Solved Examples: Using SSS Similarity to Prove Triangles

📚 Quick Study Guide

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• SSS (Side-Side-Side) Similarity Theorem: If all three sides of one triangle are proportional to the corresponding three sides of another triangle, then the two triangles are similar.
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• Proportionality: Sides are proportional if their ratios are equal. For example, if $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$, then $\triangle ABC \sim \triangle DEF$.
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• Symbol for Similarity: The symbol $\sim$ means 'is similar to'. So, $\triangle ABC \sim \triangle DEF$ is read as 'Triangle ABC is similar to triangle DEF'.
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• Checking for Similarity: To prove similarity using SSS, calculate the ratios of corresponding sides. If all the ratios are equal, the triangles are similar.
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• Important Note: Make sure you match up corresponding sides correctly! The smallest side of one triangle corresponds to the smallest side of the other, and so on.

🧪 Practice Quiz

1. Given $\triangle ABC$ with sides $AB = 4$, $BC = 6$, $CA = 8$ and $\triangle DEF$ with sides $DE = 6$, $EF = 9$, $FD = 12$. Are the triangles similar by SSS similarity?

   1. Yes, $\triangle ABC \sim \triangle DEF$
   2. No, they are not similar
   3. Cannot be determined
   4. Only if the angles are equal

2. In $\triangle PQR$, $PQ = 5$, $QR = 7$, $RP = 10$. In $\triangle XYZ$, $XY = 2.5$, $YZ = 3.5$, $ZX = 5$. Are the triangles similar?

   1. Yes, by SSS similarity
   2. No, the sides are not proportional
   3. Only if the corresponding angles are also equal.
   4. Cannot determine from the information given.

3. If $\triangle LMN$ has sides $LM = 3$, $MN = 5$, $NL = 6$, and $\triangle STU$ has sides $ST = 9$, $TU = 15$, $US = 18$, are they similar?

   1. Yes, $\triangle LMN \sim \triangle STU$
   2. No, they are not similar
   3. Only if the angles are known
   4. Similar only if the area is the same

4. $\triangle ABC$ has sides $AB=2$, $BC=3$, and $CA=4$. $\triangle DEF$ has sides $DE=4$, $EF=6$, and $FD=7$. Are these triangles similar by SSS?

   1. Yes
   2. No
   3. Cannot be determined
   4. Only if they are congruent

5. Two triangles have sides of length 5, 12, and 13. Are they similar by SSS?

   1. Yes, all such triangles are similar.
   2. No, similarity cannot be determined from side lengths alone.
   3. Yes, if both are right triangles.
   4. Not enough information is given.

6. $\triangle GHI$ has sides $GH = 8$, $HI = 10$, and $IG = 12$. $\triangle JKL$ has sides $JK = 4$, $KL = 5$, and $LJ = 6$. Are the triangles similar?

   1. Yes, by SSS similarity
   2. No, because the order of sides is reversed
   3. Not enough information.
   4. Only if the area is the same

7. Given $\triangle UVW$ with $UV = 2$, $VW = 4$, $WU = 5$ and $\triangle XYZ$ with $XY = 6$, $YZ = 12$, $ZX = 15$. Are the triangles similar?

   1. Yes, by SSS similarity.
   2. No, they are not similar.
   3. Only if the angles are congruent.
   4. Cannot be determined.