Similar Triangles

Similar triangles have the same shape, but the size may be different.

Remember "" means "is congruent to" and "~" is "similar to". Examples

Corresponding Triangles Corresponding Congruent Angles

Corresponding Proportional Sides

a/f = b/d = c/e = factor

ΔABC ~ ΔFDE

<A = <F

<B = <D

<C = <E

a/f = 6/3 = 2

b/d = 8/4 = 2

c/e = 10/5 = 2

ΔABC ΔFDE

<A = <F

<B = <D

<C = <E

a/f = 3/3 = 1

b/d = 4/4 = 1

c/e = 5/5 = 1

Two triangles are similar if:

  • two pairs of corresponding angles are congruent (therefore the third pair of corresponding angles are also congruent).
  • the three pairs of corresponding sides are proportional.

Notice the corresponding angles for the two triangles in the applet are the same. The corresponding sides lengths are the same only when the scale factor slider is set at 1.0. Study the side lengths closely and you will find that the corresponding sides are proportional.

Check out this applet

If you know triangles are similar, you can use the proportion of corresponding sides to help determine an unknown dimension.

Study the object below.  You can change the triangles by dragging on the slider or dragging vertex A or B.  The proportions of the corresponding pairs of sides changes as the scale slider position changes.  Each time you stop dragging, look at the proportions.  If one of the sides was unknown you could use two pairs of corresponding sides to calculate the missing dimension.

 

Congruent triangles are a special type of similar triangles. Congruent triangles have the same shape (similar triangles) and size.

Increase/decrease <A, <B and <C by clicking and dragging the vertices of the left triangle below. Notice the corresponding angles (<D <E and <F) remain congruent. Since the sum of three angles must be 180o, the third pair of corresponding angles must also be congruent when the first two pairs of corresponding angles are congruent. 

In the applet above:

<A = <D, <B = <E, <C = <F

Notice the size and shape of the new pairs of triangles remains the same. The patterns you may have observed and need to know for congruent triangles are displayed below.   Remember "" means "is congruent to".

Two triangles are congruent if:

  • all 6 pairs of corresponding angles and sides are congruent.
Corresponding Triangles Corresponding Congruent Angles

Corresponding Congruent Sides

factor = 1

a/f = b/d = c/e = 1

ΔABC ΔDEF <A = <D

<B = <E

<C = <F

AB = DE

BC = EF

AC = DF

 

The factor for congruent triangles is 1. Remember this fact for future problem solving activities.

 

If all 3 pairs of corresponding sides in two triangles are the same, then the triangles are congruent.

Move the corners A, B or C of the triangle above. Watch how ΔDEF changes.  Since the 3 pairs of corresponding sides of the two triangles are equal each time you manipulate the applet, the SSS (Side, Side, Side) Congruence Relation proves each pair of triangles are congruent.

 

 

 

Triangle Congruence Relations

 

Always form congruent triangles

        

These files may be slow loading on some computers.

May not form congruent triangles:

  

These files may be slow loading on some computers.

SSS

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If three sides of one triangle are congruent to three sides of a second triangle, the two triangles are congruent.

 

ASA

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If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent.

SAS

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If two sides and the included angle are congruent to two sides and the included angle of a second triangle, the two triangles are congruent.

AAS

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If two angles and a non included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, the two triangles are congruent.

Hyp-S

 

If the hypotenuse and the leg of one right triangle are congruent to the corresponding parts of the second right triangle, the two triangles are congruent

hyps.gif (1777 bytes)

SSA

ssa.gif (1812 bytes)

Two triangles with two sides and a non-included angle equal may or may not be congruent.

AAA

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If two angles on one triangle are equal, respectively, to two angles on another triangle, then the triangles are similar, but not necessarily congruent.

Comments to:  Jim Reed
Started September, 1998. Copyright 1999, 2000, 2001, 2002, 2003, 2004