High School

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Given:

- If a triangle is isosceles, then it has two congruent sides.
- If a triangle has two congruent sides, then it has two congruent angles.
- Triangle \( \triangle KLM \) is isosceles.

Select the valid conclusion(s) from the given statements. Select all that apply.

a. \( \triangle KLM \) has two congruent angles.
b. \( \triangle KLM \) has three congruent angles.
c. \( \triangle KLM \) has three congruent sides.
d. \( \triangle KLM \) is equilateral.

Answer :

Final answer:

The valid conclusion from the statements provided, considering that triangle ΔKLM is isosceles, is that it has two congruent angles. This is based on the property that in an isosceles triangle, the angles opposite the equal sides are equal. The conclusion focused solely on option a being correct.( Option A is the correct answer)

Explanation:

The question revolves around the properties of an isosceles triangle and the conclusions we can draw from knowing that triangle ΔKLM is isosceles. According to the given information, if a triangle is isosceles, then it has two congruent sides. Furthermore, if a triangle has two congruent sides, then it also has two congruent angles. Since ΔKLM is isosceles, it naturally follows that ΔKLM has two congruent sides. Based on Theorem 4, in an isosceles triangle, the angles opposite the equal sides are equal. Therefore, we can conclude option a: Triangle ΔKLM has two congruent angles. However, the other options b, c, and d are not necessarily valid based on the given information. An isosceles triangle does not necessarily have all three angles congruent (which would make it equiangular), nor does it have all three sides congruent (which would make it equilateral). Therefore, the valid conclusion from the given statements is only option a.

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