Geometry Ch. 2 Of Prentice Hall

1) Use the Law of Detachment and the Law of Syllogism to draw a conclusion from the given statements.
Franz is taller than Isabel.
Isabel is shorter than Ellen.
Janina is taller than Franz.

Janina is shortest.

Janina is taller than Isabel.

Franz is the tallest.

Ellen is shorter than Isabel.

Based on the given statements, we can use the Law of Syllogism to conclude that Janina is taller than Isabel. The first statement tells us that Franz is taller than Isabel, and the second statement tells us that Isabel is shorter than Ellen. By applying the Law of Syllogism, we can infer that Franz is taller than Ellen. Finally, since the third statement states that Janina is taller than Franz, we can conclude that Janina is taller than Isabel.

Explanation

Based on the given statements, we can use the Law of Syllogism to conclude that Janina is taller than Isabel. The first statement tells us that Franz is taller than Isabel, and the second statement tells us that Isabel is shorter than Ellen. By applying the Law of Syllogism, we can infer that Franz is taller than Ellen. Finally, since the third statement states that Janina is taller than Franz, we can conclude that Janina is taller than Isabel.

2) Select the appropriate property of equality for the statement. If a = b, then a • c = b • c.

Symmetric Property

Substitution Property

Multiplication Property

Reflexive Property

The correct answer is Multiplication Property because it states that if two quantities are equal (a = b), then multiplying both sides of the equation by the same number (c) will still result in equality (a • c = b • c). This property allows us to perform the same operation on both sides of an equation without changing its truth value.

Explanation

The correct answer is Multiplication Property because it states that if two quantities are equal (a = b), then multiplying both sides of the equation by the same number (c) will still result in equality (a • c = b • c). This property allows us to perform the same operation on both sides of an equation without changing its truth value.

3) Select the appropriate property of equality for the statement a = a.

Addition Property

Reflexive Property

Substitution Property

Transitive Property

The reflexive property states that any quantity is equal to itself. In this case, the statement "a = a" shows that the quantity 'a' is equal to itself, which aligns with the reflexive property.

Explanation

The reflexive property states that any quantity is equal to itself. In this case, the statement "a = a" shows that the quantity 'a' is equal to itself, which aligns with the reflexive property.

4) One way to show that a statement is NOT a good definition is to find a _____.

Biconditional

Conditional

Converse

Counterexample

A counterexample is a specific example or case that contradicts or disproves a statement. In this context, if a statement is claimed to be a good definition but a counterexample can be found that does not fit the definition, it demonstrates that the statement is not a good definition. Therefore, a counterexample is a valid way to show that a statement is NOT a good definition.

Explanation

A counterexample is a specific example or case that contradicts or disproves a statement. In this context, if a statement is claimed to be a good definition but a counterexample can be found that does not fit the definition, it demonstrates that the statement is not a good definition. Therefore, a counterexample is a valid way to show that a statement is NOT a good definition.

5) Which is a counterexample that proves this conditional is false?
All prime numbers are odd.

14

10

7

2

The number 2 is a counterexample that proves the conditional "All prime numbers are odd" false because 2 is a prime number but it is not odd.

Explanation

The number 2 is a counterexample that proves the conditional "All prime numbers are odd" false because 2 is a prime number but it is not odd.

6) Which is the Law of Syllogism in symbolic form?

If p → q and q → r are true statements, then p → r is a true statement.

If p → q is a true statement and p is true, then q is true.

If p → q is a false statement and p is false, then q is false.

If p → q and q → r are false statements, then p → r is a false statement.

The Law of Syllogism in symbolic form states that if p → q and q → r are true statements, then p → r is also a true statement. This means that if the first statement implies the second statement, and the second statement implies the third statement, then the first statement implies the third statement as well.

Explanation

The Law of Syllogism in symbolic form states that if p → q and q → r are true statements, then p → r is also a true statement. This means that if the first statement implies the second statement, and the second statement implies the third statement, then the first statement implies the third statement as well.

7) Two angles whose sides are opposite rays are called _____. Two coplanar angles with a common side, a common vertex, and no common interior points are called _____.

Adjacent angles; vertical angles

Vertical angles; adjacent angles

Adjacent angles; complementary angles

Vertical angles; supplementary angles

Vertical angles are formed by two intersecting lines and are opposite each other. They share a common vertex but do not have any common interior points. Adjacent angles, on the other hand, share a common side and a common vertex, but do not have any common interior points. Therefore, the correct answer is vertical angles; adjacent angles.

