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54935_title

Similarity Theorems

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Introduction

This lesson is about similar triangles and similarity theorems. As you go over the exercises you will develop your skills in determining if two triangles are similar and finding the length of a side or measure of an angle of a triangle.

Goals


GEOMETRY

Students will be able to understand and apply geometric concepts and relations in a variety of forms.

  • understand simple geometric figures and patterns of relationships in two and three dimensions

  • apply symmetry and transformations

  • apply the concepts of congruence and similarity

  • apply formulas and construct arguments and proofs to solve geometric problems

  • define common geometric figures and use deductive reasoning to relate properties of those figures

Problem

What are the four similarity theorems?

Vocabulary

  • similar
  • triangles
  • polygons
  • proportional
  • congruent

Learn

AAA Similarity Theorem

If in two triangles the corresponding angles are congruent, then the two triangles are similar.

If ΔBOS <-> ΔVIC

∠B ≅ ∠V , ∠O ≅ ∠I , ∠S ≅ ∠C

then ΔBOS ~ ΔVIC

AA Similarity

If two angles of one triangle are congruent to the corresponding two angles of another triangle, the triangles are similar.

Given: ∠A ≅ ∠O

∠J ≅ ∠B

Then: ΔJAM ~ ΔBON

SAS Similarity Theorem

If in two triangles two pairs of corresponding sides are proportional and the included angles are congruent, then the triangles are similar.

If ΔICE <-> ΔBOX

CI
OB
=
CE
OX

and ∠C ≅ ∠O

Then ΔICE ~ΔBOX

Examples:

SSS Similarity Theorem

If the two triangles three corresponding sides are proportional, then the triangles are similar.

If ΔSUN <-> ΔBLK

BL
SU
=
UN
LK
=
SN
BK

Then ΔSUN ~ ΔBLK

Example:

1.

SN
OB
=
NA
PE
=
SA
DE
4
6
=
6
9
=
8
12
Since the corresponding sides are proportional then ΔPOE ~ ΔNSA by SSS Similarity.

2. Explain: Any two congruent triangles are similar.

Resources

Think

A. Tell whether the two triangles are similar. Cite the Similarity Postulate or Theorem to justify your answer. (Identical marks indicate ≅ parts.)

B. Determine whether each pair of the following triangles are similar by SAS, AAA, SSS or not at all.

Explore

C. Answer the following:

1. Two isosceles triangles have an angle of 50 degrees. Does it follows that the triangles are similar?

2. Two angles of ΔBEL have measures of 20 and 50. Two angles of ΔJAY have measures of 30 and 100. IS ΔBEL ~ ΔJAY ?

3. Is it possible for two triangles to be similar if two angles of one have measures 50 and 75, where as two angles of the other have measures 55 and 70?

4. Two angles of have measures 40 and 80, where as the two angles of the other have measures 60 and 80, are the two triangles similar?

5 - 6. The lengths of the sides of a triangle are 12 and 15. If the length of the shortest side of a similar triangle is 12, find the lengths of the other two sides.

7- 8. In the figure, if AE = 8, AB=4, BC=10, ED=3. Find BD and DC.

Quiz

  1. 1.
    1. A.
      AAA
    2. B.
      SSS
    3. C.
      SAS
    4. D.
      AA
    5. E.
      ASA
  2. 2.
    1. A.
      yes; ΔADF ~ ΔCRT; ASA Similarity Postulate
    2. B.
      no; ΔADF and ΔCRT are not similar
    3. C.
      yes; ΔADF ~ ΔCRT; SSS Similarity Postulate
    4. D.
      yes; ΔADF ~ ΔCRT; SAS Similarity Postulate
    5. E.
      yes; ΔADF ~ ΔCRT; AA Similarity Postulate
  3. 3.
    1. A.
      ΔBEC
    2. B.
      ΔAED
    3. C.
      ΔCED
    4. D.
      ΔBDC
    5. E.
      ΔACD
  4. 4.
    1. A.
      45
    2. B.
      18
    3. C.
      10
    4. D.
      12
    5. E.
      20
  5. 5.
    1. A.
      12
    2. B.
      10
    3. C.
      4
    4. D.
      6
    5. E.
      8
  6. 6.
    1. A.
      VW = 8; UW = 10
    2. B.
      VW = 6; UW = 8
    3. C.
      VW = 7; UW = 9
    4. D.
      VW = 12; UW =14
    5. E.
      VW = 10; UW = 9
  7. 7.
    1. A.
      none of the triangles
    2. B.
      isosceles triangles
    3. C.
      all triangles
    4. D.
      right triangles
    5. E.
      equilateral triangles
  8. 8.
    1. A.
      SAS
    2. B.
      ASA
    3. C.
      AA
    4. D.
      SSS
    5. E.
      AAA
  9. 9.
    1. A.
      SAS Similarity Thm.
    2. B.
      SSS Similarity Thm.
    3. C.
      AA Similarity Thm.
    4. D.
      none of these
    5. E.
      ASA Similarity Thm.
  10. 10.
    1. A.
      ^
    2. B.
    3. C.
      ---
    4. D.
      =
    5. E.
      ~
  11. 11.
    1. A.
      6
    2. B.
      9
    3. C.
      12
    4. D.
      3
    5. E.
      5
  12. 12.
    1. A.
      14 cm
    2. B.
      6 cm
    3. C.
      8 cm
    4. D.
      9 cm
    5. E.
      10 cm
  13. 13.
    1. A.
      ASA Similarity Theorem; ΔQNM~ ΔONP
    2. B.
      AA Similarity Theorem; ΔQNM~ ΔONP
    3. C.
      AAA Similarity Theorem; ΔQNM~ ΔONP
    4. D.
      SAS Similarity Theorem; ΔQNM~ ΔONP
    5. E.
      SSS Similarity Theorem; ΔQNM~ ΔONP
  14. 14.
    1. A.
      1800
    2. B.
      64o
    3. C.
      24o
    4. D.
      44o
    5. E.
      136o
  15. 15.
    1. A.
      28o
    2. B.
      43o
    3. C.
      59o
    4. D.
      48o
    5. E.
      68o