Sum and Difference of Angles: Using variables

Since in an equilateral triangle all sides are equal and all angles are equal. It is also known that in a triangle the sum of angles is 180°, we can calculate X in the following way:

$X+5+X+5+X+5=180$

$3X+15=180$

$3X=180-15$

$3X=165$

Let's divide both sides by 3:

$\frac{3X}{3}=\frac{165}{3}$

$X=55$

Remember that the sum of angles in a triangle is equal to 180.

In an equilateral triangle, all sides and all angles are equal to each other.

Therefore, we will calculate as follows:

$x+x+x=180$

$3x=180$

We divide both sides by 3:

$x=60$

Since this is an equilateral triangle, all angles are also equal.

As the sum of angles in a triangle is 180 degrees, each angle is equal to 60 degrees. (180:3=60)

From this, we can conclude that: $60=8x$

Let's divide both sides by 8:

$\frac{60}{8}=\frac{8x}{8}$

$7.5=x$

To find the size of angle $\alpha$, we proceed as follows:

• Step 1: Establish the relationship for the angles as supplementary: $\alpha + (2x + 30) = 180^\circ$.
• Step 2: Substitute $\alpha = x + 30$ into the equation:

Substituting, we get:

Combine like terms:

Step 3: Solve for $x$:
Subtract 60 from both sides:

Divide both sides by 3:

• Step 4: Substitute $x = 40$ back into $\alpha = x + 30$ to find $\alpha$:

$\alpha = 40 + 30 = 70$

Thus, the size of the angle $\alpha$ is 70.

As we know that triangle ABC is isosceles.

$B=C=2X$

It is known that in a triangle the sum of the angles is 180.

Therefore, we can calculate in the following way:

$2X+2X+4X=180$

$4X+4X=180$

$8X=180$

We divide the two sections by 8:

$\frac{8X}{8}=\frac{180}{8}$

$X=22.5$

Let's remember that the sum of angles in a triangle is equal to 180.

Therefore, we will use the following formula:

$A+B+C=180$

We'll substitute the known data:

$115+8x+20+x=180$

We'll combine similar terms:

$9x+135=180$

We'll move terms to one side and maintain the appropriate sign:

$9x=180-135$

$9x=45$

We'll divide both sides by 9:

$x=5$

Let's remember that the sum of angles in a triangle is equal to 180 degrees.

Therefore, we will use the following formula:

$A+B+C=180$

Now let's input the known data:

$\alpha+49.6+38=180$

$\alpha+87.6=180$

We'll move the term to the other side and keep the appropriate sign:

$\alpha=180-87.6$

$\alpha=92.4$

Let's remember that the sum of angles in a triangle equals 180 degrees.

Therefore, we will use the following formula:

$A+B+C=180$

Let's input the known data:

$11x+40x+80-x=180$

We'll combine the x terms:

$50x+80=180$

We'll move terms to one side and maintain the appropriate sign:

$50x=180-80$

$50x=100$

We'll divide both sides by 50:

$x=2$

Let's remember that the sum of angles in a triangle is equal to 180 degrees.

Therefore, we will use the following formula:

$A+B+C=180$

Now let's input the known data:

$89+5\frac{1}{2}+\alpha=180$

$\alpha+94.5=180$

We'll move the term to the other side and keep the appropriate sign:

$\alpha=180-94.5$

$\alpha=85.5$

Let's remember that the sum of angles in a triangle equals 180 degrees.

Therefore, to find $x$, we will use the following equation:

$A+B+C=180$

Now let's substitute in the known data:

$\frac{1}{2}x+6+120.2-x+0.88x+50=180$

We'll factor out $x$ as a common term in the equation:

$x(\frac{1}{2}-1+0.88)+176.2=180$

Let's next expand the parentheses:

$\frac{1}{2}x-x+0.88x+176.2=180$

Then, we'll combine the $x$ terms:

$0.38x+176.2=180$

Now we'll move terms to opposite sides and maintain the appropriate sign:

$\text{0}.38x=180-176.2$

$0.38x=3.8$

Finally, we will divide both sides by 0.38 to get our answer:

$x=10$

To solve this problem, let's calculate $∢A$ with the steps outlined below:

• Step 1: Write the equations for each angle based on the given conditions: $∢B = 2A$$∢C = 3B = 3(2A) = 6A$

• Step 2: Use the sum of angles in a triangle: $∢A + ∢B + ∢C = 180^\circ$ Substitute the expressions: $A + 2A + 6A = 180$

• Step 3: Simplify the equation: $9A = 180$ Divide both sides by 9 to solve for $A$: $A = \frac{180}{9} = 20$

Therefore, the solution to the problem is $∢A = 20^\circ$.

Angle Y complements 180 and we can calculate it since we know the adjacent angle.

Let's calculate it as follows:

$180-105=75$

Now that we found angle Y, we can calculate angle X since we have the other two angles in the same triangle: 72 and 75.

