and Degrees Right Angle (90°) Enzo Exposyto 24
• 25. Types of Angles and Degrees Right Angle (90°) and Perpendicularity perpendicularity sign ... is at 90° (90 degrees) to ... ... forms a right angle with ... AB CD a line segment (AB) drawn so that it forms 2 right angles (90°) with a line (CD) Enzo Exposyto 25
• 26. Types of Angles and Degrees α - Acute Angle (less than 90°) β - Obtuse Angle (greater than 90° and less than 180°) Enzo Exposyto 26
• 27. Identifying Triangles NAME PROPERTY EQUILATERAL 3 sides have equal length ISOSCELES 2 sides have equal length SCALENE 3 sides have different lengths ACUTE 3 angles are acute RIGHT 1 angle is right OBTUSE 1 angle is obtuse Enzo Exposyto 27
• 28. Grouping Triangles by … (SIDES) (ANGLES) Equilateral Acute Isosceles Right Scalene Obtuse Enzo Exposyto 28
• 31. Grouping Triangles by … (ACUTE) (RIGHT) (OBTUSE) Equilateral Isosceles Isosceles Isosceles Scalene Scalene Scalene  Enzo Exposyto 31
• 32. Grouping Triangles by …   Enzo Exposyto 32 equilateral acute isosceles right
• 34. Sum of the Interior Angles In an Euclidean Plane, the sum of the measures of the interior angles of a triangle is ALWAYS 180° (180 DEGREES). Enzo Exposyto 34
• 35. Sum of the Interior Angles Proof 1 - Step 1 Let’s see the image. Let's draw the red line on C which is parallel to side AB. Now, we can see that the angles with the same colours are congruent: this means that they have the same measure Enzo Exposyto 35
• 36. Sum of the Interior Angles Proof 1 - Step 2 Let’s see the image: - the green colour represents the measure of the angle C, - the red colour represents the measure of the angle A, - the black colour represents themeasureoftheangleB. The three angles together form, clearly, a straight angle and, then, the sum of their measures is 180° Enzo Exposyto 36
• 37. Sum of the Interior Angles Proof 2 Let’s see the image. Let's draw the green line segment on B which is parallel to side AC. Now, we can see that the angle α ‘in’ B and the angle α which's in A are congruent: they have the same measure; besides, the angle γ ‘in’ B and the angle γ that's in C are congruent: they have the same measure. The 3 angles together in B form a straight angle and the sum of their measures is α + β + γ = 180° Enzo Exposyto 37
• 38. Sum of the Interior Angles If we know the measures of two angles of a triangle, we can determine the measure of the third angle, subtracting the known measures from 180°. For example: a triangle ha 2 angles with 2 known measures: 70° and 80°. The measure of the third angle is: 180° - (70° + 80°) = 180° - 150° = 30° Enzo Exposyto 38
• 39. Sum of the Interior Angles Examples (image from https://www.ck12.org/geometry/triangle-angle-sum-theorem/) a. EQUILATERAL ACUTE: 60° + 60° + 60° = 180° b. ISOSCELES RIGHT : 90° + 45° + 45° = 180° c. SCALENE ACUTE : 70° + 30° + 80° = 180° d. SCALENE OBTUSE : 25° + 120° + 35° = 180° Enzo Exposyto 39
• 40. Sum of the Interior Angles In Euclidean Geometry, the sum of the interior angles of a triangle is ALWAYS 180°. This is equivalent to the Euclid's Parallel Postulate. In Hyperbolic Geometry, the sum of the interior angles of a hyperbolic triangle is less than 180°. In Elliptic Geometry, the sum of the interior angles of an elliptic triangle is greater than 180°. Enzo Exposyto 40
