practice
angles of a triangle
find the value of each variable.
*please show your work and
circle your answers.
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Question

Accepted Answer

Let's solve problem 1 first (we can solve others similarly using the triangle angle - sum property which states that the sum of the interior angles of a triangle is $180^{\circ}$):

### Problem 1
# Explanation:
## Step 1: Recall the triangle angle - sum property
The sum of the interior angles of a triangle is $180^{\circ}$. In the first triangle, the given angles are $50^{\circ}$, $45^{\circ}$, and $x^{\circ}$. So we can write the equation: $50 + 45+x=180$
## Step 2: Simplify and solve for $x$
First, add $50$ and $45$: $50 + 45=95$. Then the equation becomes $95+x = 180$. Subtract $95$ from both sides: $x=180 - 95$
# Answer:
$x = 85$

### Problem 2
# Explanation:
## Step 1: Apply the triangle angle - sum property
The angles of the triangle are $x^{\circ}$, $2x^{\circ}$, and $3x^{\circ}$. So $x + 2x+3x=180$
## Step 2: Combine like terms and solve
Combine the $x$ terms: $x+2x + 3x=6x$. Then $6x=180$. Divide both sides by $6$: $x=\frac{180}{6}$
# Answer:
$x = 30$

### Problem 3
# Explanation:
## Step 1: Identify the right - triangle property
This is a right triangle (one angle is $90^{\circ}$). The sum of the non - right angles in a right triangle is $90^{\circ}$. The angles are $(2x - 2)^{\circ}$ and $(x + 5)^{\circ}$, so $(2x-2)+(x + 5)=90$
## Step 2: Simplify and solve for $x$
First, expand the left - hand side: $2x-2+x + 5=90$. Combine like terms: $3x + 3=90$. Subtract $3$ from both sides: $3x=90 - 3=87$. Divide both sides by $3$: $x=\frac{87}{3}$
# Answer:
$x = 29$

### Problem 4
# Explanation:
## Step 1: Use the exterior angle property (or triangle angle - sum)
The sum of the interior angles of a triangle is $180^{\circ}$. The two given angles are $10^{\circ}$ and $23^{\circ}$, and the third angle is supplementary to $x$ (since they form a linear pair). Wait, actually, the sum of the interior angles: let the interior angles be $10^{\circ}$, $23^{\circ}$, and $180 - x$ (because $x$ and the third interior angle are supplementary). So $10+23+(180 - x)=180$
## Step 2: Simplify and solve
Simplify the left - hand side: $10 + 23+180-x=213 - x$. Then $213 - x=180$. Subtract $180$ from both sides: $33 - x=0$, so $x = 147$ (Alternatively, using the exterior angle theorem: the exterior angle $x$ is equal to the sum of the two non - adjacent interior angles. So $x=180 - 10-23$? No, wait, the two angles given are the non - adjacent interior angles to the exterior angle $x$. Wait, the triangle has two angles: if we consider the exterior angle, the exterior angle is equal to the sum of the two remote interior angles. Wait, the angles inside the triangle: one angle is $180 - 10=170$? No, no. Wait, the triangle has angles: let's correct. The two angles marked are $10^{\circ}$ (exterior) and $23^{\circ}$ (exterior). Wait, actually, the interior angles are $180 - 10$ and $180 - 23$ and $x$. Wait, no, the sum of interior angles is $180$. So $(180 - 10)+(180 - 23)+x=180$? No, that's wrong. Wait, the correct approach: the two angles adjacent to the $10^{\circ}$ and $23^{\circ}$ exterior angles are the interior angles. Let the interior angles be $a = 180 - 10$ and $b=180 - 23$ and $x$. But $a + b+x=180$. But $a = 170$, $b = 157$, $170+157+x=180$ gives $x=180-(170 + 157)=180 - 327=- 147$, which is wrong. Wait, I made a mistake. The correct way: the triangle has two angles: the angles that are supplementary to $10^{\circ}$ and $23^{\circ}$ are the interior angles. Wait, no, the triangle has three interior angles. Let's look at the diagram again. The triangle has two exterior angles marked as $10^{\circ}$ and $23^{\circ}$, and the third angle is $x$ (interior). Wait, the sum of exterior angles of a triangle is $360^{\circ}$. So $10+23+x = 360$? No, that's for polygons. Wait, no, each interior angle and its corresponding exterior angle sum to $180^{\circ}$. Let the interior angles be $A$, $B$, $C$. Then $A=180 - 10$, $B = 180 - 23$, $C=x$. And $A + B + C=180$. So $(180 - 10)+(180 - 23)+x=180$. $170+157+x=180$, $x=180 - 327=- 147$, which is impossible. So I must have misinterpreted the diagram. Wait, the diagram shows a very thin triangle with two exterior angles (the small angles outside) of $10^{\circ}$ and $23^{\circ}$, and the third angle (the large angle inside) is $x$. So the sum of the interior angles: the two interior angles adjacent to the $10^{\circ}$ and $23^{\circ}$ exterior angles are $180 - 10$ and $180 - 23$, and the third interior angle is $x$. But that can't be. Wait, the correct formula: the sum of the interior angles of a triangle is $180^{\circ}$. If the two angles are $10^{\circ}$ and $23^{\circ}$ (interior angles), then $10 + 23+x=180$, $x = 147$. Ah, that's it! I misread the diagram. The $10^{\circ}$ and $23^{\circ}$ are interior angles. So $x=180-(10 + 23)=147$
# Answer:
$x = 147$

