Sum and Difference of Angles: Using ratios for calculation

To solve this problem, we need to understand the relationship between the angles $\alpha$ and $\beta$ given in the diagram.

Step 1: Determine the relationship between $\alpha$ and $\beta$. The angles $\alpha$ and $\beta$ appear to be supplementary, forming a straight line. Hence, their sum is $180^\circ$.

Step 2: Use the given information to calculate $\alpha$. Given $\beta = 100^\circ$, we have:

$\alpha = 180^\circ - \beta = 180^\circ - 100^\circ = 80^\circ.$

Step 3: Calculate the ratio $\alpha:\beta$.

$\text{Ratio of } \alpha \text{ to } \beta = \frac{\alpha}{\beta} = \frac{80}{100} = \frac{4}{5}.$

Step 4: Express the ratio in the required form, which is a multiple-choice problem comparing given choices:

Therefore, the relationship between angle $\alpha$ and $\beta$ according to the drawing is 4:5.

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 $∢A = 20^\circ$.

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, let's follow these steps:

• Step 1: Identify the given information; one angle is $90^\circ$, and the other two angles are in a 2:1 ratio.
• Step 2: Set up an equation using the relationship between the angles: $90^\circ + 2x + x = 180^\circ$.
• Step 3: Solve the equation for $x$.
• Step 4: Calculate the measures of the two angles.

Now, let's work through each step.
Step 1: Given a right triangle, one angle is $90^\circ$, and the other two angles are in the ratio 2:1.

Step 2: Express the other two angles in terms of $x$:
- One angle is $2x$.
- The second angle is $x$.

Step 3: Use the sum of angles in a triangle to write an equation:
$90^\circ + 2x + x = 180^\circ$.

Step 4: Simplify and solve the equation:
Combine like terms:
$90^\circ + 3x = 180^\circ$.

Subtract $90^\circ$ from both sides:
$3x = 90^\circ$.

Divide both sides by 3:
$x = 30^\circ$.

Now we can find the measures of the other angles:
The angle expressed as $x$ is $30^\circ$.
The angle expressed as $2x$ is $2 \times 30^\circ = 60^\circ$.

Therefore, the solution is that the other two angles are $60^\circ$ and $30^\circ$.