4.2: Similar Triangles
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Two triangles are said to be similar if they have equal sets of angles. In Figure 4.2.1, △ABC is similar to △DEF. The angles which are equal are called corresponding angles. In Figure 4.2.1, ∠A corresponds to ∠D, ∠B corresponds to ∠E, and ∠C corresponds to ∠F. The sides joining corresponding vertices are called corresponding sides. In Figure 4.2.1, AB corresponds to DE, BC corresponds to EF, and AC corresponds to DF. The symbol for similar is ∼. The similarity statement △ABC∼△DEF will always be written so that corresponding vertices appear in the same order.
For the triangles in Figure 4.2.1, we could also write △BAC∼△BDF or △ACB∼△DFE but never △ABC∼△EDF nor △ACB∼△DEF.


We can tell which sides correspond from the similarity statement. For example, if △ABC∼△DEF, then side AB corresponds to side DE because both are the first two letters. BC corresponds to EF because both are the last two letters, AC corresponds to DF because both consist of the first and last letters.
Determine if the triangles are similar, and if so, write the similarity statement:


Solution
∠C=180∘−(65∘+45∘)=180∘−110∘=70∘
∠D=180∘−(65∘+45∘)=180∘−110∘=70∘
Therefore both triangles have the same angles and △ABC∼△EFD.
Answer: △ABC∼△EFD.
Example A suggests that to prove similarity it is only necessary to know that two of the corresponding angles are equal:
Two triangles are similar if two angles of one equal two angles of the other (AA=AA).
In Figure 4.2.2, △ABC∼△DEF because ∠A=∠D and ∠B=∠E.


- Proof
-
△C=180∘−(∠A+∠B)=180∘−(∠D+∠E)=∠F.
Determine which triangles are similar and write a similarity statement:
Solution
∠A=∠CDE because they are corresponding angles of parallel lines. ∠C=∠C because of identity. Therefore △ABC∼△DEC by AA=AA.
Answer: △ABC∼△DEC.
Determine which triangles are similar and write a similarity statement:
Solution
∠A=∠A identity. ∠ACB=∠ADC=90∘. Therefore
Also ∠B=∠B, identity, ∠BDC=∠BCA=90∘. Therefore
Answer: △ABC∼△ACD∼△CBD.
Similar triangIes are important because of the following theorem:
The corresponding sides of similar triangles are proportional. This means that if △ABC∼△DEF then
ABDE=BCEF=ACDF.
That is, the first two letters of △ABC are to the first two letters of △DEF as the last two letters of △ABC are to the last two letters of △DEF as the first and last letters of △ABC are to the first and last letters of △DEF.
Before attempting to prove Theorem 4.2.2, we will give several examples of how it is used:
Find x:
Solution
∠A=∠D and ∠B=∠E so △ABC∼△DEF. By Theorem 4.2.2,
ABDE=BCEF=ACDF.
We will ignore ABDE here since we do not know and do not have to find either AB or DE.
BCEF=ACDF8x=2324=2x12=x
Check:
Answer: x=12.
Find x:
Solution
∠A=∠A,∠ADE=∠ABC, so △ADE∼△ABC by AA=AA.
ADAB=DEBC=AEAC.
We ignore ADAB.
DEBC=AEAC515=1010+x5(10+x)=15(10)50+5x=1505x=150−505x=100x=20
Check:
Answer: x=20.
Find x:
Solution
∠A=∠CDE because they are corresponding angles of parallel lines. ∠C=∠C because of identity. Therefore △ABC∼△DEC by AA=AA.
ABDE=BCEC=ACDC
We ignore BCEC:
ABDE=ACDCx+54=x+33(x+5)(3)=(4)(x+3)3x+15=4x+1215−12=4x−3x3=x
Check:
Answer: x=3.
Find x:
Solution
∠A=∠A, ∠ACB=∠ADC=90∘, △ABC∼△ACD.
ABAC=ACADx+128=8x(x+12)(x)=(8)(8)x2+12x=64x2+12x−64=0(x−4)(x+16)=0x=4 x=−16
We reject the answer x=−16 because AD=x cannot be negative.
Check, x=4
Answer: x=4.
A tree casts a shadow 12 feet long at the same time a 6 foot man casts a shadow 4 feet long. What is the height of the tree?
Solution
In the diagram AB and DE are parallel rays of the sun. Therefore ∠A=∠D because they are corresponding angles of parallel lines with respect to the transversal AF. Since also ∠C=∠F=90∘, we have △ABC∼△DEF by AA=AA.
ACDF=BCEF412=6x4x=72x=18
Answer: x=18 feet.
Proof of Theorem 4.2.2 ("The corresponding sides of similar triangles are proportional"):
We illustrate the proof using the triangles of Example 4.2.4 (Figure 4.2.3). The proof for other similar triangles follows the same pattern. Here we will prove that x=12 so that 23=8x.


First draw lines parallel to the sides of △ABC and △DEF as shown in Figure 4.2.4. The corresponding angles of these parallel lines are equal and each of the parallelograms with a side equal to 1 has its opposite side equal to 1 as well, Therefore all of the small triangles with a side equal to 1 are congruent by AAS=AAS. The corresponding sides of these triangles form side BC=8 of △ABC (see Figure 4.2.5). Therefore each of these sides must equal 4 and x=EF=4+4+4=12 (Figure 4.2.6).


(Note to instructor: This proof can be carried out whenever the lengths of the sides of the triangles are rational numbers. However, since irrational numbers can be approximated as closely as necessary by rationals, the proof extends to that case as well.)
Thales (c. 600 B.C.) used the proportionality of sides of similar triangles to measure the heights of the pyramids in Egypt. His method was much like the one we used in Example 4.2.8 to measure the height of trees.

In Figure 4.2.7, DE represent the height of the pyramid and CE is the length of its shadow. BC represents a vertical stick and AC is the length of its shadow. We have △ABC∼△CDE. Thales was able to measure directly the lengths AC,BC, and CE. Substituting these values in the proportion BCDE=ACCE, he was able to find the height DE.
Problems
1 - 6. Determine which triangles are similar and write the similarity statement:
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2.
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7 - 22. For each of the following
(1) write the similarity statement
(2) write the proportion between the corresponding sides
(3) solve for x or x and y.
7.
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23. A flagpole casts a shadow 80 feet long at the same time a 5 foot boy casts a shadow 4 feet long. How tall is the flagpole?
24. Find the width AB of the river: