Question

△acf and △vxz are shown. ac = 10, cf = 12, af = 20, vx = 9, xz = 7.5, zv = 15, m∠acf = 131°, m∠xzv = 27°, and m∠zvx = 22°. 6. what similarity theorem or postulate could be used to prove the triangles are similar? 7. write a similarity statement for △acf and △vxz. complete the table. 8. m∠caf = ac/zv = 9. m∠cfa = cf/xv = 10. m∠zxv = fa/vz =

Explanation:

Step1: Find the similarity theorem

• Calculate the ratios of the corresponding sides:
• \(\frac{AC}{ZX}=\frac{10}{7.5}=\frac{4}{3}\), \(\frac{CF}{XV}=\frac{12}{9}=\frac{4}{3}\), \(\frac{AF}{VZ}=\frac{20}{15}=\frac{4}{3}\)
• Since the ratios of the corresponding sides are equal (\(\frac{AC}{ZX}=\frac{CF}{XV}=\frac{AF}{VZ}\)), by the Side - Side - Side (SSS) similarity theorem, \(\triangle ACF\sim\triangle VXZ\)

Step2: Write the similarity statement

• Corresponding vertices: \(A\) corresponds to \(V\), \(C\) corresponds to \(X\), \(F\) corresponds to \(Z\)
• So the similarity statement is \(\triangle ACF\sim\triangle VXZ\)

Step3: Find \(m\angle CAF\)

• In \(\triangle VXZ\), \(m\angle VXZ = 180^{\circ}-27^{\circ}-22^{\circ}=131^{\circ}\)
• Since \(\triangle ACF\sim\triangle VXZ\), \(m\angle CAF=m\angle VXZ = 22^{\circ}\)

Step4: Find \(m\angle CFA\)

• Since \(\triangle ACF\sim\triangle VXZ\), \(m\angle CFA=m\angle XZV = 27^{\circ}\)

Step5: Find \(m\angle ZXV\)

• Since \(\triangle ACF\sim\triangle VXZ\), \(m\angle ZXV=m\angle ACF = 131^{\circ}\)

Step6: Calculate the side proportions

• \(\frac{AC}{ZX}=\frac{10}{7.5}=\frac{4}{3}\)
• \(\frac{CF}{XV}=\frac{12}{9}=\frac{4}{3}\)
• \(\frac{FA}{VZ}=\frac{20}{15}=\frac{4}{3}\)

Answer:

1. SSS (Side - Side - Side) similarity theorem
2. \(\triangle ACF\sim\triangle VXZ\)
3. \(m\angle CAF = 22^{\circ}\), \(\frac{AC}{ZX}=\frac{4}{3}\)
4. \(m\angle CFA = 27^{\circ}\), \(\frac{CF}{XV}=\frac{4}{3}\)
5. \(m\angle ZXV = 131^{\circ}\), \(\frac{FA}{VZ}=\frac{4}{3}\)