Respuesta :
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., [tex]\boxed{ \ \angle A + \angle B + \angle C = 180^0 \ }[/tex]
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:
[tex]\boxed{ \ 90^0 + x + 2x = 180^0 \ }[/tex]
Both sides subtracted by 90°.
[tex]\boxed{ \ x + 2x = 180^0 - 90^0 \ }[/tex]
[tex]\boxed{ \ 3x = 90^0 \ }[/tex]
Both sides divided by 3.
We get [tex]\boxed{ \ x = 30^0 \ }[/tex]
We substitute the value of x back into B and C.
[tex]\boxed{\boxed{ \ \angle B = x\ } \rightarrow \boxed{ \ \angle B = 30^0 \ }} \\\boxed{\boxed{ \ \angle C = 2x\ } \rightarrow \boxed{ \ \angle C = 60^0 \ }}[/tex]
We have succeeded in getting the measures of the two angles.
Learn more
- Undefined terms needed to define angles https://brainly.com/question/3717797
- What is 270° converted to radians https://brainly.com/question/3161884
- A triangle is rotated 90° about the origin https://brainly.com/question/2992432
Keywords: the ratio, the measure, the acute angle, a right triangle, 1/2, 180°, 90°, 30°, 60°, the interior angles
The two acute angles of a right angled triangle are [tex]\boxed{30^{\circ}}[/tex] and [tex]\boxed{60^{\circ}}[/tex].
Further explanation:
It is given that the ratio of the measure of the acute angles of a right angled triangle is [tex]\frac{1}{2}[/tex].
We know that for any triangle the summation of the all three angle is [tex]180^{\circ}[/tex] and as the triangle is right angled that means one angle is of [tex]90^{\circ}[/tex] and the other two angles are acute angles.
So the summation of the other two acute angle is [tex]90^{\circ}[/tex].
Suppose the two acute angle is denoted as [tex]x[/tex] and [tex]y[/tex]. So in equation form it can be written as follows,
[tex]\boxed{x+y=90}[/tex] …… (1)
The ratio of the measure of the two acute angle is [tex]\dfrac{1}{2}[/tex]. This can be written in equation form as follows,
[tex]\begin{aligned}\dfrac{x}{y}&=\dfrac{1}{2}\\2x&=y\end{aligned}[/tex]
After rearranging the above equation we get,
[tex]\boxed{y=2x}[/tex] …… (2)
Now substitute the above calculated value of [tex]y[/tex] in equation (1) to obtain the value of [tex]x[/tex] as follows,
[tex]\begin{aligned}x+2x&=90\\3x&=90\\x&=\dfrac{90}{3}\\x&=30\end{aligned}[/tex]
Substitute this value of [tex]x[/tex] in equation (2) to obtain the value of [tex]y[/tex] as follows,
[tex]\begin{aligned}y&=2\times 30\\&=60\end{aligned}[/tex]
Therefore, the value of [tex]x[/tex] and [tex]y[/tex] is [tex]30[/tex] and [tex]60[/tex] respectively.
Thus, the two acute angles of a right angled triangle are [tex]\boxed{30^{\circ}}[/tex] and [tex]\boxed{60^{\circ}}[/tex].
Learn more:
1. Problem on rules of transformation of triangles: https://brainly.com/question/2992432
2. Problem on definition of an angle uses the undefined term: https://brainly.com/question/3413207
3. Problem on the triangle to show on the graph with coordinates: https://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.
