Math 9 similar triangles intro

Gilbert Joseph Abueg

The document is a lecture on similar triangles. It defines similar triangles as having the same shape but different sizes, and discusses how similar triangles have corresponding angles that are congruent and corresponding sides that are proportional. It provides examples of similar triangles and statements showing their similarity. It also covers using proportions of corresponding sides to solve for missing sides in similar triangles and several proportionality principles related to similar triangles, including the basic proportionality theorem involving parallel lines cutting across a triangle. Read less

powerpoint presentation on similarity of triangles

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  • 1. The session shall begin shortly…
  • 3. Similar Triangles A Mathematics 9 Lecture 3
  • 4. 4 Similar Triangles What do these pairs of objects have in common? SAME SHAPES BUT DIFFERENT SIZES
  • 5. 5 Similar Triangles What do these pairs of objects have in common? They are also called SIMILAR objects
  • 6. The Concept of Similarity Similar Triangles Two objects are called similar if they have the same shape but possibly different sizes.
  • 7. The Concept of Similarity Similar Triangles You can think of similar objects as one one being a enlargement or reduction of the other.
  • 8. The Concept of Similarity Similar Triangles You can think of similar objects as one being an enlargement or reduction of the other (zoom in, zoom out). The degree of enlargement or reduction is called the SCALE FACTOR
  • 10. The Concept of Similarity Similar Triangles Enlargements and Projection
  • 11. 11 Similar Triangles QUESTION! If a polygon is enlarged or reduced, which part changes and which part remains the same?
  • 12. The Concept of Similarity Similar Triangles Two polygons are SIMILAR if they have the same shape but not necessarily of the same size. Symbol used: ~ (is SIMILAR to) A C B DE F In the figure, ABC is similar to DEF. Thus ,we write ABC ~ DEF
  • 13. The Concept of Similarity Similar Triangles Two polygons are SIMILAR if they have the same shape but not necessarily of the same size. If they are similar, then 1. The corresponding angles remain the same (or are CONGRUENT) 2. The corresponding sides are related by the same scale factor (or, are PROPORTIONAL)
  • 14. The Concept of Similarity Similar Triangles Q1 Q2 These two are similar. Corresponding angles are congruent A  E B  F C  G D  H Corresponding sides are proportional: 1 2 EH EF FG GH AD AD BC CD     Scale factor from Q1 to Q2 is ½
  • 15. The Concept of Similarity Similar Triangles T1 T2 These two are similar. Corresponding angles are congruent A  D B  E C  F Corresponding sides are proportional: 2 DE EF DF AB BC AC    Scale factor from T1 to T2 is 2
  • 16. Similar Triangles The Concept of Similarity Which pairs are similar? If they are similar, what is the scale factor?
  • 17. Similar Triangles Similar Triangles Two triangles are SIMILAR if they have the same shape but not necessarily of the same size. Symbol used: ~ (is SIMILAR to) A C B DE F In the figure, ABC is similar to DEF. Thus ,we write ABC ~ DEF
  • 18. Similar Triangles Similar Triangles http://wps.pearsoned.com.au/wps/media/objects/7029/7198491/opening/c10.gif
  • 19. Similar Triangles Two triangles are SIMILAR if all of the following are satisfied: 1. The corresponding angles are CONGRUENT. 2. The corresponding sides are PROPORTIONAL. Similar Triangles
  • 20. Similar Triangles  The two triangles shown are similar because they have the same three angle measures.  The order of the letters is important: corresponding letters should name congruent angles.  For the figure, we write 20 ABC DEF  Similar Triangles
  • 21. Similar Triangles 21 ABC DEF  Similar Triangles A B C D E F Congruent Angles A D   B E   C F  
  • 22.  Let’s stress the order of the letters again. When we write note that the first letters are A and D, and The second letters are B and E, and The third letters are C and F, and 22 ABC DEF  .A D   .B E   .C F   Similar Triangles Similar Triangles
  • 23.  We can also write the similarity statement as 23 ACB DFE   BAC EDF or CAB FDE  Similar Triangle Notation Similar Triangles Why?
  • 24.  BCA DFE Similar Triangle Notation Similar Triangles  We CANNOT write the similarity statement as  BAC EFD Why?
