Geometry - Deductive Reasoning

What does the Law of Syllogism in geometry state?

• If two true conditional statements have the conclusion of the first as the hypothesis of the second, the conclusion of the second is unknown.

• If two true conditional statements have the conclusion of the first as the hypothesis of the second, the conclusion of the second is false.

• If two true conditional statements have the conclusion of the first as the hypothesis of the second, the conclusion of the first is also true.

• If two true conditional statements have the conclusion of the first as the hypothesis of the second, the conclusion of the second is also true.

Answer explanation

The Law of Syllogism in geometry states that if two true conditional statements have the conclusion of the first as the hypothesis of the second, then the conclusion of the second is also true. This is the correct choice because it accurately describes the law's principle.

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.

Kai notices that two lines are parallel and Angle P and Angle Q are corresponding angles. What can be concluded about the angles?

• The corresponding angles P and Q are congruent.

• The corresponding angles P and Q are supplementary.

• The lines are not parallel.

• Angle P and Angle Q are not congruent.

Answer explanation

When two lines are parallel, their corresponding angles are congruent. In this case, Angle P and Angle Q are corresponding angles of parallel lines, so they must be congruent.

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.

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: 1) 'If two kites are similar, their corresponding angles are congruent.' and 2) '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.

Liam says: 'If two triangles are congruent, their corresponding sides are congruent.' Sophia points to Triangle ABC and Triangle DEF and says they are congruent. What can Isla conclude using the Law of Detachment?

• The corresponding angles of Triangle ABC and Triangle DEF are congruent.

• The corresponding sides of Triangle ABC and Triangle DEF are congruent.

• Triangle ABC and Triangle DEF are not congruent.

• The corresponding sides of Triangle ABC and Triangle DEF are not congruent.

Answer explanation

Isla can conclude that the corresponding sides of Triangle ABC and Triangle DEF are congruent. Since Liam stated that if two triangles are congruent, their corresponding sides are congruent, and Sophia confirmed that Triangle ABC and Triangle DEF are congruent, Isla can apply the Law of Detachment to deduce that their corresponding sides must be congruent.