Question

period: 1 date: 09/10/25
2-4 classwork
page 1
if-then statements and counterexamples
geometry
olympia hs 2025-26
directions: for each statement, underline the hypothesis (after if), box the conclusion (after then), write the converse, and decide if the converse is true or false. if it is false, provide a counterexample. at the end, create your own statements.

1. if it is snowing, then it is cold.

• converse: if it is cold, then it is snowing
• true/false? ______
• counterexample: ______

1. if you brush your teeth, then your breath will be fresh.

• converse: if you breath will be fresh, then you brush your teeth
• true/false? ______
• counterexample: ______

1. if you drop your phone, then it might break.

• converse: if it might break, then you drop your phone
• true/false? ______
• counterexample: ______

1. if you practice basketball, then you will improve.

• converse: if you will improve, then you practice basketball
• true/false? ______
• counterexample: ______

1. if it is saturday, then there is no school.

• converse: if there is school, then it is saturday
• true/false? ______
• counterexample: ______

1. if you water plants, then they grow.

• converse: if they grow, then you water plants
• true/false? ______
• counterexample: ______

1. if a person is asleep, then their eyes are closed.

• converse: if their eyes are closed, then a person is asleep
• true/false? ______
• counterexample: ______

Explanation:

Problem 1

Step1: Analyze the converse

The original statement is "If it is snowing, then it is cold." The converse swaps hypothesis and conclusion: "If it is cold, then it is snowing."

Step2: Determine truth value

Cold weather doesn't always mean snow (e.g., cold and dry days). So the converse is false.

Step3: Provide counterexample

A day with temperature \( 30^\circ\text{F} \) (cold) but no snowfall (e.g., clear, cold winter day).

Step1: Analyze the converse

Original: "If you brush your teeth, then your breath will be fresh." Converse: "If your breath will be fresh, then you brush your teeth."

Step2: Determine truth value

Breath can be fresh from other methods (e.g., mouthwash, eating mint). So converse is false.

Step3: Provide counterexample

Using mouthwash to make breath fresh without brushing teeth.

Step1: Analyze the converse

Original: "If you drop your phone, then it might break." Converse: "If it might break, then you drop your phone."

Step2: Determine truth value

A phone might break from other causes (e.g., water damage, overheating) not just dropping. So converse is false.

Step3: Provide counterexample

A phone gets water-damaged (might break) without being dropped.

Answer:

• Converse: If it is cold, then it is snowing.
• True/False? False
• Counterexample: A cold, clear winter day with no snow (e.g., \( 30^\circ\text{F} \), sunny, no snow).

Problem 2