  • 25. Kaibigan, sa similar triangles, the correspondence of the vertices matters!!! Similar Triangles
  • 26. 26 ABC DEF  Similar Triangles A B C D E F Corresponding Sides AB DE BC EF AC DF Proportions from Similar Triangles
  • 27. 27 ABC DEF  Similar Triangles Corresponding Sides AB DE BC EF AC DF Proportions from Similar Triangles Ratios of Corresponding Sides AB DE BC EF AC DF
  • 28.  Suppose Then the sides of the triangles are proportional, which means: 28 .ABC DEF  AB AC BC DE DF EF   Notice that each ratio consists of corresponding segments. Similar Triangles Proportions from Similar Triangles
  • 29. The Similarity Statements Based on the definition of similar triangles, we now have the following SIMILARITY STATEMENTS: 29 Congruent Angles .A D   .B E   .C F   Proportional Sides Similar Triangles   AB BC AC DE EF DF
  • 30. 30O N E P K I 110 110 30 30 40 40 Similar Triangles Give the congruence and proportionality statements and the similarity statement for the two triangles shown. The Similarity Statements
  • 31. The Similarity Statements 31O N E P K I 110 110 30 30 40 40 Similar Triangles Give the congruence and proportionality statements and the similarity statement for the two triangles shown. Congruent Angles P O   I N   K E   Corresponding SidesPI ON IK NE PK OE
  • 32. 32O N E P K I 110 110 30 30 40 40 Similar Triangles Give the congruence and proportionality statements and the similarity statement for the two triangles shown. Congruent Angles   P I   I N   K E Proportional Sides   PI IK PK ON NE OE Similarity Statement  PIK ONE The Similarity Statements
  • 33. Similar Triangles Given the triangle similarity LMN ~ FGH determine if the given statement is TRUE or FALSE. M G   true FHG NLM   false N M   false LN MN FG GH  false MN LN GH FH  true GF HG ML NM  true The Similarity Statements
  • 34. In the figure, Enumerate all the statements that will show that 34 .SA ON S A L O N . SAL NOL Similar Triangles The Similarity Statements Note: there is a COMMON vertex L, so you CANNOT use single letters for angles!
  • 35. In the figure, Enumerate all the statements that will show that 35 .SA ON S A L O N . SAL NOL Similar Triangles The Similarity Statements Congruent Angles   SAL LON   ASL LNO   OLN SLA Proportional Sides   SA AL SL ON OL NL Note: there is a COMMON vertex L, so you CANNOT use single letters for angles!
  • 36. Similar Triangles In the figure, Enumerate all the statements that will show that .KO AB . KOL ABL O B L K A Hint: SEPARATE the two right triangles and determine the corresponding vertices. Similar Triangles The Similarity Statements
  • 37. O B L K A Similar TrianglesSimilar Triangles The Similarity Statements O L K Congruent Angles   KOL ABL   LKO LAB   KLO ALB Proportional Sides   KO KL OL AB AL BL
  • 38. Similar Triangles Solving for the Sides The proportionality of the sides of similar triangles can be used to solve for missing sides of either triangle. For the two triangles shown, the statement 38   AB BC AC DE EF DF can be separated into the THREE proportions  AB AC DE DF  BC AC EF DF  AB BC DE EF
  • 39. Similar Triangles Solving for the Sides Note The ratios can also be formed using any of the following: 39 a b c d e f   a b c d e f   d e f a b c    a d b e a d or or b e c f c f
  • 40. Given that If the sides of the triangles are as marked in the figure, find the missing sides. 40 A B C D E F ,ABC DEF  68 7 12 Similar Triangles Solving for the Sides
  • 41. 41 A B C D E F 68 7 12 9  DF FE AC CB Similar Triangles Solving for the Sides Set up the proportions of the corresponding sides using the given sides For CB: 8 6 12  CB 8 72CB 9CB
  • 42. 42 A B C D E F 68 7 12 9 10.5 Similar Triangles Solving for the Sides Set up the proportions of the corresponding sides using the given sides  DF DE AC AB For AB: 8 7 12  AB 8 84AB 21 10.5 2 AB or
  • 43. S A L O N 8 10 16 x y Similar Triangles Solving for the Sides In the figure shown, solve for x and y. Solution 15 16 8 10  x For x: 8 160x 20x 8 15 10  y For y: 10 120y 12y
  • 44. Check your understanding The triangles are similar. Solve for x and z. 3 4 12  x 9x 5 4 12  z 15z
  • 45. Similar Triangles The Proportionality Principles A line parallel to a side of a triangle cuts off a triangle similar to the given triangle. This is also called the BASIC PROPORTIONALITY THEOREM BC DE cuts ABC into two similar triangles: DE ~ ADE ABC A B C D E
