solve each of the problems below. these represent the types of questions on your test. be sure to ask questions if you need more help with a topic.
i can classify angle relationships.
1. the relationship between ∠1 and ∠2 is…
image of ∠1 and ∠2 forming a linear pair
2. the relationship between ∠2 and ∠3 is…
image of angles with a right angle marker
3. the relationship between ∠3 and ∠4 is…
image of perpendicular lines forming angles
4. the relationship between ∠1 and ∠3 is…
image of intersecting lines forming vertical angles
5. the relationship between ∠1 and ∠3 is…
image of angles with a right angle marker
6. the relationship between ∠2 and ∠3 is…
image of perpendicular lines forming angles
i can use complementary & supplementary angles to write & solve equations.
7.
image of a right angle split into (x - 18)° and x°
equation: _____________
x: ________
angle measure: ________
8.
image of a linear pair: x° and (2x + 24)°
equation: _____________
x: ________
angle measures: ________
9. two angles are supplementary. the first angle is (4x)° degrees. the second angle is (2x + 6)° degrees. determine the measure of each angle.
equation: _____________
x: ________
angles measures: ________
i can use vertical and adjacent angles to write and solve equations.
10.
image of a triangle with angles 37°, x°, and a 54° exterior angle
equation: _____________
11.
image of vertical angles: (2x + 10)° and 70°
equation: _____________
12.
image of angles: (2x)°, 78°, and (3x - 6)° forming a linear pair?
equation: _____________
©maneuvering the middle llc, 2016

Question

Accepted Answer

### Problem 7
# Explanation:
## Step 1: Identify angle relationship
The two angles are complementary (form a right angle, $90^\circ$). So, $(x - 18) + x = 90$.
## Step 2: Solve the equation
Combine like terms: $2x - 18 = 90$.
Add 18 to both sides: $2x = 108$.
Divide by 2: $x = 54$.
## Step 3: Find angle measures
First angle: $x - 18 = 54 - 18 = 36^\circ$.
Second angle: $x = 54^\circ$.
# Answer:
equation: $\boldsymbol{(x - 18) + x = 90}$
$x$: $\boldsymbol{54}$
angle measure: $\boldsymbol{36^\circ}$ and $\boldsymbol{54^\circ}$

### Problem 8
# Explanation:
## Step 1: Identify angle relationship
The two angles are supplementary (form a straight line, $180^\circ$). So, $x + (2x + 24) = 180$.
## Step 2: Solve the equation
Combine like terms: $3x + 24 = 180$.
Subtract 24: $3x = 156$.
Divide by 3: $x = 52$.
## Step 3: Find angle measures
First angle: $x = 52^\circ$.
Second angle: $2x + 24 = 2(52) + 24 = 128^\circ$.
# Answer:
equation: $\boldsymbol{x + (2x + 24) = 180}$
$x$: $\boldsymbol{52}$
angle measures: $\boldsymbol{52^\circ}$ and $\boldsymbol{128^\circ}$

### Problem 9
# Explanation:
## Step 1: Identify angle relationship
Supplementary angles sum to $180^\circ$. So, $4x + (2x + 6) = 180$.
## Step 2: Solve the equation
Combine like terms: $6x + 6 = 180$.
Subtract 6: $6x = 174$.
Divide by 6: $x = 29$.
## Step 3: Find angle measures
First angle: $4x = 4(29) = 116^\circ$.
Second angle: $2x + 6 = 2(29) + 6 = 64^\circ$.
# Answer:
equation: $\boldsymbol{4x + (2x + 6) = 180}$
$x$: $\boldsymbol{29}$
angles measures: $\boldsymbol{116^\circ}$ and $\boldsymbol{64^\circ}$

### Problem 10
# Explanation:
## Step 1: Identify angle relationship
The sum of the three angles is $180^\circ$ (triangle angle sum, but here it’s a straight angle? Wait, the diagram shows three angles forming a straight line? Wait, no—wait, the angles are $37^\circ$, $x^\circ$, and $54^\circ$, and they form a straight line? Wait, no, actually, the three angles add up to $180^\circ$? Wait, no, looking at the diagram: the angles are $37^\circ$, $x^\circ$, and $54^\circ$, and they are adjacent, forming a straight line? Wait, no, the diagram has a $37^\circ$, $x^\circ$, and $54^\circ$ angle, and the total is $180^\circ$? Wait, no, maybe it’s a triangle? Wait, no, the diagram shows a ray with two angles: $37^\circ$, $x^\circ$, and $54^\circ$ forming a straight line? Wait, no, the correct equation is $37 + x + 54 = 180$? Wait, no, maybe it’s a straight angle (180°), so $37 + x + 54 = 180$. Wait, no, let’s check: $37 + 54 + x = 180$.
## Step 2: Solve the equation
$91 + x = 180$.
Subtract 91: $x = 89$.
Wait, no, maybe the diagram is a straight line with three angles? Wait, the original diagram: “$37^\circ$”, “$x^\circ$”, and “$54^\circ$” adjacent, forming a straight line (180°). So equation: $37 + x + 54 = 180$.
Simplify: $x + 91 = 180$ → $x = 89$.
# Answer:
equation: $\boldsymbol{37 + x + 54 = 180}$
$x$: $\boldsymbol{89}$

