learning target: i can apply the properties of real numbers
1) for each of the following, identify the property that applies to the statement.
a) ( mangle1 = 134^{circ} ) and ( mangle2 = 134^{circ} ), so ( mangle1 = mangle2 )
b) ( angle efgcongangle lmn ) and ( angle lmncongangle spt ), so ( angle efgcongangle spt )
c) ( qt = tu ), so ( qt + wx = tu + wx )
learning target: i can identify and apply the properties of angle relationships formed from parallel lines.
2) name all the pairs of corresponding angles in the picture.
3) name all the pairs of same - side interior angles in the picture.
4) what is the relationship between ( angle6 ) and ( angle8 )?
5) how are alternate interior angles related?
6) if ( mangle1 = 19x + 21 ) and ( mangle6 = 12x - 27 ), what is the measure of ( mangle6 )?

Question

Accepted Answer

### 1a)
# Explanation:
## Step1: Identify the property
The statement says if two quantities (angle measures here) are equal to the same quantity, then they are equal to each other. This is the **Transitive Property of Equality**? Wait, no, actually, when two things are equal to the same value, so $ m\angle1 = 134^\circ $ and $ m\angle2 = 134^\circ $, so $ m\angle1 = m\angle2 $ is the **Substitution Property** or more accurately, the **Reflexive? No, Symmetric? Wait, no. If $ a = c $ and $ b = c $, then $ a = b $ is the Transitive Property? Wait, no, Transitive is if $ a = b $ and $ b = c $, then $ a = c $. Here, $ m\angle1 = 134^\circ $ and $ m\angle2 = 134^\circ $, so $ m\angle1 = m\angle2 $ is the **Substitution** or the **Equality of Angles with Same Measure**. Actually, it's the **Transitive Property of Equality**? Wait, no, let's recall: The Transitive Property of Equality states that if $ a = b $ and $ b = c $, then $ a = c $. But here, $ m\angle1 = 134^\circ $ (let $ c = 134^\circ $), $ m\angle2 = 134^\circ $ (so $ m\angle2 = c $), so $ m\angle1 = m\angle2 $ (so $ a = b $). So it's the Transitive Property? Wait, maybe the **Substitution Property** or the **Property of Equality for Angles with Same Measure**. Alternatively, it's the **Reflexive? No. Symmetric? No. Wait, the correct property here is the **Transitive Property of Equality**? Wait, no, let's think again. If two angles have the same measure, they are equal. So $ m\angle1 = 134^\circ $ and $ m\angle2 = 134^\circ $, so by substitution, $ m\angle1 = m\angle2 $. But more accurately, it's the **Transitive Property of Equality**? Wait, maybe the **Property of Equality: If $ a = c $ and $ b = c $, then $ a = b $**. So the property is the **Transitive Property of Equality**? Wait, no, Transitive is $ a = b $, $ b = c $, so $ a = c $. Here, it's $ a = c $, $ b = c $, so $ a = b $, which is also a form of transitive? Wait, maybe the **Substitution Property** or the **Equality of Angles with Same Measure**. Alternatively, it's the **Reflexive Property**? No. Wait, the correct property is the **Transitive Property of Equality**? Wait, I think I made a mistake. Let's check: The Reflexive Property is $ a = a $. Symmetric is if $ a = b $, then $ b = a $. Transitive is if $ a = b $ and $ b = c $, then $ a = c $. So in this case, $ m\angle1 = 134^\circ $ (let $ a = m\angle1 $, $ c = 134^\circ $), $ m\angle2 = 134^\circ $ (let $ b = m\angle2 $, so $ b = c $), then $ a = b $ (so $ m\angle1 = m\angle2 $). So this is the Transitive Property? Wait, maybe the **Substitution Property** where we substitute $ 134^\circ $ with $ m\angle2 $ in the first equation. So the property is the **Transitive Property of Equality**? Wait, maybe the answer is the **Transitive Property of Equality**? Wait, no, let's look up: The Transitive Property of Equality states that if $ a = b $ and $ b = c $, then $ a = c $. But here, it's $ a = c $ and $ b = c $, so $ a = b $, which is a valid application of transitive (since equality is transitive and symmetric). So yes, Transitive Property of Equality.

## Step2: Confirm
So the property is the Transitive Property of Equality? Wait, no, maybe the **Substitution Property** or the **Property of Equality for Angles with Same Measure**. Alternatively, it's the **Reflexive Property**? No. Wait, the correct property is the **Transitive Property of Equality**? Wait, I think I was overcomplicating. The statement is: If two angles have the same measure, then they are equal. So $ m\angle1 = 134^\circ $ and $ m\angle2 = 134^\circ $, so $ m\angle1 = m\angle2 $. This is the **Transitive Property of Equality** (since $ m\angle1 = 134^\circ $, $ m\angle2 = 134^\circ $, so $ m\angle1 = m\angle2 $ by transitivity).