  • 46. A B C D E Similar Triangles The Proportionality Principles The Basic Proportionality Theorem A D E B C A BC DE
  • 47. A B C D E Similar Triangles The Proportionality Principles The Basic Proportionality Theorem A D E B C A BC DE   AD AE DE AB AC BC Proportions:
  • 48. A B C D E Similar Triangles The Proportionality Principles The Basic Proportionality Theorem BC DE  AD AE DB EC Note The two sides cut by the line segment are also cut proportionally; thus we have
  • 49. Similar Triangles The Proportionality Principles The Basic Proportionality Theorem Find the value of x. Solution 28 12 14  x 2 12  x 24x
  • 50. Similar Triangles The Proportionality Principles The Basic Proportionality Theorem O B L K A 12 6 9 In the figure, Find OL and OB. .KO AB Solution 12 9 6  OL For OL: 9 72OL 8OL For OB:  OB OL BL 8 6  2OB
  • 51. Similar Triangles The Proportionality Principles The Basic Proportionality Theorem Find BU and SB if .BC ST
  • 52. Similar Triangles The Proportionality Principles The Basic Proportionality Theorem Find BU and SB if .BC ST Solution 6 24 12  BUFor BU: 24 72BU 3BU  SB SU BU For SB: 12 3  9SB
  • 53. Check your understandingIf , find PQ, PV, and PW.VW QR 22 12 6  PQ For PQ: 22 2 PQ 2 22PQ 11PQ For PV: 11 9 PV 2PV 22 11 2  PW For PW: 11 44PW 4PW
  • 54. Similar Triangles The Proportionality Principles A bisector of an angle of a triangle divides the opposite side into segments which are proportional to the adjacent sides. is the angle bisector of C. CD  CB BD CA DA
  • 55. Similar Triangles The Proportionality Principles Angle Bisectors Find the value of x. Solution 15 10 18  x 5 10 6  x 5 60x 12x
  • 56. Similar Triangles The Proportionality Principles Angle Bisectors Find the value of x. Solution 21 30 15  x 7 30 5  x 5 210x 42x x
  • 57. Similar Triangles The Proportionality Principles Three or more parallel lines divide any two transversals proportionally. AB EF CD and are transversals. AC BD  a b c d
  • 58. Similar Triangles The Proportionality Principles Three or more parallel lines divide any two transversals proportionally. Note The cut segment and the length of the segment themselves are also proportional; thus we have    a c a b c d    b d a b c d
  • 59. Similar Triangles The Proportionality Principles Parallel Lines and Transversals Find the value of x. Solution 8 28 16  x 1 28 2  x 2 28x 14x x
  • 60. Similar Triangles The Proportionality Principles Parallel Lines and Transversals Find the value of x. Solution 6 9 4  x 6 36x 6x
  • 61. Similar Triangles The Proportionality Principles Parallel Lines and Transversals Find the value of x. Solution 3 10 5  x x 10 3 2 5 30x 6x
  • 62. Check your understandingSolve for the indicated variable. 2. for x and y y 20 5 28 7 7 5 12       x x y 1. for a a15 25 10 6   a a x
  • 63. Similar Triangles The Proportionality Principles The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original and each other A B C D
  • 64. Similar Triangles The Proportionality Principles A B C D C D B A D C A C B ∆CBD ~ ∆ACD ∆ACD ~ ∆ABC ∆CBD ~ ∆ABC Similar Right Triangles
  • 65. Similar Triangles The Proportionality Principles A B C D C D B A D C A C B Similar Right Triangles h ab y x c x h h y a b a b c  h y x h  a x c a  b y c b Proportions ∆CBD ~ ∆ACD ∆ACD ~ ∆ABC ∆CBD ~ ∆ABC
  • 66. Similar Triangles The Proportionality Principles A B C D h ab y x c 2 h xy h xy 2 a xc a xc 2 b yc b yc Similar Right Triangles This result is also called the GEOMETRIC MEAN THEOREM for similar right triangles
  • 67. Similar Triangles The Proportionality Principles Similar Right Triangles The GEOMETRIC MEAN of two positive numbers a and b is GM ab The geometric mean of 16 and 4 is 16 4GM 64 8
  • 68. Similar Triangles The Proportionality Principles Similar Right Triangles Find the value of x. Solution 36 x   6 3x 18 3 2
  • 69. Similar Triangles The Proportionality Principles Similar Right Triangles Find JM, JK and JL. Solution 8 2 8 2 16 4  JM 8 10 80 4 5  JK 2 10 20 2 5  JL
  • 70. Similar Triangles The Proportionality Principles Similar Right Triangles Find the value of x. Solution 9 25x 3 5 15
  • 71. Similar Triangles The Proportionality Principles Similar Right Triangles Find x. Solution 12 16 x 144 16 x 9x
  • 72. Similar Triangles The Proportionality Principles Similar Right Triangles Find x, y and z. Solution 6 9 x 36 9 x 4x 4 13y 2 13y 9 13z 3 13z
  • 74. Thankyou!