### Problem 5
# Explanation:
## Step 1: Apply the triangle angle - sum property
The triangle has three equal angles (it's an equilateral triangle, since all angles are $x^{\circ}$). So $x+x+x=180$
## Step 2: Solve for $x$
Combine like terms: $3x=180$. Divide both sides by $3$: $x = 60$
# Answer:
$x = 60$

### Problem 6
# Explanation:
## Step 1: Use the triangle angle - sum property
The angles are $54^{\circ}$, $(x + 40)^{\circ}$, and $(2x-5)^{\circ}$. So $54+(x + 40)+(2x-5)=180$
## Step 2: Simplify and solve
First, expand the left - hand side: $54+x + 40+2x-5=89 + 3x$. Then $89+3x=180$. Subtract $89$ from both sides: $3x=180 - 89 = 91$. Wait, no, $54+40 - 5=89$, $x+2x=3x$. So $3x=180 - 89=91$? Wait, $54 + 40=94$, $94-5 = 89$. Then $3x=180 - 89=91$, $x=\frac{91}{3}\approx30.33$? Wait, no, I made a mistake. $54+(x + 40)+(2x-5)=54+x + 40+2x-5=(54 + 40-5)+(x + 2x)=89 + 3x$. Then $89+3x=180$, $3x=180 - 89 = 91$, $x=\frac{91}{3}\approx30.33$. Wait, but maybe the angle is $54^{\circ}$, $(x + 40)^{\circ}$, $(2x - 5)^{\circ}$. Let's check again: $54+x + 40+2x-5=180$, $3x+89 = 180$, $3x=91$, $x=\frac{91}{3}\approx30.33$
# Answer:
$x=\frac{91}{3}$ (or approximately $30.33$)

### Problem 7
# Explanation:
## Step 1: Use vertical angles and triangle angle - sum
The two triangles are vertical (the angles formed by the intersection of two lines are vertical angles). So the angle $x$ in the first triangle is equal to the angle in the second triangle. The second triangle has angles $43^{\circ}$ and $57^{\circ}$. The sum of angles in a triangle is $180^{\circ}$, so the third angle of the second triangle is $180-(43 + 57)=80^{\circ}$. Since vertical angles are equal, $x = 80$
# Answer:
$x = 80$

### Problem 8
# Explanation:
## Step 1: Analyze the right triangle
The large triangle is a right triangle (has a right angle). The angle of $65^{\circ}$ and $y$ are complementary (since they are in a right triangle), so $y=90 - 65=25$. Also, the small triangle is a right triangle, and $x$ and $65^{\circ}$ are complementary? Wait, no. Wait, the large triangle has angles $x$, $65^{\circ}$, and $90^{\circ}$. Wait, no, the line inside the triangle is perpendicular to the base, so we have two right triangles. In the large right triangle, one angle is $x$, one is $65^{\circ}$, and one is $90^{\circ}$. Wait, no, $x+65 + 90=180$? No, $x+65=90$ (since the other angle is $90^{\circ}$). So $x=90 - 65 = 25$. And $z$ is equal to $x$ (since the two small triangles are similar or by angle - sum). Wait, $z = 25$, $y=25$
# Answer:
$x = 25$, $y = 25$, $z = 25$

