Question

which conclusion can be drawn based on the true statements shown?
if a triangle is equilateral, then all the sides in the triangle are congruent.
if all the sides in a triangle are congruent, then all the angles are congruent.
based on the law of syllogism, if all the angles in a triangle are congruent, then the triangle is equilateral.
based on the law of detachment, if all the angles in a triangle are congruent, then the triangle is equilateral.
based on the law of syllogism, if a triangle is equilateral, then all the angles are congruent.
based on the law of detachment, if a triangle is equilateral, then all the angles are congruent.

Explanation:

The law of syllogism states that if \(p
ightarrow q\) and \(q
ightarrow r\), then \(p
ightarrow r\). Here, \(p\) is "a triangle is equilateral", \(q\) is "all the sides in the triangle are congruent", and \(r\) is "all the angles are congruent". So, combining the two given statements (\(p
ightarrow q\) and \(q
ightarrow r\)) using the law of syllogism gives \(p
ightarrow r\), which is "if a triangle is equilateral, then all the angles are congruent".
The law of detachment requires a statement \(p
ightarrow q\) and the fact that \(p\) is true to conclude \(q\). But the problem is about combining two conditional statements, not applying detachment. Also, the first two options reverse the order of the implications wrongly.

Answer:

Based on the law of syllogism, if a triangle is equilateral, then all the angles are congruent.