$180-75-72=33$

We can calculate angle Z since we have two angles in the triangle: 25 and 105

The sum of angles in a triangle is 180, so we'll calculate Z as follows:

$180-105-25=50$

First, let's note that in triangle ACB we are given two angles.

Angle ABC equals 47 degrees, angle ACB equals 90 degrees.

Since the sum of angles in a triangle equals 180, we can calculate angle BAC and find the value of Y as follows:

$180-90-47=43$

Now let's look at triangle ACD, where we are also given two angles.

Angle CAD equals 43 degrees, angle ACD equals 90 degrees.

Since the sum of angles in a triangle equals 180, we can calculate angle ADC and find the value of X as follows:

$180-90-43=47$

To find the value of $\angle C$, follow these steps:

Step 1: Set up the equations.
We know:
- $\angle A = \alpha$
- $\angle B = 3\alpha$

Using the given condition $\angle A + \angle B = 2\angle C$:
$\alpha + 3\alpha = 2\angle C \implies 4\alpha = 2\angle C \implies \angle C = 2\alpha$

Step 2: Use the triangle angle sum property.
From the triangle angle sum, we have:
$\angle A + \angle B + \angle C = 180^\circ$ Substituting the expressions for the angles:
$\alpha + 3\alpha + 2\alpha = 180^\circ$ $6\alpha = 180^\circ$ Solving for $\alpha$:
$\alpha = \frac{180^\circ}{6} = 30^\circ$

Step 3: Calculate $\angle C$.
Since $\angle C = 2\alpha$:
$\angle C = 2 \times 30^\circ = 60^\circ$ Therefore, the size of angle $\angle C$ is $\boxed{60^\circ}$.

To solve this problem, we'll use the properties of a triangle and given ratio:

• Step 1: Let $∢B = x$ and $∢C = 3x$ as per the given ratio $\frac{∢B}{∢C} = \frac{1}{3}$.
• Step 2: Use the triangle sum property: $∢A + ∢B + ∢C = 180^\circ$.
• Step 3: Substitute known values: $70^\circ + x + 3x = 180^\circ$.
• Step 4: Simplify: $4x + 70^\circ = 180^\circ$.
• Step 5: Solve for $x$: $4x = 110^\circ$.
• Step 6: Determine $x$: $x = 27.5^\circ$.
• Step 7: Calculate $∢C$: $∢C = 3x = 3 \times 27.5^\circ = 82.5^\circ$.

Therefore, the measure of angle $∢C$ is $82.5^\circ$.

To solve this problem, we will follow these steps:

• Step 1: Express all angles in terms of a single variable.
• Step 2: Apply the angle sum property of triangles.
• Step 3: Calculate the measures of the individual angles.

Now, let's proceed with the detailed solution:

Step 1: We know that:

• $B = 5A$
• $C = 2B = 2(5A) = 10A$

Thus, all angles are expressed in terms of $A$.

Step 2: Use the angle sum property:

$A + B + C = 180^\circ$

Substituting for $B$ and $C$:

$A + 5A + 10A = 180^\circ$

$16A = 180^\circ$

Solve for $A$:

$A = \frac{180^\circ}{16} = 11.25^\circ$

Step 3: Calculate $B$ and $C$:

$B = 5A = 5 \times 11.25^\circ = 56.25^\circ$

$C = 2B = 2 \times 56.25^\circ = 112.5^\circ$

Therefore, the measure of angle $C$ is $56\frac{1}{4}°$, which matches the provided correct answer.

To approach the problem, follow these steps:

• Step 1: Establish and simplify the given equation.

• Step 2: Use properties of triangle angles to form additional equations.

• Step 3: Solve the equations to find $\angle A$.

Step 1: We're given $\angle C + \angle A = 2(\angle A + \angle B)$.

Substitute $\angle B = \angle A$:

$\angle C + \angle A = 2(\angle A + \angle A) = 4\angle A$.

Step 2: Use the triangle angle sum property:

$\angle A + \angle B + \angle C = 180^\circ$.

Since $\angle B = \angle A$, we have:

$\angle A + \angle A + \angle C = 180^\circ$, simplifying to $2\angle A + \angle C = 180^\circ$.

Step 3: Solve the System:

• From $2\angle A + \angle C = 180^\circ$, express $\angle C$ as:

• $\angle C = 180^\circ - 2\angle A$.

• Substitute into the equation $\angle C + \angle A = 4\angle A$:

• $(180^\circ - 2\angle A) + \angle A = 4\angle A$.

• Simplify: $180^\circ - \angle A = 4\angle A$.

• Add $\angle A$ to both sides: $180^\circ = 5\angle A$.

• Solving for $\angle A$, we get: $\angle A = \frac{180^\circ}{5}$.

• Thus, $\angle A = 36^\circ$.