Parts of a Triangle: Using angles in a triangle

To solve this problem, we'll follow these steps:

• Step 1: Use the angle sum property of triangle $ABC$.
• Step 2: Set up an equation using the condition that $∢A = 3(∢B + ∢C)$.
• Step 3: Solve the equations to find $∢A$.

Now, let's work through each step:
Step 1: According to the angle sum property of a triangle:
$∢A + ∢B + ∢C = 180^\circ$ Step 2: We are given $∢A = 3(∢B + ∢C)$. Substitute this relationship into the equation from Step 1:
$3(∢B + ∢C) + ∢B + ∢C = 180^\circ$ Step 3: Simplify the equation:
Start by letting $∢B + ∢C = x$, then $∢A = 3x$. Substitute into the equation:
$3x + x = 180^\circ$ $4x = 180^\circ$ Solving for $x$:
$x = \frac{180^\circ}{4} = 45^\circ$ So, $∢B + ∢C = 45^\circ$. Since $∢A = 3x$:
$∢A = 3 \times 45^\circ = 135^\circ$

Therefore, the measure of angle $∢A$ is $\mathbf{135^\circ}$.

To solve this problem, we'll systematically go through the following steps:

• Use the sum of angles in a triangle to relate $∢A$ and $∢B$.
• Substitute the given relationship between $∢A$ and $∢B$ into the equation.
• Solve for $∢B$, and then find $∢A$.

First, note that since triangle $ABC$ is right-angled, $∢C = 90^\circ$. Therefore:

$∢A + ∢B + ∢C = 180^\circ$

$∢A + ∢B + 90^\circ = 180^\circ$

This simplifies to:

$∢A + ∢B = 90^\circ$

Given $∢A = 4∢B$, substitute it into the equation:

$4∢B + ∢B = 90^\circ$

Simplify to:

$5∢B = 90^\circ$

Divide by 5:

$∢B = 18^\circ$

Now, using $∢A = 4∢B$:

$∢A = 4 \times 18^\circ = 72^\circ$

Therefore, the calculated angles are $∢B = 18^\circ$ and $∢A = 72^\circ$.

The correct answer is choice 3: $∢B = 18^\circ, ∢A = 72^\circ$.

To determine the angle $\angle B$ in triangle ABC with given $\angle A = 20^\circ$ and $\angle B > 90^\circ$, consider these facts:

The sum of all angles in any triangle is $180^\circ$.

With $\angle A = 20^\circ$, and knowing that $\angle B$ should be greater than $90^\circ$, mathematically, the sum of $\angle B$ and $\angle C$ should be $160^\circ$. However, without specific value for $\angle C$, multiple combinations of $\angle B$ and $\angle C$ that satisfy this condition exist.

To determine a unique value for $\angle B$, more information about $\angle C$ or any other angles or conditions is needed.

Thus, with the current information, it is not possible to calculate an exact measure for $\angle B$. Hence, the answer is No.

To solve this problem, we need to determine the measure of angle $∢C$ in the right triangle $\triangle ABC$ where angle $∢A = 20°$.

Since $\triangle ABC$ is a right triangle, we know that one angle, $∢B$, is $90°$. Hence, the other two angles, $∢A$ and $∢C$, must sum to $90°$ as well.

We are given that $∢A = 20°$. Therefore, we can set up the equation:$∢A + ∢C = 90°$

Substitute the given value of $∢A$ into the equation:
$20° + ∢C = 90°$

To solve for $∢C$, subtract $20°$ from both sides:
$∢C = 90° - 20°$

Thus, we find that:
$∢C = 70°$

Therefore, the size of angle $∢C$ is $\textbf{70°}$.

To solve for $\angle A$ in triangle $\triangle ABC$, we proceed as follows:

• First, note that the sum of angles in any triangle is $180^\circ$. Therefore, $\angle A + \angle B + \angle C = 180^\circ$.
• We know that $\angle B = 3 \angle A$ and $\angle C = \frac{1}{2} \angle A$.
• Substitute these expressions into the triangle sum equation: $\angle A + 3\angle A + \frac{1}{2}\angle A = 180^\circ$.
• Combine like terms: $\angle A + 3\angle A + \frac{1}{2}\angle A = 4\angle A + \frac{1}{2}\angle A = \frac{9}{2}\angle A$.
• The equation becomes $\frac{9}{2} \angle A = 180^\circ$.
• To solve for $\angle A$, multiply both sides by $\frac{2}{9}$:
$\angle A = \frac{2}{9} \times 180^\circ = 40^\circ$.
• Check consistency: $\angle A = 40^\circ$ leads to $\angle B = 120^\circ$ and $\angle C = 20^\circ$.
• Verify that $\triangle ABC$ is consistent with being obtuse: Indeed, the triangle has $\angle B = 120^\circ$ which is greater than $90^\circ$, confirming the triangle is obtuse.

Therefore, it is possible to calculate $\angle A$, and the solution is $\angle A = 40^\circ$.