Question

1. the two conditional statements below are true.

if ∠3 and ∠4 form a linear pair, then they are supplementary.
if ∠3 and ∠4 are supplementary, then m∠3 + m∠4 = 180°.
based on these conditional statements, which statement must also be true?
options:

• if ∠3 and ∠4 are supplementary, then ∠3 and ∠4 form a linear pair.
• if ∠3 and ∠4 form a linear pair, then m∠3 + m∠4 = 180°.
• if ∠3 and ∠4 form a linear pair, then m∠3 = 90°, and m∠4 = 90°.
• if m∠3 + m∠4 = 180°, then ∠3 and ∠4 form a linear pair.

Explanation:

Step1: Analyze the given conditionals

We have two true conditionals:

1. If \( \angle 3 \) and \( \angle 4 \) form a linear pair, then they are supplementary.
2. If \( \angle 3 \) and \( \angle 4 \) are supplementary, then \( m\angle 3 + m\angle 4 = 180^\circ \).

Step2: Apply the Law of Syllogism

The Law of Syllogism states that if we have two conditionals \( p \to q \) and \( q \to r \), then we can conclude \( p \to r \).
Let \( p \): \( \angle 3 \) and \( \angle 4 \) form a linear pair,
\( q \): \( \angle 3 \) and \( \angle 4 \) are supplementary,
\( r \): \( m\angle 3 + m\angle 4 = 180^\circ \).

From the first conditional, \( p \to q \); from the second, \( q \to r \). By syllogism, \( p \to r \), which translates to: If \( \angle 3 \) and \( \angle 4 \) form a linear pair, then \( m\angle 3 + m\angle 4 = 180^\circ \).

Step3: Evaluate other options

• First option: A linear pair requires adjacent angles, but supplementary angles need not be adjacent. So "If supplementary, then linear pair" is false.
• Third option: A linear pair sums to \( 180^\circ \), but angles don’t have to be \( 90^\circ \) each. False.
• Fourth option: Supplementary angles (sum \( 180^\circ \)) need not be a linear pair (adjacent). False.

Answer:

The statement "If \( \angle 3 \) and \( \angle 4 \) form a linear pair, then \( m\angle 3 + m\angle 4 = 180^\circ \)" (the second option in the list).