Question

(c) problem 11: (first taught in lesson 48) read the true statement below and then tell whether the converse, inverse, and contrapositive are also true. if a triangle is not equilateral, then it is not equiangular. converse inverse contrapositive if a triangle is not equiangular, then it is not equilateral. true false

Explanation:

Step1: Recall definitions

Let \(p\): a triangle is not equilateral, \(q\): a triangle is not equiangular. The original statement is \(p
ightarrow q\).
The converse is \(q
ightarrow p\) (If a triangle is not equiangular, then it is not equilateral).
The inverse is \(
eg p
ightarrow
eg q\) (If a triangle is equilateral, then it is equiangular).
The contrapositive is \(
eg q
ightarrow
eg p\) (If a triangle is equiangular, then it is equilateral).

Step2: Analyze the converse

An equilateral triangle is equiangular (\(60^{\circ}\) each angle). But a triangle that is not equiangular (e.g., a triangle with angles \(30^{\circ},60^{\circ},90^{\circ}\)) is not equilateral. So the converse is True.

Step3: Analyze the inverse

An equilateral triangle has all sides equal. By the property of triangles (equal sides correspond to equal angles), if a triangle is equilateral (\(
eg p\)), then it is equiangular (\(
eg q\)). So the inverse is True.

Step4: Analyze the contrapositive

Since the original statement \(p
ightarrow q\) and its contrapositive \(
eg q
ightarrow
eg p\) are logically equivalent. The contrapositive is True.

Answer:

Converse: True; Inverse: True; Contrapositive: True