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Proving Triangles Similar

Proving Triangles Similar

Author: Michele Harris
Description:

Using new corollaries and theorems, the student will be able to prove triangles similar.

Using knowledge of theorems and corollaries, student will be able to prove triangles similar.

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Tutorial

Theorems and Corollaries

​Theorem 57 - If 2 triangles have the 3 angles of one = respectively to the 3 angles of the other, then the triangles are ~.  

​Corollary 57-1 - If 2 angles of one triangle are = respectively to 2 angles of another, then the triangles are ~.

​Corollary 57 -2 - 2 right triangles are similar if an acute angle of one is = to an acute angle of the other.

​Theorem 58 - If 2 triangles have 2 pairs of sides proportional and the included angles = respectively, then the 2 triangles are ~.  (s.a.s.)

​Corollary 58-1 - If the legs of one triangle are proportional to the legs of another, the triangles are similar. (l.l.)

​Theorem 59 - If 2 triangles have their sides respectively proportional, then the triangles are ~.  (s.s.s.)

*Read through the proofs for T's and c's in your text before moving on to the exercises.

Source: ABEKA Plane Geometry 2006

exercise #1

Prove that 2 isosceles triangles are similar if any angle of one equals the corresponding angle of the other.

case 1:  equal vertex angles

Given:  Isosceles triangle ABC, isosceles triangle DEF

            AC=BC; DF=EF; <C=<F

Prove:  triangle ABC ~ triangle DEF

​Draw 2 isosc. triangles, different sizes for your diagrams.

​statements for case 1 proof:

​2. <A = <D

​3.  triangle ABC ~ triangle DEF

​case 2:  equal base angles

​statements for case 2 proof:

​2.  <A = <B; <D = <E

​3.  <B = <E

​4.  triangle ABC ~ triangle DEF

 

 

 

Source: ABEKA Plane Geometry 2006

Reasons for proofs, exercise #1

case 1:

​2.  isosc. triangles with = vertex angles have = base angles also.

​3.  a.a.

 

​case 2:

 

​2.  in any isosc. triangle, the angles opposite the = sides are =.

​3.  quantities that are = to = quantities are =.

​4.  a.a.

 

 

Source: ABEKA Plane Geometry 2006

exercise #2

If AC = 9, AE = 3, AB = 12, AD = 4, prove: triangle ABC ~ triangle ADE

*diagram in text

given:  triangle ABC; DE; AC = 9; AE = 3; AB = 12; AD = 4

prove:  triangle ABC ~ triangle ADE

statements for proof:

2.  12:4 = 9:3

3.  AB:AD = AC:AE

4.  <A = <A

5.  triangle ABC ~ triangle ADE 

Source: ABEKA Plane Geometry 2006

Reasons for proof, exercise #2

2.  definition of proportion

​3.  substitution

​4.  identity

​5.  S.A.S.

Source: ABEKA Plane Geometry 2006

exercise #3

​Prove that if a line is drawn parallel to the base of a triangle, it cuts off a triangle similar to the given triangle

*look at what the problem tells you, and draw the needed diagram.  Remember, the line is drawn parallel to the base of the triangle, this line will be the transversal to the sides of the triangle.  If 2 parallel lines are cut by a transversal, what do you know about the corresponding angles?

​given:  triangle ABC; DE parallel AB

​prove:  triangle DEC ~ triangle ABC

​statements for proof:

​2.  <CDE = <A;  <CED = <B

​3.  triangle ABC ~ triangle DEC

 

Source: ABEKA Plane Geometry 2006

Reasons for proof, exercise #3

​2.  if 2 parallel lines are cut by a transversal, then the corresponding angles are =.

​3.  a.a.

Source: ABEKA Plane Geometry 2006

exercise #4

​In the triangle ABC, AE is the altitude on BC and CD is the altitude on AB.  Prove:  triangle CBD ~ triangle AEB

​given:  triangle ABC; AE altitude to BC; CD altitude to AB

*draw your diagram

statements for proof:

​2.  <B = <B

​3.  AE perpendicular to BC; CD perpendicular to AB

​4.  <AEB, <CDB are right angles

​5.  triangle CDB and triangle AEB are right triangles

​6.  triangle CDB ~ triangle AEB

 

Source: ABEKA Plane Geometry 2006