# Answer: Transitive Property of Equality (or more accurately, the Property of Equality stating that if two quantities are equal to the same quantity, they are equal to each other)

### 1b)
# Explanation:
## Step1: Identify the property
The statement is: $ \angle EFG \cong \angle LMN $ and $ \angle LMN \cong \angle SPT $, so $ \angle EFG \cong \angle SPT $. This is the **Transitive Property of Congruence**. The Transitive Property of Congruence states that if $ \angle A \cong \angle B $ and $ \angle B \cong \angle C $, then $ \angle A \cong \angle C $.

## Step2: Confirm
In this case, $ \angle EFG \cong \angle LMN $ (let $ \angle A = \angle EFG $, $ \angle B = \angle LMN $) and $ \angle LMN \cong \angle SPT $ (let $ \angle B = \angle LMN $, $ \angle C = \angle SPT $), so by the Transitive Property of Congruence, $ \angle EFG \cong \angle SPT $.

# Answer: Transitive Property of Congruence

### 1c)
# Explanation:
## Step1: Identify the property
The statement is: $ QT = TU $, so $ QT + WX = TU + WX $. This is the **Addition Property of Equality**. The Addition Property of Equality states that if $ a = b $, then $ a + c = b + c $ for any real number $ c $. Here, $ a = QT $, $ b = TU $, and $ c = WX $.

## Step2: Confirm
Since $ QT = TU $ (given), adding $ WX $ to both sides of the equation preserves the equality, so $ QT + WX = TU + WX $. This is the Addition Property of Equality.

# Answer: Addition Property of Equality

### 2)
# Explanation:
## Step1: Recall Corresponding Angles
Corresponding angles are pairs of angles that are in the same relative position at each intersection where a straight line crosses two others (usually parallel lines). In the diagram (with lines $ l $, $ m $, and transversal $ t $), the corresponding angles are:

- $ \angle 1 $ and $ \angle 5 $
- $ \angle 2 $ and $ \angle 6 $
- $ \angle 3 $ and $ \angle 7 $
- $ \angle 4 $ and $ \angle 8 $

Wait, but the handwritten answer has $ \angle 1 \cong \angle 5 $, $ \angle 4 \cong \angle 8 $, $ \angle 6 \cong \angle 2 $, $ \angle 7 \cong \angle 3 $. Let's check:

- $ \angle 1 $ and $ \angle 5 $: same position (top-left) at each intersection.
- $ \angle 4 $ and $ \angle 8 $: bottom-left at each intersection.
- $ \angle 2 $ and $ \angle 6 $: top-right at each intersection.
- $ \angle 3 $ and $ \angle 7 $: bottom-right at each intersection.

So the correct pairs of corresponding angles are $ \angle 1 $ and $ \angle 5 $, $ \angle 2 $ and $ \angle 6 $, $ \angle 3 $ and $ \angle 7 $, $ \angle 4 $ and $ \angle 8 $.

## Step2: List them
So the pairs are:

1. $ \angle 1 $ and $ \angle 5 $
2. $ \angle 2 $ and $ \angle 6 $
3. $ \angle 3 $ and $ \angle 7 $
4. $ \angle 4 $ and $ \angle 8 $

# Answer: The pairs of corresponding angles are $ \angle 1 $ and $ \angle 5 $, $ \angle 2 $ and $ \angle 6 $, $ \angle 3 $ and $ \angle 7 $, $ \angle 4 $ and $ \angle 8 $.

### 3)
# Explanation:
## Step1: Recall Same-Side Interior Angles
Same-side interior angles are two angles that lie between the two lines (interior) and on the same side of the transversal. In the diagram, the same-side interior angles are:

- $ \angle 2 $ and $ \angle 5 $
- $ \angle 3 $ and $ \angle 8 $

Wait, the handwritten answer has $ \angle 2 \cong \angle 5 $ and $ \angle 3 \cong \angle 8 $. Let's check:

- $ \angle 2 $ and $ \angle 5 $: between the two lines (interior) and on the same side of the transversal (right side for $ \angle 2 $ and $ \angle 5 $? Wait, no, let's see the diagram. Lines $ l $ and $ m $, transversal $ t $. $ \angle 2 $ is between $ l $ and $ m $, on the right side of $ l $; $ \angle 5 $ is between $ l $ and $ m $, on the right side of $ m $? Wait, maybe I got it wrong. Wait, same-side interior angles: for two parallel lines cut by a transversal, same-side interior angles are supplementary, but the pairs are $ \angle 2 $ and $ \angle 5 $, $ \angle 3 $ and $ \angle 8 $. Let's confirm:

- $ \angle 2 $ and $ \angle 5 $: both inside the two lines, same side of the transversal.
- $ \angle 3 $ and $ \angle 8 $: both inside the two lines, same side of the transversal.

