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Similarity, Proof, and Trigonometry

The Conditions for Triangle Similarity

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.

Challenge

Using Similarity of Triangles to Solve Problems

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

Regular pentagon ABCDE with all five diagonals drawn

Explore

Investigating Triangles With Two Pairs of Congruent Angles

In the applet, move points and The applet copies and on and labels the intersection of the corresponding rays by

Discussion

Angle-Angle Similarity Theorem

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.

Triangles ABC and DEF with congruent angles marked at A and D and B and E.

If and then

Proof

Consider two triangles and whose two corresponding angles are congruent.

Triangles ABC and DEF.

These triangles can be proven to be similar by identifying a similarity transformation that maps one triangle onto the other. First, can be dilated with the scale factor about forming the new triangle

Since a dilation is a similarity transformation, it can be concluded that and are similar triangles. Next, it has to be proven that a rigid motion that maps onto exists. The corresponding angles of similar figures are congruent, so and are congruent angles. Additionally, since is congruent to by the Transitive Property of Congruence, is congruent to The ratios of the corresponding side lengths of similar polygons are the same and equal to the scale factor. In this case, the scale factor is Applying the Transitive Property of Equality, an equation can be formed and simplified. It has been obtained that the two angles and the included side of are congruent to the corresponding two angles and the included side of

Therefore, by the Angle-Side-Angle (ASA) Congruence Theorem, the two triangles are congruent. Since congruent figures can be transformed into each other using rigid motions, and and are congruent triangles, there is a rigid motion placing onto

The combination of this rigid motion and the dilation performed earlier forms a similarity transformation that maps onto

Therefore, it can be concluded that and are similar triangles.

The proof is now complete.

Example

Solving Problems Using Angle-Angle-Similarity Theorem

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

Reapershadow.jpg

Hint

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

Solution

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

A right triangle with vertical leg marked as x. On the horizontal leg two segments are marked with 16 and 8. A vertical line of length 5 cuts the right triangle.

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 feet.

Two right triangles, one with legs 5 and 8, the other with legs x and 16+8=24.

Since the triangles are similar, the ratios between corresponding side lengths are the same. The solution of this equation defines the value of — the height of the street lamp.

Solve for

The street lamp at feet high towers over The Grimp Reaper.

Pop Quiz

Practice Solving Problems Using Similar Triangles

For the given diagram, find the missing length.

Discussion

Side-Side-Side Similarity Theorem

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.

Triangles ABC and DEF.

If then

Proof

Consider two triangles and whose corresponding sides are proportional.

Triangles ABC and DEF.

These triangles can be proven to be similar by identifying a similarity transformation that maps one triangle onto the other. First, can be dilated with the scale factor about forming the new triangle

Because dilation is a similarity transformation, it can be concluded that and are similar triangles. Now, it has to be proven that a rigid motion that maps onto exists. The ratios of the corresponding side lengths of similar polygons are the same and equal to the scale factor. In this case, the scale factor is Since all of the sides of and are proportional, the scale factor can be expressed by any of the following ratios. Applying the Transitive Property of Equality, three equations can be formed and simplified. These relations imply that the three sides of are congruent to the three sides of Therefore, by the Side-Side-Side (SSS) Congruence Theorem, the two triangles are congruent. Since congruent figures can be transformed into each other using rigid motions, and and are congruent triangles, there is a rigid motion placing onto

The combination of this rigid motion and the dilation performed earlier forms a similarity transformation that maps onto

Therefore, it can be concluded that and are similar triangles.

The proof is now complete.

Example

Solving Problems Using Similar Triangles

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

Triangle ACE and point B on AC and D on AE. Labelled distances: AB=459, BC=1275, AD=405, DB=360, DE=1125, BE=1275, EC=1360

Answer

The marked congruent angles are ABD, DBE, BEC, ECB

Hint

Look for similar triangles and an isosceles triangle.

Solution

Step 1

First, notice that segments and are equal in length.

Triangle BCE highlighted and the angles at C and E are marked as congruent

These are two sides of so by the Isosceles Triangle Theorem, the opposite angles are congruent.

Step 2

Two of the triangles, and look similar.

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

Ratio Expression Simplification

The last column of the table shows that the corresponding sides of and are proportional. According to the Side-Side-Side (SSS) Similarity Theorem, the two triangles are similar. Corresponding angles of similar triangles are congruent.

One pair of these angles is marked on the figure.