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Triangle Similarity: AA, SSS, SAS

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triangle similarity

Triangle Similarity

Aug 27, 2014

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Triangle Similarity. Keystone Geometry. F. C. A. B. D. E. Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional. Similar Polygons. ~ means “is similar to”. Congruence vs. Similarity.

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Triangle Similarity Keystone Geometry

F C A B D E Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional. Similar Polygons ~ means “is similar to”

Congruence vs. Similarity • If two triangles are congruent, then they are exactly the same. • Methods: SAS, SSS, ASA, AAS, and HL • If two triangle are similar, then they will have congruent angles and their sides will be proportional. • Methods: AA, SSS, SAS

If two angles of one triangle are congruent (equal) to two angles of another triangle, then the triangles are similar. Note: If you know two angles, the third angle is not negotiable AA Similarity Postulate

Example 1: • Example 2: C 60º F 62º 58º 60º 50º 40º G H D E 61º 59º YES- 2 angles of triangle CDE are congruent to 2 angles of triangle FHG NO- only 1 angle is congruent in both triangles Examples: Tell whether the triangles are similar or not. *You need two angles to be congruent to prove the triangles are similar!

*If two triangles share an angle, then they share a congruent angle y y 4 4 6 6 3 2 y+3 4+2=6 x x Example: Find the values of x and y. The triangle on the inside is similar to the larger triangle on the outside because of AA similarity.

M J ΔLMN N L K Example: Complete the following statement: ΔJKN ~ _______ *Make sure you match up corresponding angles *It matters how you name your triangle, just like with congruence!!