• 42. INTERIOR and EXTERIOR ANGLES with TWO SETS of EXTERIOR ANGLES  Enzo Exposyto 42
• 43. 1st SET of EXTERIOR ANGLES 1st set Enzo Exposyto 43
• 44. 2nd SET of EXTERIOR ANGLES exterior angles - 2nd set Enzo Exposyto 44
• 45. SUM of an INTERIOR ANGLE and ITS EXTERIOR From the last ﬁgure, we can see that α + α exterior = 180° β + β exterior = 180° γ + γ exterior = 180° We get the same result from the 2nd ﬁgure at page 43 The SUM of an INTERIOR ANGLE and ITS EXTERIOR ANGLE IS ALWAYS 180° Enzo Exposyto 45
• 46. EXTERIOR ANGLES THEOREM - 1 Besides, we can state the following theorem: EVERY EXTERIOR ANGLE of a TRIANGLE ALWAYS EQUALS the SUM of the OTHER TWO FAR AWAY INTERIOR ANGLES α exterior = β + γ β exterior = α + γ γ exterior = α + β Enzo Exposyto 46
• 47. EXTERIOR ANGLES THEOREM - 2 EXAMPLE γ exterior = α + β   Enzo Exposyto 47 γ = 180° - (α + β) = 180° - (70° + 50°) = 180° - 120° = 60° γ exterior = α + β = 70° + 50° = 120°
• 48. EXTERIOR ANGLES THEOREM - 3 PROOF - Premise 1 Let's see the images: if we call a, b, c respectively the interior angles α, β, γ and A, B, C the respective exterior angles, we can write: γ exterior = C Therefore, the thesis which we must prove γ exterior = α + β becomes C = a + b Enzo Exposyto 48
• 49. EXTERIOR ANGLES THEOREM - 4 PROOF - Premise 2 Besides, how we can see, the sum of the interior angles α + β + γ = 180° becomes a + b + c = 180° and, now, the sum γ exterior + γ = 180° becomes C + c = 180° Enzo Exposyto 49
• 50. EXTERIOR ANGLES THEOREM - 5 PROOF - 1 We must prove that C = a + b Since (let's see the image) C + c = 180° and a + b + c = 180° we get C + c = a + b + c If we simplify, it gives us C = a + b Q.E.D. Enzo Exposyto 50
• 51. EXTERIOR ANGLES THEOREM - 6 PROOF - 2 We must prove that C = a + b Since this equality is true (let's see the image): C + c = 180° we obtain C = 180° - c From the sum of the interior angles, a + b + c = 180° we get the c value: c = 180° - (a + b) If we substitute the c value in the equality with C, we get C = 180° - [180° - (a + b)] Simpliﬁed, it becomes C = a + b Q.E.D. Enzo Exposyto 51
• 52. EXTERIOR ANGLES THEOREM - 7 PROOF - 3 Since C = a + b is equivalent to γ exterior = α + β we also proved this last thesis. In similar ways, we can prove that A = b +c namely α exterior = β + γ and B = a + c namely β exterior = α + γ Enzo Exposyto 52
• 54. the SUM of the EXTERIOR ANGLES of a TRIANGLE IS EQUAL to 360° Enzo Exposyto 54
• 55. the SUM of the EXTERIOR ANGLES of a TRIANGLE IS EQUAL to 360° PROOF - STEP 1 If the EXTERIOR ANGLES of a TRIANGLE ARE CALLED A, B, C and the SUM of these EXTERIOR ANGLES is CALLED S we get S = A + B + C Now (page 52), A = b + c B = a + c C = a + b where a, b and c are the corresponding interior angles of the exterior angles A, B, C.  Enzo Exposyto 55
• 56. the SUM of the EXTERIOR ANGLES of a TRIANGLE IS EQUAL to 360° PROOF - STEP 2 If we substitute the values of A, B and C in the ﬁrst equality, we get: S = (b + c) + (a + c) + (a + b) = (a + a) + (b + b) + (c + c) = 2 * a + 2 * b + 2 * c = 2 * (a + b + c) = 2 * 180° = 360° Q.E.D. Enzo Exposyto 56
Current LanguageEnglish
Español
Portugues
Français
Deutsche