Question

determine whether each conclusion is valid based on the given information. write valid or invalid. explain your reasoning.

1. given: right angles are congruent. ∠1 and ∠2 are right angles.

conclusion: ∠1 ≅ ∠2

1. given: if a figure is a square, then it has four right angles. figure abcd has four right angles.

conclusion: figure abcd is a square

1. given: if you leave your lights on while your car is off, then your battery is dead. your battery is dead.

conclusion: you left your lights on while your car was off

1. given: if dennis gets a part - time job, then he can afford a car payment. dennis can afford a car payment.

conclusion: dennis got a part - time job

1. given: if 75% of the prom tickets are sold, then the prom will be held at the country club. 75% of the prom tickets were sold.

conclusion: the prom will be held at the country club

Explanation:

To solve these problems, we analyze each one using deductive reasoning (checking if the conclusion follows logically from the given information):

Problem 7

Given: Right angles are congruent. $\angle 1$ and $\angle 2$ are right angles.
Conclusion: $\angle 1 \cong \angle 2$

Step 1: Identify the general rule

The given general rule is: All right angles are congruent.

Step 2: Apply the rule to the specific case

$\angle 1$ and $\angle 2$ are both right angles (specific case). By the general rule, they must be congruent.

Answer:

**: Valid

Problem 8

Given: If a figure is a square, then it has four right angles. Figure $ABCD$ has four right angles.
Conclusion: Figure $ABCD$ is a square

Step 1: Analyze the logical structure

The general rule is: Square $\implies$ Four right angles (if $P$, then $Q$). The conclusion assumes: Four right angles $\implies$ Square (if $Q$, then $P$).

Step 2: Check for logical validity

The converse of a true statement is not always true. A rectangle (e.g., a non - square rectangle) also has four right angles but is not a square. So the conclusion does not follow.