Identify the similar triangles and find x. Then find the measures of the indicated sides.

Answer:The similar triangles are Δ KMJ and Δ NML The value of x is 3KM = 6 and NM = 3Step-by-step explanation:* Lets revise the cases of similarity1) AAA similarity : two triangles are similar if all three angles in the first   triangle equal the corresponding angle in the second triangle  - Example : In ΔABC and ΔDEF, m∠A = m∠D, m∠B = m∠E and    m∠C= m∠F then ΔABC ≈ ΔDEF by AAA  2) AA similarity : If two angles of one triangle are equal to the    corresponding angles of the other triangle, then the two triangles      are similar.- Example : In ΔPQR and ΔDEF, m∠P = m∠D, m∠R = m∠F then     ΔPQR ≈ ΔDEF by AA  3) SSS similarity : If the corresponding sides of two triangles are    proportional, then the two triangles are similar.- Example : In ΔXYZ and ΔLMN, if     then the two triangles are similar by SSS  4) SAS similarity : In two triangles, if two sets of corresponding sides      are proportional and the included angles are equal then the two      triangles are similar.- Example : In triangle ABC and DEF, if m∠A = m∠D and     then the two triangles are similar by SAS* Now lets solve the problem- ∠KMJ is a aright angle and M is on JL∴ m∠JML = 180° ⇒ straight angle∵ m∠JMK + m∠LMN = m∠JML∴ 90° + m∠NML = 180° ⇒ subtract 90° from both sides∴ m∠NML = 90°- In Δ KMJ and ΔNML∵ m∠KMJ = m∠NML ⇒ proved∵ m∠KJM = m∠NLM ⇒ given- By using the second case above (AA similarity)∴ Δ KMJ ≈ Δ NML * The similar triangles are Δ KMJ and Δ NML - From similarity∴ Their sides are proportion∴ [tex]\frac{KM}{NM}=\frac{MJ}{ML}=\frac{KJ}{NL}[/tex]∵ KJ = 10 and NL = 5∵ KM = 3 + x and NM = x- Substitute these values in the proportion relation∵ [tex]\frac{KM}{NM}=\frac{KJ}{NL}[/tex]∴ [tex]\frac{3+x}{x}=\frac{10}{5}[/tex]- By using cross multiplication∴ 5(3 + x) = 10(x) ⇒ simplify∴ 5(3) + 5(x) = 10x∴ 15 + 5x = 10x ⇒ subtract 5x from both sides∴ 15 = 5x ⇒ divide both sides by 5∴ 3 = x* The value of x is 3∵ KM = 3 + x∵ x = 3∴ KM = 3 + 3 = 6∵ NM = x∴ NM = 3* KM = 6 and NM = 3- Check the ratio∵ KM/NM = 6/3 = 2∵ KJ/NL = 10/5 = 2∴ The sides are proportion