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.

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2.

Construct the angle bisector of the following angles.

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4.

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

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6.

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.

Student 1	Student 2
$m ∠ 1 + m ∠ 2 = 180 °$ because they are a linear pair $m ∠ 3 + m ∠ 2 = 180 °$ because they are a linear pair So, $m ∠ 1 + m ∠ 2 = m ∠ 3 + m ∠ 2$ by substitution. And then $m ∠ 1 = m ∠ 3$ by the subtraction property of equality. Just take $m ∠ 2$ from both sides. This shows that the vertical angles are congruent.	You can rotate angle $1$ so that it lands on angle $3$ . Since the lines are straight we know that is $180$ degree angle. So, rotating $180$ degrees will cause the lines to land back on themselves. This confirms that angle $1$ will land on angle $3$ after a $180$ degree rotation.

Student 1

Student 2

$m ∠ 1 + m ∠ 2 = 180 °$ because they are a linear pair $m ∠ 3 + m ∠ 2 = 180 °$ because they are a linear pair

So, $m ∠ 1 + m ∠ 2 = m ∠ 3 + m ∠ 2$ by substitution.

And then $m ∠ 1 = m ∠ 3$ by the subtraction property of equality.

Just take $m ∠ 2$ from both sides.

This shows that the vertical angles are congruent.

You can rotate angle $1$ so that it lands on angle $3$ .

Since the lines are straight we know that is $180$ degree angle.

So, rotating $180$ degrees will cause the lines to land back on themselves.

This confirms that angle $1$ will land on angle $3$

after a $180$ 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.

Given: $\overset{\leftrightarrow}{B F} ∥ \overset{\leftrightarrow}{A D}$

Prove: Alternate interior angles $∠ 3$ and $∠ 6$ are congruent.

Student 1
$\overset{\leftrightarrow}{B F} ∥ \overset{\leftrightarrow}{A D}$	Given
$∠ 3 ≅ ∠ 2$	Vertical Angles
$∠ 6 ≅ ∠ 2$	Corresponding Angles
$∠ 3 ≅ ∠ 6$	Substitution

Student 2
Since $\overset{\leftrightarrow}{B F} ∥ \overset{\leftrightarrow}{A D}$ , we know from corresponding angles that $∠ 1$ and $∠ 5$ are congruent. So, $m ∠ 1 = m ∠ 5$ . $∠ 1$ and $∠ 3$ form a linear pair, so $m ∠ 1 + m ∠ 3 = 180 °$ . Substituting we have, $m ∠ 5 + m ∠ 3 = 180 °$ . $∠ 5$ and $∠ 6$ form a linear pair, so $m ∠ 5 + m ∠ 6 = 180 °$ . This means we have $m ∠ 5 + m ∠ 3 = m ∠ 5 + m ∠ 6$ . And you take $m ∠ 5$ off from both sides to get $m ∠ 3 = m ∠ 6$ , and therefore alternate interior angles $∠ 3$ and $∠ 6$ are congruent.

Student 2

Since $\overset{\leftrightarrow}{B F} ∥ \overset{\leftrightarrow}{A D}$ , we know from corresponding angles that

$∠ 1$ and $∠ 5$ are congruent. So, $m ∠ 1 = m ∠ 5$ .

$∠ 1$ and $∠ 3$ form a linear pair, so $m ∠ 1 + m ∠ 3 = 180 °$ .

Substituting we have, $m ∠ 5 + m ∠ 3 = 180 °$ .

$∠ 5$ and $∠ 6$ form a linear pair, so $m ∠ 5 + m ∠ 6 = 180 °$ .

This means we have $m ∠ 5 + m ∠ 3 = m ∠ 5 + m ∠ 6$ .

And you take $m ∠ 5$ off from both sides to get $m ∠ 3 = m ∠ 6$ ,

and therefore alternate interior angles $∠ 3$ and $∠ 6$ 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.

Given: $\overset{\leftrightarrow}{B F} ∥ \overset{\leftrightarrow}{A D}$

Prove: Alternate interior angles $∠ 3$ and $∠ 6$ are congruent.

11.

Given: $\overset{\leftrightarrow}{B F} ∥ \overset{\leftrightarrow}{A D}$

Prove: Same-side interior angles $∠ 3$ and $∠ 5$ are supplementary.

Go

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

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B.

supplementary