Question

1. two angles supplementary to the same angle must be congruent. 15. all state names have at least two syllables. 16. squaring a number and adding one will always result in an even number. 17. write your own conjectures! then trade with your partner and determine if the conjecture is true or false. if false, provide a counterexample. conjecture: ______ t/f: ____ 18. conjecture: ____ t/f: ______

Explanation:

Question 14: Two angles supplementary to the same angle must be congruent.

Step1: Recall the definition of supplementary angles.

If two angles \( \angle A \) and \( \angle B \) are supplementary to \( \angle C \), then \( \angle A + \angle C = 180^\circ \) and \( \angle B + \angle C = 180^\circ \).

Step2: Solve for \( \angle A \) and \( \angle B \).

From \( \angle A + \angle C = 180^\circ \), we get \( \angle A = 180^\circ - \angle C \). From \( \angle B + \angle C = 180^\circ \), we get \( \angle B = 180^\circ - \angle C \).

Step3: Compare \( \angle A \) and \( \angle B \).

Since \( \angle A = 180^\circ - \angle C \) and \( \angle B = 180^\circ - \angle C \), by the transitive property, \( \angle A = \angle B \). So the two angles are congruent.

Question 15: All state names have at least two syllables.

Step1: Recall state names.

Consider the state name "Utah".

Step2: Analyze the syllables in "Utah".

"Utah" is pronounced as /ˈjuːtɑː/ or /ˈjuːtə/, which has only one syllable. Since there exists a state name with less than two syllables, the statement is false.

Question 16: Squaring a number and adding one will always result in an even number.

Step1: Choose a number to test.

Let's take the number \( 2 \).

Step2: Square the number and add one.

Square of \( 2 \) is \( 2^2 = 4 \). Then add one: \( 4 + 1 = 5 \).

Step3: Check if the result is even.

\( 5 \) is an odd number. Since we found a number (2) for which squaring and adding one does not result in an even number, the statement is false.

Question 17: (Example Conjecture) Conjecture: All prime numbers are odd.

Answer:

True