### Problem 11
# Explanation:
## Step 1: Identify angle relationship
Vertical angles are equal? Wait, no, the angle $(2x + 10)^\circ$ and the vertical angle to $70^\circ$? Wait, no, the diagram shows two intersecting lines, so vertical angles. Wait, the angle given is $70^\circ$, and the angle $(2x + 10)^\circ$ is vertical to... Wait, no, adjacent angles? Wait, no, when two lines intersect, vertical angles are equal. Wait, the angle $(2x + 10)^\circ$ and the angle opposite to $70^\circ$? Wait, no, maybe $(2x + 10)$ and $70^\circ$ are vertical angles? Wait, no, the diagram: “$(2x + 10)^\circ$” and “$70^\circ$” are vertical angles? Wait, no, vertical angles are equal. Wait, maybe $(2x + 10) = 70$? Wait, no, that would be if they are vertical. Wait, let’s check: if two lines intersect, vertical angles are equal. So $(2x + 10) = 70$.
## Step 2: Solve the equation
$2x + 10 = 70$.
Subtract 10: $2x = 60$.
Divide by 2: $x = 30$.
# Answer:
equation: $\boldsymbol{2x + 10 = 70}$
$x$: $\boldsymbol{30}$

### Problem 12
# Explanation:
## Step 1: Identify angle relationship
The two angles $(2x)^\circ$ and $(3x - 6)^\circ$ and $78^\circ$ form a straight line? Wait, no, the diagram shows $(2x)^\circ$, $78^\circ$, and $(3x - 6)^\circ$ adjacent, forming a straight line (180°). So equation: $2x + 78 + (3x - 6) = 180$.
## Step 2: Solve the equation
Combine like terms: $5x + 72 = 180$.
Subtract 72: $5x = 108$? Wait, no: $2x + 3x - 6 + 78 = 5x + 72$. So $5x + 72 = 180$.
Subtract 72: $5x = 108$? Wait, 180 - 72 = 108? 180 - 72 = 108? Yes. Then $x = 108 / 5 = 21.6$? Wait, no, maybe the diagram is a straight line with two angles: $(2x)^\circ$ and $(3x - 6)^\circ$ and $78^\circ$ sum to 180. Wait, let’s recheck: $2x + (3x - 6) + 78 = 180$ → $5x + 72 = 180$ → $5x = 108$ → $x = 21.6$. But maybe the diagram is a straight line with $(2x)^\circ$ and $(3x - 6)^\circ$ and $78^\circ$ as adjacent angles. Alternatively, maybe $(2x) + (3x - 6) = 180 - 78$? Wait, no, the total is 180. So:
$2x + 3x - 6 + 78 = 180$
$5x + 72 = 180$
$5x = 108$
$x = 21.6$. But maybe the diagram is different. Wait, the problem says “$(2x)^\circ$”, “$78^\circ$”, “$(3x - 6)^\circ$” forming a straight line. So equation: $2x + 78 + 3x - 6 = 180$.
Simplify: $5x + 72 = 180$ → $5x = 108$ → $x = 21.6$.
# Answer:
equation: $\boldsymbol{2x + 78 + 3x - 6 = 180}$
$x$: $\boldsymbol{21.6}$

(Note: For angle relationship classification problems (1–6), we analyze the diagrams:
1. $\angle 1$ and $\angle 2$: supplementary (form a straight line).
2. $\angle 2$ and $\angle 3$: complementary (form a right angle with $\angle 1$ and $\angle 4$).
3. $\angle 3$ and $\angle 4$: complementary (right angle).
4. $\angle 1$ and $\angle 3$: vertical angles (equal).
5. $\angle 1$ and $\angle 3$: complementary (form a right angle with $\angle 2$).
6. $\angle 2$ and $\angle 3$: vertical angles (equal).)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 7

Step 1: Identify angle relationship

Step 2: Solve the equation

Step 3: Find angle measures

Step 1: Identify angle relationship

Step 2: Solve the equation

Step 3: Find angle measures

Step 1: Identify angle relationship

Step 2: Solve the equation

Step 3: Find angle measures

Answer:

Problem 8