mvusd, math ii
name
per
angle & triangle properties
6. draw and label a diagram that demonstrates the given types of angles.
a) vertical b) linear pair c) corresponding
d) alternate interior e) alternate exterior f) same - side interior
7 - 8) for each diagram, solve for x and the measures the two designated angles.
7.
8.
9. ∠a & ∠b are complementary. if the m∠a = (6x + 40)° and the m∠b = (4x + 30)°, solve for x and the two angles.
10. for the given diagram, place the measures of angles a, b, c appropriately on the number line, according to location of x.

Question

mvusd, math ii
name
per
angle & triangle properties
6. draw and label a diagram that demonstrates the given types of angles.
a) vertical  b) linear pair  c) corresponding
d) alternate interior  e) alternate exterior  f) same - side interior
7 - 8) for each diagram, solve for x and the measures the two designated angles.
7.
8.
9. ∠a & ∠b are complementary. if the m∠a = (6x + 40)° and the m∠b = (4x + 30)°, solve for x and the two angles.
10. for the given diagram, place the measures of angles a, b, c appropriately on the number line, according to location of x.

Accepted Answer

### Problem 7:
# Explanation:
## Step1: Identify angle relationship
The two angles are vertical angles, so they are equal. Thus, $2x + 15 = x + 30$.
## Step2: Solve for $x$
Subtract $x$ from both sides: $2x - x + 15 = 30$ → $x + 15 = 30$. Then subtract 15: $x = 30 - 15 = 15$.
## Step3: Find angle measures
Substitute $x = 15$ into $2x + 15$: $2(15)+15 = 30 + 15 = 45^\circ$. For $x + 30$: $15 + 30 = 45^\circ$ (both angles are equal, as expected for vertical angles).

# Answer:
$x = 15$, both angles measure $45^\circ$

### Problem 8:
# Explanation:
## Step1: Identify angle relationship
The two angles form a linear pair, so their sum is $180^\circ$. Thus, $(2x + 3)+(x - 6)=180$.
## Step2: Solve for $x$
Combine like terms: $2x + x + 3 - 6 = 180$ → $3x - 3 = 180$. Add 3 to both sides: $3x = 183$. Divide by 3: $x = 61$.
## Step3: Find angle measures
Substitute $x = 61$ into $2x + 3$: $2(61)+3 = 122 + 3 = 125^\circ$. For $x - 6$: $61 - 6 = 55^\circ$. Check: $125 + 55 = 180^\circ$ (correct for a linear pair).

# Answer:
$x = 61$, angles are $125^\circ$ and $55^\circ$

### Problem 9:
# Explanation:
## Step1: Recall complementary angles
Complementary angles sum to $90^\circ$, so $m\angle A + m\angle B = 90^\circ$. Substitute the given expressions: $(6x + 40)+(4x + 30)=90$.
## Step2: Solve for $x$
Combine like terms: $6x + 4x + 40 + 30 = 90$ → $10x + 70 = 90$. Subtract 70: $10x = 20$. Divide by 10: $x = 2$.
## Step3: Find angle measures
Substitute $x = 2$ into $m\angle A$: $6(2)+40 = 12 + 40 = 52^\circ$. For $m\angle B$: $4(2)+30 = 8 + 30 = 38^\circ$. Check: $52 + 38 = 90^\circ$ (correct for complementary angles).

# Answer:
$x = 2$, $m\angle A = 52^\circ$, $m\angle B = 38^\circ$

### Problem 10:
(Note: Since the diagram for angles $a, b, c$ and the number line with $x$ is provided, we analyze the angle relationships. Typically, vertical angles are equal, linear pairs sum to $180^\circ$, and adjacent angles may have specific relationships. Assuming $x$ is an acute angle (between $0^\circ$ and $90^\circ$) from the number line:
- $a$ and $x$ might be vertical angles (equal), $b$ and $x$ form a linear pair (so $b = 180^\circ - x$), and $c$ is vertical to $b$ (so $c = b = 180^\circ - x$) or adjacent. But without the exact diagram, we assume standard angle relationships. If $x$ is on the left part of the number line (small angle), then:
- $a = x$ (vertical angles), $b = 180^\circ - x$ (linear pair with $x$), $c = b$ (vertical angles). So on the number line, $x$ is near $0^\circ$, $a$ is equal to $x$ (same position as $x$ or close), $b$ and $c$ are near $180^\circ$ (since $180^\circ - x$ is obtuse).

But since the problem says "place the measures... according to location of $x$", we infer:
- If $x$ is between $0^\circ$ and $90^\circ$, $a$ (equal to $x$) is near $x$, $b$ (obtuse, $180 - x$) is near $180^\circ$, and $c$ (equal to $b$) is also near $180^\circ$.

However, without the exact diagram, we can state the relationships:
- Vertical angles: $a = x$, $c = b$
- Linear pair: $b + x = 180^\circ$ → $b = 180^\circ - x$, so $c = 180^\circ - x$

Thus, on the number line, $x$ and $a$ are close to $0^\circ$, $b$ and $c$ are close to $180^\circ$.

# Answer:
(Assuming standard diagram) $a$ is equal to $x$ (placed near $x$ on the number line), $b$ and $c$ are equal to $180^\circ - x$ (placed near $180^\circ$ on the number line).

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 7:

Step1: Identify angle relationship

Step2: Solve for \(x\)

Step3: Find angle measures

Step1: Identify angle relationship

Step2: Solve for \(x\)

Step3: Find angle measures

Step1: Recall complementary angles

Step2: Solve for \(x\)

Step3: Find angle measures

Answer:

Problem 8: