Question

The ratio of the measure of the acute angle in a right triangle is 1/2 . Find the measures of the two angles

Answer 1

30° and 60°.

Further explanation

We will solve the problem of the measures of angles in the triangle.

Recall this condition:

• The acute angle ⇒ an angle of less than 90°.
• The right angle ⇒ an angle of exactly 90°.
• A right triangle ⇒ a triangle in which one angle is a right angle.
• The interior angles ⇒ the angles inside a triangle.
• All the interior angles in a triangle , i.e., $\boxed{ \ \angle A + \angle B + \angle C = 180^0 \ }$

Given that:

The ratio of the measure of the acute angle in a right triangle is ¹/₂.

Question:

Find the measures of the two angles.

The Process:

We call it the triangle ABC. An interior angle inside is a right angle, e.g., ∠A = 90°.

From the ratio of two other acute angles, i.e., 1: 2, we call it ∠B = x and ∠C = 2x.

Let's arrange the three angles in ABC triangle as follows:

$\boxed{ \ 90^0 + x + 2x = 180^0 \ }$

Both sides subtracted by 90°.

$\boxed{ \ x + 2x = 180^0 - 90^0 \ }$

$\boxed{ \ 3x = 90^0 \ }$

Both sides divided by 3.

We get $\boxed{ \ x = 30^0 \ }$

We substitute the value of x back into B and C.

$\boxed{\boxed{ \ \angle B = x\ } \rightarrow \boxed{ \ \angle B = 30^0 \ }} \\\boxed{\boxed{ \ \angle C = 2x\ } \rightarrow \boxed{ \ \angle C = 60^0 \ }}$

We have succeeded in getting the measures of the two angles.

Learn more

1. Undefined terms needed to define angles  brainly.com/question/3717797
2. What is 270° converted to radians brainly.com/question/3161884
3. A triangle is rotated 90° about the origin brainly.com/question/2992432

Keywords: the ratio, the measure, the acute angle, a right triangle, 1/2, 180°, 90°, 30°, 60°, the interior angles

Answer 2

The two acute angles of a right angled triangle are $\boxed{30^{\circ}}$ and $\boxed{60^{\circ}}$.

Further explanation:

It is given that the ratio of the measure of the acute angles of a right angled triangle is $\frac{1}{2}$.

We know that for any triangle the summation of the all three angle is $180^{\circ}$ and as the triangle is right angled that means one angle is of $90^{\circ}$ and the other two angles are acute angles.

So the summation of the other two acute angle is $90^{\circ}$.

Suppose the two acute angle is denoted as $x$ and $y$. So in equation form it can be written as follows,

$\boxed{x+y=90}$    …… (1)

The ratio of the measure of the two acute angle is $\dfrac{1}{2}$. This can be written in equation form as follows,

$\begin{aligned}\dfrac{x}{y}&=\dfrac{1}{2}\\2x&=y\end{aligned}$

After rearranging the above equation we get,

$\boxed{y=2x}$  …… (2)

Now substitute the above calculated value of $y$ in equation (1) to obtain the value of $x$ as follows,

$\begin{aligned}x+2x&=90\\3x&=90\\x&=\dfrac{90}{3}\\x&=30\end{aligned}$

Substitute this value of $x$ in equation (2) to obtain the value of $y$ as follows,

$\begin{aligned}y&=2\times 30\\&=60\end{aligned}$

Therefore, the value of $x$ and $y$ is $30$ and $60$ respectively.

Thus, the two acute angles of a right angled triangle are $\boxed{30^{\circ}}$ and $\boxed{60^{\circ}}$.

Learn more:  

1. Problem on rules of transformation of triangles: brainly.com/question/2992432

2. Problem on definition of an angle uses the undefined term: brainly.com/question/3413207

3. Problem on the triangle to show on the graph with coordinates: brainly.com/question/7437053

Answer details:

Grade: Middle School

Subject: Mathematics

Chapter: Angles and Triangles

Keywords:  Angle, triangle, ratio, measure, right angled triangle, 30 degree, 60 degree, acute angle, summation, equation, rearrangement, 90 degree, perpendicular, vertical, horizontal, equation, value.