Mathematics, 11.11.2019 09:31 sxbenn

Provide reasons for the statements

given: < 1 and < 3 are vertical angles.
prove: < 1 ~= < 3

each of these will be used once in the reasoning section.
definition of supplementary angles, definition of a linear pair, substitution, subtraction property of equality, and definition of congruence.

Provide reasons for the statements given: < 1 and < 3 are vertical angles.

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Answer from: XAINEE

3. Definition of linear pair.

4. Definition of supplementary angles.

5. Substitution.

6. Subtraction property of equality.

7. Definition of congruence.

Step-by-step explanation:

Reason 3:

The angles are supplementary because the angles 1 and 2, 2 and 3 form a linear pair and thus because of the definition of linear pair, they are supplementary angles.

Reason 4:

The sum of angles of 1 and 2, 2 and 3 is equal to 180 because these pair of angles are supplementary angles. Supplementary angles are those angles whose sum is 180 degrees.

Reason 5:

$m\angle 1+m\angle 2=180\\m\angle 2+m\angle 3=180$

Substitute $m\angle 2+m\angle 3$ in place of 180 for the first sum.

So, $m\angle 1+m\angle 2=m\angle 2+m\angle 3$

Reason 6:

$m\angle 1+m\angle 2=m\angle 2+m\angle 3$ .

Since $m\angle 2$ is on either side of equality, we can subtract it using the subtraction property of equality and thus we get:

$m\angle 1=m\angle 3$

Reason 7:

As angle 1 is equal to angle 3, therefore by definition of congruence, these angles are congruent.

So, $m\angle 1\cong m\angle 3$

Answer from: saintsfan2004

The reasons that justify the given statements that proves that $\angle 1 \cong \angle 3$ are:

Reason 3: Definition of linear angles.

Reason 4: Definition of supplementary angles

Reason 5: Substitution.

Reason 6: Subtraction property of equality

Reason 7: Definition of congruence

Statement 3:

$\angle 1 $ and $ \angle 2$ are supplementary

$\angle 2 $ and $ 3$ are supplementary

Reason 3:

Linear angles on a straight line are supplementary angles. Therefore the reason that justifies statement 3 is: Definition of linear angles.

Statement 4:

$\angle 1 + \angle 2 = 180^{\circ}\\\\\angle 2 + \angle 3 = 180^{\circ}$

Reason 4:

The sum of two angles that are supplementary to each other equals 180 degrees. Therefore the reason for statement 4 is: Definition of supplementary angles

Statement 5:

$\angle 1 + \angle 2 = \angle 2 + \angle 3$

Reason:

Since $\angle 1 + \angle 2 = 180$ and $\angle 2 + \angle 3 = 180$ ,

therefore,

substituting $\angle 2 + \angle 3$ for 180 into $\angle 1 + \angle 2 = 180$ will give us:

$\angle 1 + \angle 2 = \angle 2 + \angle 3$

We simply substituted here.

Therefore the reason for statement 5 is: Substitution.

Statement 6: $\angle 1 = \angle 3$

Reason:

We have, $\angle 1 + \angle 2 = \angle 2 + \angle 3$

Subtract $\angle 2$ 0 from both sides of the equation

$\angle 1 + \angle 2 - \angle 2 = \angle 2 + \angle 3 - \angle 2\\\\\angle 1 = \angle 3$ (subtraction property of equality)

Therefore, the reason that justifies statement 6 is: Subtraction property of equality

Statement 7: $\angle 1 \cong \angle 3$

Reason:

Definition of congruence states that if two angles are equal in measure, therefore, they are congruent to each other.

The reason that justifies statement 7 is: Definition of congruence

In summary, the reasons that justify the given statements are:

Reason 3: Definition of linear angles.

Reason 4: Definition of supplementary angles

Reason 5: Substitution.

Reason 6: Subtraction property of equality

Reason 7: Definition of congruence

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Answer from: Quest

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Answer from: Quest

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Provide reasons for the statements

given: < 1 and < 3 are vertical angles.
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