Question

question 2(multiple choice worth 1 points) (04.01 mc) which characteristics will prove that △def is an acute, isosceles triangle? de and ef are congruent to each other but not to df, and their slopes are not related. de and ef are congruent, and their slopes are opposite reciprocals. de is larger than ef, and their slopes are not related. de is larger than ef, and their slopes are opposite reciprocals.

Explanation:

Step1: Recall isosceles - triangle property

An isosceles triangle has two congruent sides. So, for $\triangle DEF$ to be isosceles, two of its sides should be congruent. Options with one side larger than the other (C and D) are incorrect as they don't meet the isosceles - triangle side - length requirement.

Step2: Recall acute - triangle property

If the slopes of two sides are opposite reciprocals, the angle between them is a right - angle, and the triangle is a right - triangle, not an acute - triangle. So, option B is incorrect as it would make $\triangle DEF$ a right - isosceles triangle. Option A satisfies the condition of having two congruent sides (isosceles) and non - perpendicular sides (acute) since the slopes are not related.

Answer:

A. $\overline{DE}$ and $\overline{EF}$ are congruent to each other but not to $\overline{DF}$, and their slopes are not related.