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In this lesson some conditions will be developed that guarantee the similarity of triangles.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

- The concept of similarity.
- Conditions for similarity of polygons.
- The concept of congruence.
- Conditions that guarantee the congruence of triangles.

Challenge

Find the ratio of the length of a diagonal and a side of a regular pentagon.

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Explore

Discussion

Two polygons are similar if corresponding angles are congruent and corresponding sides are proportional. For triangles, the congruence of two angles already implies similarity.

If two angles of a triangle are congruent to two angles of another triangle, then the triangles are similar.

If $∠A≅∠D$ and $∠B≅∠E,$ then $△ABC∼△DEF.$

Consider two triangles $△ABC$ and $△DEF,$ whose two corresponding angles are congruent.

These triangles can be proven to be similar by identifying a similarity transformation that maps one triangle onto the other. First, $△DEF$ can be dilated with the scale factor $k=DEAB $ about $D,$ forming the new triangle $△DE_{′}F_{′}.$

Since a dilation is a similarity transformation, it can be concluded that $△DE_{′}F_{′}$ and $△DEF$ are similar triangles. Next, it has to be proven that a rigid motion that maps $△DE_{′}F_{′}$ onto $△ABC$ exists. The corresponding angles of similar figures are congruent, so $∠E_{′}$ and $∠E$ are congruent angles.$∠E_{′}≅∠E $

Additionally, since $∠E$ is congruent to $∠B,$ by the Transitive Property of Congruence, $∠E_{′}$ is congruent to $∠B.$
$∠E_{′}≅∠B $

The ratios of the corresponding side lengths of similar polygons are the same and equal to the scale factor.
$DEDE_{′} =k $

In this case, the scale factor $k$ is $DEAB .$ Applying the Transitive Property of Equality, an equation can be formed and simplified. $DEDE_{′} =DEAB ⇕DE_{′}=AB $

It has been obtained that the two angles and the included side of $△DE_{′}F_{′}$ are congruent to the corresponding two angles and the included side of $△ABC.$
Therefore, by the Angle-Side-Angle (ASA) Congruence Theorem, the two triangles are congruent.
$△ABC≅△DE_{′}F_{′} $

Since congruent figures can be transformed into each other using rigid motions, and $△ABC$ and $△DE_{′}F_{′}$ are congruent triangles, there is a rigid motion placing $△DE_{′}F_{′}$ onto $△ABC.$
The combination of this rigid motion and the dilation performed earlier forms a similarity transformation that maps $△DEF$ onto $△ABC.$

Therefore, it can be concluded that $△ABC$ and $△DEF$ are similar triangles.

$△ABC∼△DEF$

The proof is now complete.

Example

The Grim Reaper, who is $5$ feet tall, stands $16$ feet away from a street lamp at night. The Grim Reaper's shadow cast by the streetlamp light is $8$ feet long. How tall is the street lamp?

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Both the lamp post and the Grim Reaper stand vertically on horizontal ground.

A sketch of the situation is helpful for finding the solution. Under the assumption that the lamp post and the Grim Reaper make right angles in relation to the ground, two right triangles can be drawn. The unknown height of the lamp post is labeled as $x.$

As these triangles both have a right angle and share the angle on the right-hand side, they are similar by the Angle-Angle (AA) Similarity Theorem. Notice that the base of the larger triangle measures to be $24$ feet.

Since the triangles are similar, the ratios between corresponding side lengths are the same.$5x =824 $

The solution of this equation defines the value of $x$ — the height of the street lamp.
The street lamp at $15$ feet high towers over The Grimp Reaper.

Pop Quiz

For the given diagram, find the missing length.

Discussion

A second theorem allows for determining triangle similarity when only the lengths of corresponding sides are known.

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

If $DEAB =EFBC =FDCA ,$ then $△ABC∼△DEF.$

Consider two triangles $△ABC$ and $△DEF,$ whose corresponding sides are proportional.

These triangles can be proven to be similar by identifying a similarity transformation that maps one triangle onto the other. First, $△DEF$ can be dilated with the scale factor $k=DEAB $ about $D,$ forming the new triangle $△DE_{′}F_{′}.$

Because dilation is a similarity transformation, it can be concluded that $△DE_{′}F_{′}$ and $△DEF$ are similar triangles. Now, it has to be proven that a rigid motion that maps $△DE_{′}F_{′}$ onto $△ABC$ exists. The ratios of the corresponding side lengths of similar polygons are the same and equal to the scale factor.$DEDE_{′} =DFDF_{′} =EFE_{′}F_{′} =k $

In this case, the scale factor $k$ is $DEAB .$ Since all of the sides of $△ABC$ and $△DEF$ are proportional, the scale factor can be expressed by any of the following ratios.
$k=DEAB =EFBC =DFCA $