So the pairs are $ \angle 2 $ and $ \angle 5 $, $ \angle 3 $ and $ \angle 8 $.

## Step2: List them
So the pairs of same-side interior angles are $ \angle 2 $ and $ \angle 5 $, $ \angle 3 $ and $ \angle 8 $.

# Answer: The pairs of same-side interior angles are $ \angle 2 $ and $ \angle 5 $, $ \angle 3 $ and $ \angle 8 $.

### 4)
# Explanation:
## Step1: Recall Vertical Angles
Vertical angles are opposite angles formed by the intersection of two lines. $ \angle 6 $ and $ \angle 8 $ are formed by the intersection of lines $ m $ and $ t $. $ \angle 6 $ and $ \angle 8 $ are opposite each other (vertical angles), so they are congruent (equal in measure).

## Step2: Confirm
Vertical angles are always congruent. So $ \angle 6 $ and $ \angle 8 $ are vertical angles, so their relationship is that they are congruent (vertical angles).

# Answer: $ \angle 6 $ and $ \angle 8 $ are vertical angles (and thus congruent).

### 5)
# Explanation:
## Step1: Recall Alternate Interior Angles
Alternate interior angles are two angles that lie between the two lines (interior) and on opposite sides of the transversal. When the two lines are parallel, alternate interior angles are congruent (equal in measure).

## Step2: State the Relationship
If two parallel lines are cut by a transversal, then alternate interior angles are congruent. If the lines are not parallel, alternate interior angles are not necessarily congruent. But in the context of angle relationships formed from parallel lines (as per the learning target), alternate interior angles are congruent.

# Answer: Alternate interior angles are congruent when the two lines cut by the transversal are parallel.

### 6)
# Explanation:
## Step1: Identify Angle Relationship
From the diagram, $ \angle 1 $ and $ \angle 6 $: Wait, $ \angle 1 $ and $ \angle 5 $ are corresponding angles, and $ \angle 5 $ and $ \angle 6 $ are supplementary (since they form a linear pair)? Wait, no. Wait, $ \angle 1 $ and $ \angle 6 $: Let's see, $ \angle 1 $ and $ \angle 5 $ are corresponding angles (so $ \angle 1 \cong \angle 5 $ if lines are parallel, but maybe here $ \angle 1 $ and $ \angle 6 $ are supplementary? Wait, no, the problem says $ m\angle1 = 19x + 21 $ and $ m\angle6 = 12x - 27 $. Wait, maybe $ \angle 1 $ and $ \angle 6 $ are supplementary? Wait, no, let's check the diagram. $ \angle 1 $ is on line $ l $, $ \angle 6 $ is on line $ m $. If lines $ l $ and $ m $ are parallel, then $ \angle 1 $ and $ \angle 6 $ might be same-side interior angles (supplementary) or alternate interior angles (congruent). Wait, no, $ \angle 1 $ and $ \angle 5 $ are corresponding, $ \angle 5 $ and $ \angle 6 $ are adjacent (linear pair), so $ \angle 5 + \angle 6 = 180^\circ $. Since $ \angle 1 \cong \angle 5 $ (corresponding angles), then $ \angle 1 + \angle 6 = 180^\circ $. So:

$ m\angle1 + m\angle6 = 180^\circ $

Substitute the given expressions:

$ (19x + 21) + (12x - 27) = 180 $

## Step2: Solve for $ x $
Combine like terms:

$ 19x + 12x + 21 - 27 = 180 $

$ 31x - 6 = 180 $

Add 6 to both sides:

$ 31x = 186 $

Divide both sides by 31:

$ x = 6 $

## Step3: Find $ m\angle6 $
Substitute $ x = 6 $ into $ m\angle6 = 12x - 27 $:

$ m\angle6 = 12(6) - 27 = 72 - 27 = 45^\circ $

# Answer: The measure of $ \angle 6 $ is $ 45^\circ $.

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QUESTION IMAGE

Question

Explanation:

1a)

Step1: Identify the property

Step2: Confirm

Step1: Identify the property

Step2: Confirm

Step1: Identify the property

Step2: Confirm

Answer:

1b)