Respuesta :
Using the triangle similarity theorems, the given pairs of triangles can be proven to be similar:
1. [tex]\triangle STR $ and $ \triangle PQN[/tex] are similar by the SSS similarity theorem
2. [tex]\triangle EFG $ and $ \triangle JHG[/tex] are similar by the AA similarity theorem.
3. [tex]\triangle XYZ $ and $ \triangle XWT[/tex] are similar by SAS similarity theorem.
4. [tex]\triangle ANF $ and $ \triangle SEH[/tex] are similar by the AA similarity theorem.
5. [tex]\triangle RNQ $ and $ \triangle MNL[/tex] are similar by the SAS similarity theorem.
6. [tex]\triangle ECD $ and $ \triangle LMN[/tex] are similar by the SSS similarity theorem.
7. [tex]\triangle AEB $ and $ \triangle BDC[/tex] are not similar.
8. [tex]\triangle JLN $ and $ \triangle KLM[/tex] are similar by the SAS similarity theorem.
9. [tex]\triangle SUT $ and $ \triangle WUV[/tex] are similar by the AA similarity theorem.
10. [tex]\triangle DHK $ and $ \triangle BAE[/tex] are similar by the SAS similarity theorem.
The following are triangle similarity theorems that can be used to prove that two triangles are similar if we don't have all the information about the two triangles:
- Angle-Angle Similarity Theorem (AA~): two triangles having two congruent triangles that corresponds to each other are similar.
- Side-Side-Side Similarity Theorem (SSS~): Two triangles having all three corresponding congruent sides proportional are similar.
- Side-Angle-Side Similarity Theorem (SAS~): Two triangles having two corresponding congruent angles and a corresponding included congruent angle in both triangles are similar.
Using the above stated similarity theorems, let's prove the given pairs of triangles are similar or not:
1. [tex]\frac{ST}{PQ} = \frac{SR}{NP} = \frac{TR}{NQ}[/tex]
Plug in the values
[tex]\frac{55}{25} = \frac{44}{20} = \frac{37.4}{17} = 2.2[/tex] (All three sides of one triangle are proportional to all three sides of the other triangle).
Therefore, [tex]\triangle STR $ and $ \triangle PQN[/tex] are similar by the SSS similarity theorem.
2. <F in [tex]\triangle EFG[/tex] is congruent to <H in [tex]\triangle JHG[/tex]
<EGF in [tex]\triangle EFG[/tex] is congruent to <JGH in [tex]\triangle JHG[/tex]
Therefore, [tex]\triangle EFG $ and $ \triangle JHG[/tex] are similar by the AA similarity theorem.
3.<YXZ in [tex]\triangle XYZ[/tex] is congruent to <WXT in [tex]\triangle XWT[/tex]
[tex]\frac{XY}{XW} = \frac{XZ}{XT}[/tex]
Substitute
[tex]\frac{35}{28} = \frac{38}{30} = 1.3[/tex]
This implies that the two corresponding sides of [tex]\triangle XYZ $ and $ \triangle XWT[/tex] are proportional to each other. Also the included angles are congruent.
Therefore, [tex]\triangle XYZ $ and $ \triangle XWT[/tex] are similar by SAS similarity theorem.
4. <A = <S = 29 degrees
<N = <E = 106 degrees (Note: <E = 180 - 29 - 45)
Therefore, [tex]\triangle ANF $ and $ \triangle SEH[/tex] are similar by the AA similarity theorem.
5. <RNQ in [tex]\triangle RNQ[/tex] is congruent to <MNL in [tex]\triangle MNL[/tex] (an included angle is congruent to each other)
[tex]\frac{MN}{NR} = \frac{LN}{NQ} = 1.25[/tex] (two sides of both triangles are proportional)
Therefore, [tex]\triangle RNQ $ and $ \triangle MNL[/tex] are similar by the SAS similarity theorem.
6. [tex]\frac{EC}{LM} = \frac{ED}{LN} = \frac{CD}{MN} = 1.8[/tex] (three sides of both triangles are proportional).
Therefore, [tex]\triangle ECD $ and $ \triangle LMN[/tex] are similar by the SSS similarity theorem.
7. The information given is not enough to prove that the two triangles are similar by any of the similarity theorems.
Therefore, [tex]\triangle AEB $ and $ \triangle BDC[/tex] are not similar.
8. <JLN in [tex]\triangle JLN[/tex] is congruent to <KLM in [tex]\triangle KLM[/tex] (an included angle is congruent to each other)
[tex]\frac{JL}{KL} = \frac{NL}{ML} = 1.4[/tex] (two sides of both triangles are proportional)
Therefore, [tex]\triangle JLN $ and $ \triangle KLM[/tex] are similar by the SAS similarity theorem.
9. <T = <V = 72 degrees
<SUT = <WUV (Vertical angles)
Therefore, [tex]\triangle SUT $ and $ \triangle WUV[/tex] are similar by the AA similarity theorem.
10. <H = <A = 38 degrees (included angles)
[tex]\frac{AB}{HK} = \frac{AE}{HD} = 1.4[/tex] (two sides of both triangles are proportional)
Therefore, [tex]\triangle DHK $ and $ \triangle BAE[/tex] are similar by the SAS similarity theorem.
In summary, using the triangle similarity theorems, the given pairs of triangles can be proven to be similar:
1. similar by the SSS similarity theorem
2. similar by the AA similarity theorem.
3. similar by SAS similarity theorem.
4. similar by the AA similarity theorem.
5. similar by the SAS similarity theorem.
6. similar by the SSS similarity theorem.
7. not similar.
8. similar by the SAS similarity theorem.
9. similar by the AA similarity theorem.
10. similar by the SAS similarity theorem.
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