Lesson 7 Justification and Proof Practice Understanding

Ready

Construct perpendicular bisectors for the two segments. Be sure to show the circles you use. Mark the right angle and the two congruent segments created.

1.

line segment AB

2.

line segment WV

Construct the angle bisector of the following angles.

3.

Angle ABC

4.

Angle GHI

Use the properties of parallel lines cut by a transversal to determine the value of .

5.

Parallel lines and a transversal; one acute angle is marked x degrees and an obtuse angle is marked 115 degrees.

6.

Parallel lines and a transversal; alternate exterior angles marked 75 degrees and angle 2x 7

Set

7.

When it comes to proving vertical angles are congruent, two students present their reasoning and they both claim to be correct. Examine their reasoning and decide if they are both correct. Explain the logic they are using and what you think about their justifications.

Intersecting lines form angles 1-4

Student 1

Student 2

because they are a linear pair because they are a linear pair

So, by substitution.

And then by the subtraction property of equality.

Just take from both sides.

This shows that the vertical angles are congruent.

You can rotate angle so that it lands on angle .

Since the lines are straight we know that is degree angle.

So, rotating degrees will cause the lines to land back on themselves.

This confirms that angle will land on angle

after a degree rotation.

Your explanation of their logic and justifications:

8.

What is the difference between a mathematical conjecture and a mathematical proof?

9.

When it comes to proving alternate interior angles are congruent, two students present their reasoning and they both claim to be correct. Examine their reasoning and decide if they are both correct. Explain the logic they are using and what you think about their justifications.

Transversal line AB intersect lines EF and CD forming angles 1-8. 12345678

Given:

Prove: Alternate interior angles and are congruent.

Student 1

Given

Vertical Angles

Corresponding Angles

Substitution

Student 2

Since , we know from corresponding angles that

and are congruent. So, .

and form a linear pair, so .

Substituting we have, .

and form a linear pair, so .

This means we have .

And you take off from both sides to get ,

and therefore alternate interior angles and are congruent.

Your explanation of their logic and justifications:

10.

Prove alternate interior angles formed by parallel lines cut by a transversal are congruent using transformations. Provide justification for your work.

Transversal line AB intersect lines EF and CD forming angles 1-8. 12345678

Given:

Prove: Alternate interior angles and are congruent.

11.

Given:

Prove: Same-side interior angles and are supplementary.

Transversal line AB intersect lines EF and CD forming angles 1-8. 12345678

Go

Find the value of and the measure of the missing angles in each diagram.

12.

Triangle with one exterior angle. The exterior angle is marked x and the remote interior angles are labeled 23 degrees and 85 degrees.

13.

Triangle with one exterior angle. The exterior angle is marked 135 degrees and the remote interior angles are labeled x degrees and 87 degrees.

14.

Triangle with one exterior angle. The exterior angle is marked 120 degrees and the remote interior angles are labeled 6x-10 and 2x 10.

15.

Triangle with one exterior angle. The exterior angle is marked 72 degrees and the remote interior angles are labeled 4x-2 and 3x 4.

16.

Triangle with one exterior angle. The exterior angle is marked 111 degrees and the remote interior angles are labeled 8x 14 and 7x 5.

Indicate whether each pair of angles would be congruent or supplementary.

Lines and are parallel.

Transversal intersects parallel lines p and q forming angles 1-8 12345687

17.

and

A.

congruent

B.

supplementary

18.

and

A.

congruent

B.

supplementary

19.

and

A.

congruent

B.

supplementary

20.

and

A.

congruent

B.

supplementary

21.

and

A.

congruent

B.

supplementary

22.

and

A.

congruent

B.

supplementary