Explanation

Vertical angles are formed by two intersecting lines and are opposite each other. They share a common vertex but do not have any common interior points. Adjacent angles, on the other hand, share a common side and a common vertex, but do not have any common interior points. Therefore, the correct answer is vertical angles; adjacent angles.

8) Name an angle complementary to ∠COD.

∠AOC

∠BOC

∠AOE

∠DOE

An angle complementary to ∠COD is an angle that, when added to ∠COD, will result in a sum of 90 degrees. Among the given options, ∠BOC is the only angle that, when added to ∠COD, will result in a sum of 90 degrees. Therefore, ∠BOC is the correct answer.

Explanation

An angle complementary to ∠COD is an angle that, when added to ∠COD, will result in a sum of 90 degrees. Among the given options, ∠BOC is the only angle that, when added to ∠COD, will result in a sum of 90 degrees. Therefore, ∠BOC is the correct answer.

9) Use the Law of Detachment to draw a conclusion.A bisector of a line segment intersects the segment at its midpoint. bisects at D.

D is the midpoint of Segment CE.

D is the midpoint of Segment AB.

Segment AB is perpendicular to Segment CE.

Segment AB has the same length as Segment CE.

The Law of Detachment states that if a conditional statement is true and the hypothesis of the conditional statement is true, then the conclusion of the conditional statement is also true. In this case, the given information states that a bisector of a line segment intersects the segment at its midpoint. The statement "D is the midpoint of Segment CE" is a conclusion that can be drawn from this information because it matches the given condition. Therefore, the answer is D is the midpoint of Segment CE.

Explanation

The Law of Detachment states that if a conditional statement is true and the hypothesis of the conditional statement is true, then the conclusion of the conditional statement is also true. In this case, the given information states that a bisector of a line segment intersects the segment at its midpoint. The statement "D is the midpoint of Segment CE" is a conclusion that can be drawn from this information because it matches the given condition. Therefore, the answer is D is the midpoint of Segment CE.

10) Rewrite the biconditional as a conditional statement. Then write its converse and determine if the biconditional is an accurate definition.

An angle is obtuse if and only if its measure is greater than ninety degrees.

Yes, it is an accurate definition.

No, it is not an accurate definition.

it could be a straight angle (a line)

Explanation

it could be a straight angle (a line)

11) Which biconditional is a good definition?

Two angles are adjacent if and only if they share a common side.

A point is the midpoint of a segment if and only if it is between the endpoints of the segment.

Two angles are supplementary if and only if they form a linear pair.

Two lines are parallel if and only if they never intersect.

not-available-via-ai

Explanation

not-available-via-ai

12) N the figure shown m∠AED = 97. Which statement is false?

M∠AEB = 83

M∠BEC = 97

∠AEB and m∠DEC are congruent angles.

# ∠BEC and m∠CED are vertical angles.

The statement "# ∠BEC and m∠CED are vertical angles" is false. Vertical angles are formed by two intersecting lines and are always congruent. However, in the figure shown, m∠BEC and m∠CED are not congruent angles, so they cannot be vertical angles.

Explanation

The statement "# ∠BEC and m∠CED are vertical angles" is false. Vertical angles are formed by two intersecting lines and are always congruent. However, in the figure shown, m∠BEC and m∠CED are not congruent angles, so they cannot be vertical angles.

13) Use the Law of Detachment to draw a conclusion.
I will have room for dessert if and only if I do not finish my dinner. I finished my dinner.

I do not have room for dessert.

I did not finish my dinner.

I had macaroni and cheese for dinner.

No conclusion possible

need the hypothesis to be true: I will have room for dessert

Explanation

need the hypothesis to be true: I will have room for dessert

14) Complete the proof.Given: ∠A and ∠B are right angles.Prove: ∠A ≈ ∠BBy the definition of a. _____ m∠A = 90 and m∠B = 90. By the b. _____ Property m∠A = m∠B or ∠A ≅ ∠B.

By the definition of a right angle, we know that the measure of angle A is 90 degrees and the measure of angle B is also 90 degrees. By the substitution property, we can conclude that the measures of angles A and B are equal, or in other words, angle A is congruent to angle B.

Explanation

By the definition of a right angle, we know that the measure of angle A is 90 degrees and the measure of angle B is also 90 degrees. By the substitution property, we can conclude that the measures of angles A and B are equal, or in other words, angle A is congruent to angle B.

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