Respuesta :

Triangles are similar if they have two congruent angles or two sides of one are proportional to two sides of the other and the included angle are equal.

Correct responses:

9. ΔSUT ~ ΔWUV by AA similarity postulate

10. ΔKDH ~ ΔABD by SAS similarity

11. ΔKJL and ΔGJH are not similar

12. ΔSRT ~ ΔPRQ by SAS similarity

13. ΔAEB ~ ΔCED are similar by SAS similarity

14. ΔPQR and ΔPST are not similar

15. ΔMBR ~ ΔLPZ are similar by AA similarity postulate

16. ΔLNK ~ ΔJNM are similar by SAS similarity

Methods used to determine the similarities of the given triangles

9. The given properties from the diagram are;

∠STU = ∠WVU = 72°

∠STU ≅ ∠WVU by definition of congruency

∠SUT ≅ ∠WUV by vertical angles theorem

  • ΔSUT is similar to ΔWUV by Angle-Angle, AA, similarity postulate.

[tex]\displaystyle 10. \hspace{0.3 cm} \frac{\overline{AD}}{\overline{DH}} = \frac{21}{15} = \mathbf{ \frac{7}{5}}[/tex]

[tex]\displaystyle \frac{\overline{AB}}{\overline{KH}} = \frac{28}{20} = \frac{7}{5}[/tex]

Therefore, given that the two sides of triangle ΔKDH are proportional to  two sides on ΔABD and the included angles are equal, therefore;

  • ΔKDH is similar to ΔABD by Side-Angle-Side SAS similarity.

[tex]\displaystyle 11. \hspace{0.3 cm} \frac{\overline{LJ}}{\overline{JH}} = \frac{95}{72} \neq \frac{87}{68} = \mathbf{ \frac{\overline{KJ}}{\overline{GJ}}}[/tex]

ΔKJL and ΔGJH are not similar.

[tex]\displaystyle 12. \hspace{0.3 cm} \frac{\overline{PR}}{\overline{SR}} = \frac{49}{35} = \frac{7}{5} = \frac{42}{30} = \mathbf{\frac{\overline{QR}}{\overline{TR}}}[/tex]

∠PRQ ≅ ∠SRT

Therefore;

  • ΔSRT is similar to ΔPRQ by SAS similarity postulate.

[tex]\displaystyle 13. \hspace{0.3 cm} \frac{\overline{BE}}{\overline{DE}} = \frac{42}{32} =\frac{21}{16} = \mathbf{\frac{\overline{AE}}{\overline{CE}}}[/tex]

∠CED ≅ ∠BEA

Therefore;

  • ΔAEB is similar to ΔCED by SAS similarity postulate.

14. ∠PQR 101° ≠ 102° = ∠STP

Therefore;

ΔPQR and ΔPST are not similar.

15. Given that ΔMBR and ΔPLZ are isosceles triangles.

∠MBR and ∠LPZ are the base angles of the isosceles triangles ΔMBR and ΔPLZ respectively

∠MBR = ∠MRB by base angles of isosceles triangle ΔMBR

∠LPZ = ∠LZP

∠MBR ≅ ∠LPZ given

∠MBR = ∠LPZ definition of congruency

∠MRB = ∠LZP by transitive property

Therefore;

  • ΔMBR is similar to ΔLPZ by Angle-Angle A-A similarity postulate.

16. [tex]\overline{JN}[/tex] = √(30.6² - 27²) = 14.4

[tex]\displaystyle \frac{\overline{MN}}{\overline{KN}} = \frac{27}{15} = \frac{9}{5} = 1.8 = \frac{14.4}{8} = \mathbf{ \frac{\overline{JN}}{\overline{NL}}}[/tex]

∠LNK = ∠JNM = 90° by vertical angles theorem.

  • ΔLNK and ΔJNM are similar by SAS similarity postulate.

Learn more about similar triangles here:

https://brainly.com/question/2410410

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