Question

among the following conditions, the one that can make △abc and △def similar is
a. m∠c = 98°, m∠e = 98°, (\frac{ac}{bc}=\frac{de}{df})
b. ab = 1, ac = 1.5, bc = 2, ef = 8, de = 10, fd = 6
c. m∠a = m∠f = 90°, ac = 5, bc = 13, df = 10, ef = 26
d. m∠a = 46°, m∠b = 54°, m∠e = 54°, m∠f = 80°

Explanation:

Step1: Recall similarity - criteria

For two triangles to be similar, we can use AA (angle - angle), SAS (side - angle - side), SSS (side - side - side) criteria.

Step2: Analyze option A

In option A, $\frac{AC}{BC}=\frac{DE}{DF}$, but the equal angles $\angle C$ and $\angle E$ are not the included angles of the corresponding sides, so we cannot use SAS.

Step3: Analyze option B

Calculate the ratios of corresponding sides: $\frac{AB}{FD}=\frac{1}{6}$, $\frac{AC}{DE}=\frac{1.5}{10}=\frac{3}{20}$, $\frac{BC}{EF}=\frac{2}{8}=\frac{1}{4}$. The ratios of corresponding sides are not equal, so the triangles are not similar by SSS.

Step4: Analyze option C

In right - triangle $ABC$ with $\angle A = 90^{\circ}$, $AB=\sqrt{BC^{2}-AC^{2}}=\sqrt{13^{2}-5^{2}} = 12$ (by the Pythagorean theorem). In right - triangle $DEF$ with $\angle F=90^{\circ}$, $DE=\sqrt{EF^{2}-DF^{2}}=\sqrt{26^{2}-10^{2}} = 24$.
The ratios of corresponding sides are: $\frac{AC}{DF}=\frac{5}{10}=\frac{1}{2}$, $\frac{AB}{DE}=\frac{12}{24}=\frac{1}{2}$, $\frac{BC}{EF}=\frac{13}{26}=\frac{1}{2}$. Since the ratios of all three pairs of corresponding sides are equal, $\triangle ABC\sim\triangle DEF$ by SSS.

Step5: Analyze option D

In $\triangle ABC$, $\angle C=180^{\circ}-\angle A - \angle B=180^{\circ}-46^{\circ}-54^{\circ}=80^{\circ}$. In $\triangle DEF$, $\angle D=180^{\circ}-\angle E-\angle F=180^{\circ}-54^{\circ}-80^{\circ}=46^{\circ}$. Although there are some equal angles, the order of correspondence is not correct for AA similarity.

Answer:

C. $m\angle A = m\angle F = 90^{\circ},AC = 5,BC = 13,DF = 10,EF = 26$