SSS Similarity Theorem If the sides of two triangles are in proportion, then the triangles are similar. All proportions will be equal to the scale factor of the two triangles.

* Yes, they are similar by SSS theorem! 16 20 10 32 24 15 Example: Are the two triangles similar? The proportions of the sides are equal! The scale factor of the two triangles is 5:8.

SAS Similarity Theorem Remember! Proportional Side – Congruent Angle – Proportional Side! If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar.

M J 6 12 N 9 8 L K Included Angle Example: Are the two triangles similar? Side Side Yes, they are similar by SAS theorem!

20 16 20 24 32 32 24 24 16 36 36 Example: Are the two triangles similar? NO! They are NOT similar because not all of the sides are proportional!

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COMMENTS

  1. Similar triangles | PPT | Free Download - SlideShare

    This document covers several theorems regarding similar triangles: AAA, AA, and SAS similarity theorems state that if corresponding angles or sides are proportional, the triangles are similar. The SSS and L-L theorems for right triangles also make claims of similarity based on proportional sides.

  2. PPT - Similar Triangles PowerPoint Presentation, free ...

    Summary • Triangles are similar if they show AA Similarity, SSS Similarity, or SAS Similarity. • The areas of similar triangles are the ratio that equals the square of the scale factor. • To find the scale factors of two areas, just compute for the areas of each triangle and then, form the ratio of the areas.

  3. Triangles (Similarity) | PPT | Free Download - SlideShare

    It explains the different rules to determine if triangles are similar: AAA (angle-angle-angle), PPP (proportional property), PAP (proportional angles property), and RHS (right-hypotenuse-side). Examples are given applying these rules to prove triangles are similar and calculate missing side lengths or scale factors.

  4. Similar Triangles | PPT | Free Download - SlideShare

    It provides examples of similar triangles in nature, art, architecture, and mathematics. It explains the different rules to determine if triangles are similar: AAA (angle-angle-angle), PPP (proportional property), PAP (proportional angles property), and RHS (right-hypotenuse-side).

  5. PPT - SIMILAR TRIANGLES PowerPoint Presentation, free ...

    In this lesson, you will continue the study of similar polygons by looking at the properties of similar triangles. In the diagram, ∆BTW ~ ∆ETC. Write the statement of proportionality.

  6. PPT - Similar Triangles PowerPoint Presentation, free ...

    Similar Triangles and Transformations • Remember triangles remained congruent over reflection rotation and translation These transformations created congruent images. • Triangles also remain similar over reflection rotation transformation and can do under dilation.

  7. Similar Triangles - WordPress.com

    Similar Triangles. Similar shapes Are Enlargements of each other Corresponding angles are equal Sides are related by the same scale factor Similar Triangles 50º 50º 30º 30º 100º 100º Triangles are similar if matching angles remain the same size. Show that these triangles are similar 50º 50º 10º 10º 120º 120º To calculate a length 4 ...

  8. Math 9 similar triangles intro | PPT - SlideShare

    It defines similar triangles as having the same shape but different sizes, and discusses how similar triangles have corresponding angles that are congruent and corresponding sides that are proportional. It provides examples of similar triangles and statements showing their similarity.

  9. Triangle Similarity: AA, SSS, SAS - ppt download - SlidePlayer

    If ∆QRS ~ ∆XYZ, identify the pairs of congruent angles and write 3 proportions using pairs of corresponding sides. x = 8 z = ±10 Q X; R Y; S Z; 3 Objectives Prove certain triangles are similar by using AA, SSS, and SAS. Use triangle similarity to solve problems.

  10. PPT - Triangle Similarity PowerPoint Presentation, free ...

    I CAN -Use the triangle similarity theorems to determine if two triangles are similar. Use proportions in similar triangles to solve for missing sides. Recall in 7-2, to prove that two polygons are similar you had to:.