Question

2 - 85. one possible proportion for the similar triangles below is $\frac{ac}{df}=\frac{ab}{de}$. write at least three more proportions given that $\triangle abcsim\triangle def$. homework help \\\\ 2 - 88. based on the measurements provided for each triangle below, determine whether the measure of x must be more than, less than, or equal to 45. homework help \\\\ a. \\\\ 3 \\\\ 4 \\\\ b. \\\\ 18 \\\\ 18 \\\\ x

Explanation:

Step1: Recall property of similar - triangles

For similar triangles $\triangle ABC\sim\triangle DEF$, the ratios of corresponding sides are equal.

Step2: Write first proportion

Since $\triangle ABC\sim\triangle DEF$, we have $\frac{AB}{DE}=\frac{AC}{DF}$.

Step3: Write second proportion

Also, $\frac{AB}{DE}=\frac{BC}{EF}$.

Step4: Write third proportion

And $\frac{BC}{EF}=\frac{AC}{DF}$.

Step5: Analyze triangle in part a

In a right - triangle with legs 3 and 4, the tangent of an acute angle $\theta$ is $\tan\theta=\frac{3}{4}< 1$. Since $\tan45^{\circ}=1$, the angle $x$ is less than $45^{\circ}$.

Step6: Analyze triangle in part b

In a right - triangle with legs 18 and 18, the tangent of an acute angle $x$ is $\tan x = 1$. So $x = 45^{\circ}$.

Answer:

For similar triangles: $\frac{AB}{DE}=\frac{AC}{DF}$, $\frac{AB}{DE}=\frac{BC}{EF}$, $\frac{BC}{EF}=\frac{AC}{DF}$
For part a: $x<45^{\circ}$
For part b: $x = 45^{\circ}$