Applying the Transitive Property of Equality, three equations can be formed and simplified. $DEDE_{′} DFDF_{′} EFE_{′}F_{′} =DEAB =DFCA =EFBC ⇒DE_{′}DF_{′}E_{′}F_{′} =AB=CA=BC $

These relations imply that the three sides of $△DE_{′}F_{′}$ are congruent to the three sides of $△ABC.$ Therefore, by the Side-Side-Side (SSS) Congruence Theorem, the two triangles are congruent.
$△ABC≅△DE_{′}F_{′} $

Since congruent figures can be transformed into each other using rigid motions, and $△ABC$ and $△DE_{′}F_{′}$ are congruent triangles, there is a rigid motion placing $△DE_{′}F_{′}$ onto $△ABC.$
The combination of this rigid motion and the dilation performed earlier forms a similarity transformation that maps $△DEF$ onto $△ABC.$

Therefore, it can be concluded that $△ABC$ and $△DEF$ are similar triangles.

$△ABC∼△DEF$

The proof is now complete.

Example

There are four congruent angles in the figure. Try to identify them.

$∠DBA≅∠BCE≅∠BEC≅∠DBE$

Look for similar triangles and an isosceles triangle.

First, notice that segments $BE$ and $BC$ are equal in length.

These are two sides of $△BCE,$ so by the Isosceles Triangle Theorem, the opposite angles are congruent.$∠BCE≅∠BEC $

Two of the triangles, $△ABD$ and $△ACE$ look similar.

Because the lengths of the sides are given, the ratio of corresponding sides can be calculated.

Ratio | Expression | Simplification |
---|---|---|

$ABAC $ | $459459+1275 =4591734 $ | $934 $ |

$ADAE $ | $405405+1125 =4051530 $ | $934 $ |

$DBEC $ | $3601360 $ | $934 $ |

$ABAC =ADAE =DBEC $

According to the Side-Side-Side (SSS) Similarity Theorem, the two triangles are similar.
$△ACE∼△ABD $

Corresponding angles of similar triangles are congruent.
One pair of these angles is marked on the figure. $∠BCE≅∠DBA $

In addition to the proportions in **Step 2** showing that $△ACE$ and $△ABD$ are similar, they also show the two triangles are dilations of each other from the common vertex $A.$ Since dilations map a segment to a parallel segment, segments $DB$ and $EC$ are parallel.

$∠BEC≅∠DBE $

These angles are marked on the figure.
$∠DBA≅∠BCE≅∠BEC≅∠DBE $

The congruent angles are marked on the figure.

Discussion

Two theorems have been covered, now a third theorem that can be used to prove triangle similarity will be investigated. This third theorem allows for determining triangle similarity when the lengths of two corresponding sides and the measure of the included angles are known.

If two sides of a triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar.

If $DEAB =DFAC $ and $∠A≅∠D,$ then $△ABC∼△DEF.$

Consider two triangles $△ABC$ and $△DEF,$ whose two pairs of corresponding sides are proportional and the included angles are congruent.

These triangles can be proven to be similar by identifying a similarity transformation that maps one triangle onto the other. First, $△DEF$ can be dilated with the scale factor $k=DEAB $ about $D,$ forming the new triangle $△DE_{′}F_{′}.$

Because dilation is a similarity transformation, it can be concluded that $△DE_{′}F_{′}$ and $△DEF$ are similar triangles. Now, it has to be proven that a rigid motion that maps $△DE_{′}F_{′}$ onto $△ABC$ exists. The ratios of the corresponding side lengths of similar polygons are the same and equal to the scale factor.$DEDE_{′} =DFDF_{′} =k $

In this case, the scale factor $k$ is $DEAB .$ Since $AB$ and $AC$ are proportional to $DE$ and $DF$ respectively, the scale factor can be expressed by any of the following ratios.
$k=DEAB =DFAC $

Applying the Transitive Property of Equality, three equations can be formed and simplified. $DEDE_{′} =DEAB DFDF_{′} =DFAC ⇒⇒ DE_{′}=ABDF_{′}=AC $

These relations imply that the two sides of $△DE_{′}F_{′}$ are congruent to the corresponding two sides of $△ABC.$ Moreover, the included angles $∠A$ and $∠D$ are also congruent.
Therefore, by the Side-Angle-Side (SAS) Congruence Theorem, the two triangles are congruent.
$△ABC≅△DE_{′}F_{′} $

Since congruent figures can be transformed into each other using rigid motions, and $△ABC$ and $△DE_{′}F_{′}$ are congruent triangles, there is a rigid motion placing $△DE_{′}F_{′}$ onto $△ABC.$
The combination of this rigid motion and the dilation performed earlier forms a similarity transformation that maps $△DEF$ onto $△ABC.$

Therefore, it can be concluded that $△ABC$ and $△DEF$ are similar triangles.

