2.10 Final Reasoning and Proof

The argument is valid by the law of detachment.

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.

∠1 and ∠2 ​ form a linear pair, so ∠1 and ∠2 are supplementary by the linear pair postulate. Therefore, m∠1+ m∠2 = 180° by the definition of supplementary. It is given that ∠2≅∠3, so m∠2=m∠3 by the angle congruence postulate. By substitution, m∠1+m∠3=180°, so ∠1 and ∠3 are supplementary by the definition of supplementary.

A conjecture and the paragraph proof used to prove the conjecture are shown. Given: angle 2 is congruent to angle 3. Prove: angle 1 and angle 3 are supplementary. A horizontal line. Two rays extend from upper region of the line diagonally down to the left and right and intersect the line forming interior angles labeled as 2 and 3 and an exterior angle labeled as 1. Drag an expression or statement to each box to complete the proof.

In a parallelogram, consecutive angles are supplementary.

Chloe draws three parallelograms. In each figure, she measures a pair of angles, as shown. What is a reasonable conjecture for Chloe to make by recognizing a pattern and using inductive reasoning?

In a scalene triangle, none of the angles are congruent.

Jasmine draws three scalene triangles. In each figure, she measures each of the angles. What is a reasonable conjecture for Jasmine to make by recognizing a pattern and using inductive reasoning?

When a pair of parallel lines are intersected by a third line, the alternate interior angles are congruent.

Mei draws three pairs of parallel lines that are each intersected by a third line. In each figure, she measures a pair of angles. What is a reasonable conjecture for Mei to make by recognizing a pattern and using inductive reasoning?

The argument is valid by the law of detachment.

Which statement is true about this argument? Premises: If a parallelogram has a right angle, then it is a rectangle. Parallelogram PQRSParallelogram PQRS has a right angle. Conclusion: Parallelogram PQRSParallelogram PQRS is a rectangle.

The argument is valid by the law of syllogism.

Which statement is true about this argument? Premises: If a triangle is an isosceles triangle, then it has two sides of equal length. If a triangle has two sides of equal length, then it has two angles of equal measure. Conclusion: If a triangle is an isosceles triangle, then it has two angles of equal measure.