2.10 Unit Test Reasoning And Proof Part 1

Question: Kathryn draws three pairs of intersecting lines. In each figure, she measures a pair of vertical angles.

What is a reasonable conjecture for Kathryn to make by recognizing a pattern and using inductive reasoning?

Answer: When a pair of lines intersect, the vertical angles are congruent.

Question: Ricardo draws three right triangles. In each figure, he measures a pair of angles.

What is a reasonable conjecture for Ricardo to make by recognizing a pattern and using inductive reasoning?

Answer: In a right triangle, the acute angles are complementary.

(Watching Mini Ladd has ruined the name ‘Ricardo’ for me.)

Question: Ava draws three parallelograms. In each figure, she measures a pair of angles, as shown.

What is a reasonable conjecture for Ava to make by recognizing a pattern and using inductive reasoning?

Answer: In a parallelogram, opposite angles are congruent.

Question: Which statement is true about this argument?

Premises:

If an angle measure is greater than 90°, then the angle is an obtuse angle.

The measure of ∠C is 102°.

Conclusion:

∠C is an obtuse angle.

Answer: The argument is valid by the law of detachment.

Question: Which statement is true about this argument?

Premises:

If a triangle has an angle that measures 150°, then it is an obtuse triangle.

△JKL is an obtuse triangle.

Conclusion:

△JKL has an angle that measures 150°.

Answer: The argument is not valid because the conclusion does not follow from the premises.

Question: Which statement is true about this argument?

Premises:

If a parallelogram has a right angle, then it is a rectangle.

Parallelogram PQRS has a right angle.

Conclusion:

Parallelogram PQRS is a rectangle.

Answer: The argument is valid by the law of detachment.

( I don’t really get why it’s this one when if it has one right angle it has all right angles and should just be called a rectangle not a parallelogram.)

Question:

Answer: (1) Given

(2) Angle Addition Postulate

(3) 51° + 39° = m∠DEF

(4) 90° = m∠DEF

(5) ∠DEF is a right angle.

(6) Definition of a right triangle.

Question:

Answer: (1) linear pair postulate

(2) m∠2

(3) ∠3

(4) angle congruence postulate

Question: A conjecture and the flowchart proof used to prove the conjecture are shown.

Drag an expression or phrase to each box to complete the proof.

AD is parallel to BC. - (1)

m∠1 = m∠2 - (2)

(3) - Definition of supplementary

(4) - Substitution Property of Equality

∠1 and ∠2 are supplementary. - (5)

Answer: (1) Definition of parallelogram

(2) Angle Congruence Postulate

(3) m∠2 + m∠3 = 180°

(4) m∠1 + m∠3 = 180°

(5) Definition of supplementary

Question: A conjecture and the two-column proof used to prove the conjecture are shown.

Drag an expression or phrase to each box to complete the proof.

2. JK ≅ KL - (1)

3. (2) - Given

4. (3) - Definition of a midpoint

5. JK ≅ LM - (4)

Answer: (1) Definition of a midpoint

(2) l is the midpoint of KM

(3) KL ≅ LM

(4) Transitive Property of Congruence