$△ABC∼△DEF$

The proof is now complete.

Example

The diagram shows the distances between points on a figure.

Show that $△ABC$ and $△AED$ are similar triangles. Then find $DE.$

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Triangles $△ABC$ and $△AED$ have a common angle at $A.$

The table below contains the ratios of two pairs of corresponding sides of the two triangles.

Ratio | Expression | Simplified Form |
---|---|---|

$AEAB $ | $5+97 =147 $ | $21 $ |

$ADAC $ | $7+35 =105 $ | $21 $ |

$△ABC∼△AED $

In similar triangles, all pairs of corresponding sides are proportional. Therefore, the ratio of the two pairs of corresponding sides, found in the table, also applies to the third pair of corresponding sides.
$DECB =21 $

Since the length of $CB$ is given in the diagram, this equation can be solved for $DE.$
The length of $DE$ is $6$ units.
Closure

Through applying the theorems of similar triangles, the ratio of the lengths of a diagonal and the sides of a regular pentagon can be found.

Begin by determining the angle measures of the figure.

The Polygon Angle Sum Theorem identifies the sum of the interior angle measures of a pentagon.
Due to the symmetry of the regular pentagon, there are ten angles with the same measure in the diagram.
Since the measure of $∠A$ is known, this gives the measure of $∠CAD.$
This information and the measures of the angles in symmetrical position can now be labeled in the diagram.
The ratio of two lengths is positive, so the positive solution gives the ratio of the length of the diagonal and the side of a regular pentagon.

$(5−2)180=540 $

In a regular pentagon, all five angles are equal measures. Therefore, one of the measures of the angles is a fifth of the sum of the five angle measures. $m∠A=5540 =108 $

Furthermore, since the sides of a regular pentagon are congruent, $△ABE$ is an isosceles triangle.
According to the Isosceles Triangle Theorem, the base angles of $△ABE$ are congruent.
$∠ABE≅∠AEB $

Applying the information found so far to the Triangle Angle Sum Theorem, the measure of the base angles can be found.
$m∠ABE+m∠AEB+m∠A=180$

SubstituteII

$m∠AEB=m∠ABE$, $m∠A=108$

$m∠ABE+m∠ABE+108=180$

▼

Solve for $m∠ABE$

$m∠ABE=36$

$m∠BAC+m∠CAD+m∠DAE=m∠A$

SubstituteValues

Substitute values

$36+m∠CAD+36=108$

SubEqn

$LHS−(36+36)=RHS−(36+36)$

$m∠CAD=36$

Next, focus on $△ACE.$ In this triangle, $AC$ and $EC$ are diagonals of the pentagon, and $AE$ is a side.

In the diagram, a smaller triangle labeled $△AFE$ is also present. These two triangles share a common angle at $A$ and congruent angles at $C$ and $E.$$∠CAE∠ACE ≅∠EAF≅∠AEF $

According to the Angle-Angle (AA) Similarity Theorem, that means the two triangles are similar. Therefore, the corresponding sides are proportional.
$AECA =AFAE $

Continuing forward, notice that triangles $△AFE$ and $△CEF$ are isosceles. Therefore, their legs have equal lengths.
$AE=EF=FC $

Using $s$ for the length of the sides, $AE=EF=FC=s$, as indicated on the figure. Also, using $d$ for the length of the diagonal, $CE=d$ and $FA=CA−FC=d−s.$
These expressions can be substituted in the proportionality relationship previously obtained.
The question asks for the ratio of $d$ to $s.$ A new variable can be introduced for to represent this unknown ratio.
$x=sd $

The variable $d$ in the expression can then be replaced with $x.$ The result can then be simplified.
The next step is to solve this equation.
$x=x−11 $

▼

Solve for $x$

MultEqn

$LHS⋅(x−1)=RHS⋅(x−1)$

$x(x−1)=1$

MultPar

Multiply parentheses

$x_{2}−x=1$

SubEqn

$LHS−1=RHS−1$

$x_{2}−x−1=0$

UseQuadForm

Use the Quadratic Formula: $a=1,b=-1,c=-1$

$x=2(1)-(-1)±(-1)_{2}−4(1)(-1) $

NegNeg

$-(-a)=a$

$x=2(1)1±(-1)_{2}−4(1)(-1) $

CalcPowProd

Calculate power and product

$x=21±1−(-4) $

SubNeg

$a−(-b)=a+b$

$x=21±1+4 $

AddTerms

Add terms

$x=21±5 $

$Length of sideLength of diagonal =21+5 ≈1.618$

The ratio of the diagonal to the side of a regular pentagon can be used to prove that the following construction creates a regular pentagon. This is a construction created by Yosifusa Hirano in the 19th century.

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