Geometry Deductive Reasoning

Which statement correctly describes the Law of Detachment in geometry?

• If p -> q is true and p is false, then q is false.

• If p -> q is true and q is true, then p is true.

• If p -> q is true and p is true, then q is true.

• If p -> q is false and p is true, then q is true.

Answer explanation

The Law of Detachment in geometry is a rule of inference. It states that if a conditional statement (p -> q) is true and the antecedent (p) is true, then the consequent (q) is also true. This is the correct choice because it accurately describes the Law of Detachment.

Using the Law of Detachment, what can be concluded from these statements: 1) 'If two angles are complementary, their sum is 90 degrees.' and 2) 'Angle A and Angle B are complementary angles.'

• The sum of Angle A and Angle B is 180 degrees.

• The sum of Angle A and Angle B is 90 degrees.

• Angle A and Angle B are congruent.

• The sum of Angle A and Angle B is 45 degrees.

Answer explanation

Using the Law of Detachment, we can conclude that since Angle A and Angle B are complementary angles, their sum must be 90 degrees, as stated in the first statement. This matches the correct choice provided in the options.

Liam and Oliver are comparing two angles. Liam says, 'If two angles are congruent, then they have equal measures.' Oliver agrees and adds, 'Angle X and Angle Y are congruent angles.' What can be concluded using the Law of Detachment?

• Angle X and Angle Y have different measures.

• Angle X and Angle Y are not congruent angles.

• Angle X and Angle Y have equal measures.

• The Law of Detachment cannot be applied to this situation.

Answer explanation

Liam and Oliver agree that congruent angles have equal measures. They also agree that Angle X and Angle Y are congruent. Using the Law of Detachment, we can conclude that since Angle X and Angle Y are congruent, they must have equal measures.

Statement 1: 'If a quadrilateral is a rectangle, then its opposite sides are congruent.' Statement 2: 'ABCD is a rectangle.' What can be concluded using the Law of Detachment?

• If a quadrilateral is not a rectangle, then its opposite sides are congruent.

• ABCD has opposite sides that are congruent.

• ABCD has opposite sides that are not congruent.

• ABCD is not a rectangle.

Answer explanation

Using the Law of Detachment, we can conclude that since ABCD is a rectangle (Statement 2), its opposite sides are congruent (Statement 1). This matches the given correct choice, which states that ABCD has opposite sides that are congruent.

Reasoning based on facts, definitions, accepted properties, and the laws of logic is called ____________ reasoning.

• Inductive

• Deductive

Which law of logic is illustrated by the following statements?

If <1 and <2 are vertical angles, then <1 is congruent to <2. If <1 is congruent to <2, then m<1 = m<2.
If <1 and <2 are vertical angles, then m<1 = m<2.

• Law of Detachment

• Law of Syllogism

• 5th Amendment

• Not possible

Which law of deductive reasoning is being used, if any?
If a triangle is equilateral, then all three sides are congruent. 
If all the sides of a triangle are congruent, then the angles opposite them are congruent. 
Therefore, if a triangle is equilateral, then the angles opposite the sides are congruent.

• Law of Detachment

• Law of Syllogism

• Law of the Contrapositive

• Not valid

Evelyn says: 'If a triangle is equilateral, all its angles are congruent.' James shows an equilateral triangle XYZ. What can be concluded using the Law of Detachment?

• Triangle XYZ is not equilateral.

• Some angles of triangle XYZ are congruent.

• All angles of triangle XYZ are congruent.

• The Law of Detachment cannot be applied.

Answer explanation

Evelyn states that if a triangle is equilateral, all its angles are congruent. James shows an equilateral triangle XYZ. By applying the Law of Detachment, we can conclude that all angles of triangle XYZ are congruent, as it is an equilateral triangle.

Abigail learns two statements: 1: 'If a polygon is a regular hexagon, then all its angles are congruent.' and 2: 'Polygon ABCDEF is a regular hexagon.' What can be concluded using the Law of Detachment?

• Polygon ABCDEF is not a regular hexagon.

• All angles of polygon ABCDEF are congruent.

• Some angles of polygon ABCDEF are congruent.

• The Law of Detachment cannot be applied to this situation.

Answer explanation

The Law of Detachment can be applied to this situation. The question states that polygon ABCDEF is a regular hexagon. Statement 1 says that if a polygon is a regular hexagon, then all its angles are congruent. Therefore, we can conclude that all angles of polygon ABCDEF are congruent.

Using the Law of Detachment, what can Aria conclude from the statements: 'If two kites are similar, their corresponding angles are congruent.' and 'David's and Michael's kites are similar.'?

• The corresponding sides of the kites are congruent.

• The corresponding angles of the kites are not congruent.

• The corresponding sides of the kites are not congruent.

• The corresponding angles of the kites are congruent.

Answer explanation

Using the Law of Detachment, Aria can conclude that the corresponding angles of David's and Michael's kites are congruent. This is because the first statement establishes a relationship between similar kites and congruent angles, and the second statement confirms that their kites are similar.