Step 3

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

Furthermore, since is a transversal to two parallel lines, the Alternate Interior Angles Theorem guarantees that the angles at and are congruent. These angles are marked on the figure.

Answering the question

The previous three steps showed three pairs of congruent angles. The transitive property of congruence shows that all four angles mentioned in these pairs are congruent to each other. The congruent angles are marked on the figure.

The marked congruent angles are ABD, DBE, BEC, ECB


Discussion

Side-Angle-Side Similarity Theorem

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.

Triangles ABC and DEF with congruent angles marked at A and D.

If and then

Proof

Consider two triangles and whose two pairs of corresponding sides are proportional and the included angles are congruent.

Triangles ABC and DEF.

These triangles can be proven to be similar by identifying a similarity transformation that maps one triangle onto the other. First, can be dilated with the scale factor about forming the new triangle

Because dilation is a similarity transformation, it can be concluded that and are similar triangles. Now, it has to be proven that a rigid motion that maps onto exists. The ratios of the corresponding side lengths of similar polygons are the same and equal to the scale factor. In this case, the scale factor is Since and are proportional to and respectively, the scale factor can be expressed by any of the following ratios. Applying the Transitive Property of Equality, three equations can be formed and simplified. These relations imply that the two sides of are congruent to the corresponding two sides of Moreover, the included angles and are also congruent.

Therefore, by the Side-Angle-Side (SAS) Congruence Theorem, the two triangles are congruent. Since congruent figures can be transformed into each other using rigid motions, and and are congruent triangles, there is a rigid motion placing onto

The combination of this rigid motion and the dilation performed earlier forms a similarity transformation that maps onto

Therefore, it can be concluded that and are similar triangles.

The proof is now complete.

Example

Proving Similarity Between Triangles Given Sides

The diagram shows the distances between points on a figure.

Triangle ADE with point C on AE and B on AD. The distances given are AB=7, BD=3, AC=5, CE=9, BC=3.

Show that and are similar triangles. Then find

Solution

Triangles and have a common angle at

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

Ratio Expression Simplified Form
The simplified forms of each ratio are the same. That means these two pairs of sides are proportional. Since the included angle is common in both triangles, the Side-Angle-Side (SAS) Similarity Theorem implies that the triangles are similar. 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. Since the length of is given in the diagram, this equation can be solved for
Solve for
The length of is units.

Closure

Applying Triangle Similarity Theorems to Solve Problems

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.

Regular pentagon ABCDE with all five diagonals drawn

Hint

Begin by determining the angle measures of the figure.

Solution

The Polygon Angle Sum Theorem identifies the sum of the interior angle measures of a pentagon. 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. Furthermore, since the sides of a regular pentagon are congruent, is an isosceles triangle.

Regular pentagon ABCDE with triangle ABE highlighted.
According to the Isosceles Triangle Theorem, the base angles of are congruent. Applying the information found so far to the Triangle Angle Sum Theorem, the measure of the base angles can be found.
Solve for
Due to the symmetry of the regular pentagon, there are ten angles with the same measure in the diagram.
Regular pentagon ABCDE with angles ABE, CBD, BCA, DCE, CDB, ADE, DEC, BEA, and EAD marked as 36 degrees.
Since the measure of is known, this gives the measure of
This information and the measures of the angles in symmetrical position can now be labeled in the diagram.
Regular pentagon ABCDE with angles ABE, CBD, BCA, DCE, CDB, ADE, DEC, BEA, EAD, CAD, DBE, ECA, ADB, and CEB marked as 36 degrees.

Next, focus on In this triangle, and are diagonals of the pentagon, and is a side.

The intersection of AC and BE are marked as F and triangles ACE and AFE are highlighted

In the diagram, a smaller triangle labeled is also present. These two triangles share a common angle at and congruent angles at and According to the Angle-Angle (AA) Similarity Theorem, that means the two triangles are similar. Therefore, the corresponding sides are proportional. Continuing forward, notice that triangles and are isosceles. Therefore, their legs have equal lengths. Using for the length of the sides, , as indicated on the figure. Also, using for the length of the diagonal, and

These expressions can be substituted in the proportionality relationship previously obtained.
The question asks for the ratio of to A new variable can be introduced for to represent this unknown ratio. The variable in the expression can then be replaced with The result can then be simplified.
Simplify
The next step is to solve this equation.
Solve for
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.

Extra

Construction of a regular pentagon

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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