### Problem 9
# Explanation:
## Step 1: Use the exterior angle property
The exterior angle $x$ is equal to the sum of the two non - adjacent interior angles. The two non - adjacent interior angles are $44^{\circ}$ and the other interior angle (which is equal to $44^{\circ}$ if the triangle is isoceles? Wait, no, the triangle has two equal interior angles? Wait, no, the exterior angle is equal to the sum of the two remote interior angles. The two remote interior angles are $44^{\circ}$ and the other angle. Wait, the sum of interior angles: let the two base angles be $a$ and $a$ (if it's isoceles), and the vertex angle is $44^{\circ}$. Then $2a+44 = 180$, $2a=136$, $a = 68$. Then the exterior angle $x=180 - 68=112$? Wait, no, the exterior angle is adjacent to the base angle. So $x=180 - 68=112$
# Answer:
$x = 112$

### Problem 10
# Explanation:
## Step 1: Use the exterior angle property
The exterior angle is $122^{\circ}$, and the triangle is a right triangle (has a right angle $90^{\circ}$). The exterior angle is equal to the sum of the right angle and $x$. So $122=90 + x$
## Step 2: Solve for $x$
Subtract $90$ from both sides: $x=122 - 90=32$
# Answer:
$x = 32$

### Problem 11
# Explanation:
## Step 1: Use the exterior angle property (or triangle angle - sum)
First, look at the triangle with angles $55^{\circ}$, $28^{\circ}$, and the angle adjacent to $(40 + y)^{\circ}$. The sum of angles in a triangle is $180$, so the angle adjacent to $(40 + y)^{\circ}$ is $180-(55 + 28)=97^{\circ}$. Then $(40 + y)$ and $97^{\circ}$ are supplementary (since they form a linear pair), so $40 + y+97=180$, $y=180-(40 + 97)=43$. For the angle $(6x - 7)^{\circ}$, it is equal to the sum of $55^{\circ}$ and $28^{\circ}$ (exterior angle property), so $6x-7=55 + 28$
## Step 2: Solve for $x$
First, calculate $55 + 28 = 83$. Then $6x-7=83$. Add $7$ to both sides: $6x=90$. Divide by $6$: $x = 15$
# Answer:
$x = 15$, $y = 43$

### Problem 12
# Explanation:
## Step 1: Recognize the isoceles triangle
The triangle is isoceles (two angles are equal, as indicated by the marks). Let the two equal angles be $x$. The sum of angles in a triangle is $180^{\circ}$, so $x+x+\text{third angle}=180$. But since it's isoceles and the base is parallel? Wait, no, the triangle has two equal angles, so $2x+\text{third angle}=180$. But from the diagram, the two equal angles are $x$, so $x = 60$ (if it's equilateral) or other. Wait, no, the triangle is isoceles with two angles equal to $x$. Wait, the sum of angles: if we assume it's an equilateral triangle, $x = 60$. But more accurately, since the triangle has two equal angles, and if we consider the base, the two base angles are equal. So $2x+\text{vertex angle}=180$. But without more info, but from the diagram, it's an equilateral triangle, so $x = 60$
# Answer:
$x = 60$ (assuming equilateral triangle)

### Problem 13
# Explanation:
## Step 1: Analyze the isoceles triangle
The triangle has two equal sides (marked with $x$), so it's isoceles with two equal angles. The vertex angle is $55^{\circ}$. The sum of angles in a triangle is $180^{\circ}$, so the two base angles are equal. Let each base angle be $y$. Then $55+2y=180$, $2y=180 - 55 = 125$, $y=\frac{125}{2}=62.5$. The angle adjacent to $x$ (exterior angle) is equal to the sum of the two non - adjacent interior angles. Wait, the line is parallel, so by alternate interior angles

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QUESTION IMAGE

Question

Explanation:

Problem 1

Step 1: Recall the triangle angle - sum property

Step 2: Simplify and solve for \(x\)

Step 1: Apply the triangle angle - sum property

Step 2: Combine like terms and solve

Step 1: Identify the right - triangle property

Step 2: Simplify and solve for \(x\)

Answer:

Problem 2