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Grade 9:  The Learning Equation Math

32.01 Similar Triangles

3icon.gif (4368 bytes)

3D Objects and 2D Shapes

Refresher pp 74-5



similar, scale, model, congruent, corresponding angles, transversal, interior angle, exterior angle, alternate angles, ratio, proportion, parallel



Learning Outcomes:

The student will:

Similar Triangles

In this section we will be studying congruent () and similar (~) triangles.  It will be easier to remember these concepts if you start by considering size and shape.

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


<A = <F

<B = <D

<C = <E

a/f = 6/3 = 2

b/d = 8/4 = 2

c/e = 10/5 = 2


<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, B or C.  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.



Similar Triangles 2

When you learn the mathematics of similar triangles, you can you use the knowledge to determine unknown dimensions or angles without measuring.  You have already explored the angle relationships of similar triangles.  If you know an angle in one triangle, the corresponding angle in a similar triangle will be the same size.

In the past you studied, corresponding angles when a line (transfersal) intersects parallel lines.  Review the chart to refresh your memory.  Angles are usually the key to determine if triangles are similar.

two parallel lines (p & q) and a transversal (t)

parlines.gif (1476 bytes)

Before the next rules can apply, there must be two parallel lines and a transversal.


< angle     congruent

corangles.gif (1708 bytes) Each time the transversal t crosses a parallel line, two sets of opposite angles are formed.

Line p:

<1 = <4 and <2 = <3

Line q:

<5 = <8 and <6 = <7

oppangles.gif (1708 bytes) Corresponding angles are on the same side of the transversal t..

Left side of transversal t:

<1 = <5 and <3 = <7

Right side of transversal t:

<4 = <8 and <2 = <6

altintangle.gif (1687 bytes)

Interior (inside) angles are the ones between the two parallel lines p and q.

<3 = <6 and <4 = <5

altextangle.gif (1655 bytes) Exterior (outside) angles are the ones above and below the two parallel lines p and q.

<1 = <8 and <2 = <7





BBC Education - GCSE Revision Bite - similar triangles



Similar Figures

Click and drag on the red dots to adjust the size and shape of the triangles.  Study the effect on the similar and congruent triangles.

java applet or image

source:  Euclid's Elements Book VI:   Proposition 5

source file location for



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Comments to:  Jim Reed
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