Question

1. what is a valid conclusion that can be drawn from these statements?

if a quadrilateral is a rhombus, then it is a parallelogram.
if a quadrilateral is a parallelogram, then its opposite angles are congruent.
a. every quadrilateral is a rhombus.
b. every parallelogram is a rhombus.
c. opposite angles of a rhombus are congruent.
d. opposite angles of a quadrilateral are congruent.

Explanation:

Step1: Analyze the statements

We have two conditional statements:

1. If a quadrilateral is a rhombus (\(p\)), then it is a parallelogram (\(q\)): \(p

ightarrow q\)

1. If a quadrilateral is a parallelogram (\(q\)), then its opposite angles are congruent (\(r\)): \(q

ightarrow r\)

Step2: Use the law of syllogism

By the law of syllogism (\(p
ightarrow q\) and \(q
ightarrow r\) implies \(p
ightarrow r\)), if a quadrilateral is a rhombus (\(p\)), then its opposite angles are congruent (\(r\))

Step3: Evaluate each option

• Option A: The first statement does not imply that every quadrilateral is a rhombus. There are many types of quadrilaterals (e.g., rectangles, trapezoids) that are not rhombuses.
• Option B: The first statement is \(p

ightarrow q\) (rhombus \(
ightarrow\) parallelogram), not \(q
ightarrow p\) (parallelogram \(
ightarrow\) rhombus). A parallelogram is a rhombus only if its sides are equal in length.

• Option C: Since \(p

ightarrow r\) (if a quadrilateral is a rhombus, then its opposite angles are congruent), this option is correct.

• Option D: The second statement is about parallelograms (a subset of quadrilaterals). Not all quadrilaterals have congruent opposite angles. For example, a trapezoid (a non - parallelogram quadrilateral) does not have congruent opposite angles in general.

Answer:

C. Opposite angles of a rhombus are congruent.