Question

1. which is the inverse of the statement \if a triangle is equilateral, then it is equiangular.\

a. if a triangle is not equilateral, then it is not equiangular.
b. if a triangle is not equiangular, then it is not equilateral.
c. if a triangle is equiangular, then it is equilateral.
d. if a triangle is not equilateral, then it is equiangular.

1. which conclusion is not justified by the prior g

a. given: if a line is vertical, then it has an u
given: line l is vertical.
conclusion: line l has an undefined slop
b. given: if a triangle is isosceles, then it ha
given: if a triangle has two congruent si
conclusion: if a triangle is isosceles, then
c. given: if two angles are vertical, then
given: ∠1 and ∠2 are vertical angles
conclusion: ∠1 ≅ ∠2

Explanation:

Question 17

Step1: Recall Inverse Definition

The inverse of a conditional statement "If \( p \), then \( q \)" is "If not \( p \), then not \( q \)". Here, \( p \): "a triangle is equilateral", \( q \): "it is equiangular".

Step2: Apply to Options

• Option A: "If not \( p \) (not equilateral), then not \( q \) (not equiangular)" – matches inverse definition.
• Option B: "If not \( q \), then not \( p \)" – this is the contrapositive.
• Option C: "If \( q \), then \( p \)" – this is the converse.
• Option D: "If not \( p \), then \( q \)" – does not match inverse.

(Assuming) For a conclusion to be valid, it should follow logical rules (like modus ponens: if \( p
ightarrow q \) and \( p \), then \( q \); transitivity: if \( p
ightarrow q \) and \( q
ightarrow r \), then \( p
ightarrow r \)). If Option B has a structure that does not follow these (e.g., incorrect transitivity or non - matching premises), it's invalid.

Answer:

A. If a triangle is not equilateral, then it is not equiangular.

Question 19 (Note: Some text is obscured, but assuming a typical logic problem):

(Assuming the question is about valid conclusions from logical implications, e.g., modus ponens, transitivity, etc. Since text is partial, but if we assume Option B has an invalid chain or non - applicable logic):