Example Question - angle measurement
Here are examples of questions we've helped users solve.
Finding the Value of an Angle
<p> Dado que los ángulos de un triángulo suman 180 grados, podemos establecer la siguiente ecuación: </p> <p> 33° + 145° + x = 180° </p> <p> Ahora, sumamos 33° y 145°: </p> <p> 178° + x = 180° </p> <p> A continuación, restamos 178° de ambos lados: </p> <p> x = 180° - 178° </p> <p> Por lo tanto, tenemos: </p> <p> x = 2° </p>
Solving a Right Triangle
<p>Paso 1: Determinar la hipotenusa \( C \)</p> <p>\( C = \sqrt{A^2+B^2} \)</p> <p>\( C = \sqrt{3^2+5^2} \)</p> <p>\( C = \sqrt{9+25} \)</p> <p>\( C = \sqrt{34} \)</p> <p>\( C \approx 5.83 \) (2 decimales)</p> <p>Paso 2: Encontrar las medidas de los ángulos \( A \) y \( B \)</p> <p>Usando las funciones trigonométricas:</p> <p>Para \( A \):</p> <p>\( \tan(A) = \frac{\text{Opuesto}}{\text{Adyacente}} = \frac{B}{A} \)</p> <p>\( \tan(A) = \frac{5}{3} \)</p> <p>\( A = \arctan\left(\frac{5}{3}\right) \)</p> <p>\( A \approx 59.04^\circ \) (2 decimales)</p> <p>Para \( B \) (sabemos que en un triángulo rectángulo \( A + B = 90^\circ \)):</p> <p>\( B = 90^\circ - A \)</p> <p>\( B \approx 90^\circ - 59.04^\circ \)</p> <p>\( B \approx 30.96^\circ \) (2 decimales)</p> <p>Paso 3: Determinar las 6 razones trigonométricas</p> <p>Sen \( A = \frac{\text{Opuesto a } A}{\text{Hipotenusa}} = \frac{B}{C} = \frac{5}{\sqrt{34}} \approx 0.857 \) (3 decimales)</p> <p>Cos \( A = \frac{\text{Adyacente a } A}{\text{Hipotenusa}} = \frac{A}{C} = \frac{3}{\sqrt{34}} \approx 0.515 \) (3 decimales)</p> <p>Tan \( A = \frac{\text{Opuesto a } A}{\text{Adyacente a } A} = \frac{B}{A} = \frac{5}{3} \approx 1.667 \) (3 decimales)</p> <p>Sen \( B = \frac{\text{Opuesto a } B}{\text{Hipotenusa}} = \frac{A}{C} = \frac{3}{\sqrt{34}} \approx 0.515 \) (3 decimales)</p> <p>Cos \( B = \frac{\text{Adyacente a } B}{\text{Hipotenusa}} = \frac{B}{C} = \frac{5}{\sqrt{34}} \approx 0.857 \) (3 decimales)</p> <p>Tan \( B = \frac{\text{Opuesto a } B}{\text{Adyacente a } B} = \frac{A}{B} = \frac{3}{5} \approx 0.600 \) (3 decimales)</p>
Solving trigonometric equation for sine function
Para resolver la ecuación \( 2\sin(x) = \sqrt{3} \), primero dividimos ambos lados de la ecuación por 2 para aislar el seno de x: \[ \sin(x) = \frac{\sqrt{3}}{2} \] A continuación, buscamos los valores del ángulo x para los cuales el seno es igual a \( \frac{\sqrt{3}}{2} \). Esta relación corresponde a un ángulo de 60 grados o, en radianes, \( \frac{\pi}{3} \). Sin embargo, como estamos buscando un ángulo agudo y positivo, la respuesta debe estar en el primer cuadrante del círculo unitario (ya que el seno es positivo en los cuadrantes primero y segundo, pero solo es agudo en el primero). Por lo tanto, el ángulo agudo y positivo que satisface la ecuación dada es \( x = \frac{\pi}{3} \) radianes.
Solving Trigonometric Equation
La ecuación proporcionada en la imagen es \( 2 \cos(\theta) = \sqrt{3} \). Para resolver esta ecuación, primero debemos aislar \( \cos(\theta) \) dividiendo ambos lados de la ecuación entre 2: \[ \cos(\theta) = \frac{\sqrt{3}}{2} \] Ahora buscamos un ángulo agudo cuyo coseno sea \( \frac{\sqrt{3}}{2} \). Sabemos en trigonometría que el coseno de 30 grados, o \( \frac{\pi}{6} \) radianes, es igual a \( \frac{\sqrt{3}}{2} \). Por lo tanto, el ángulo agudo y positivo \(\theta\) que satisface la ecuación original es: \[ \theta = 30^\circ \] \[ \text{o} \] \[ \theta = \frac{\pi}{6} \text{ radianes} \] Dado que hemos encontrado un ángulo agudo (menor a 90 grados o \( \frac{\pi}{2} \) radianes), y que es positivo, hemos respondido a la pregunta de manera correcta.
Calculating Interior Angles of a Hexagon
The diagram provided is of a hexagon, which has six sides. The sum of the interior angles of a hexagon can be calculated using the formula: Sum of interior angles = (n - 2) × 180° where n is the number of sides in the polygon. For a hexagon, n = 6. So the sum of the interior angles of our hexagon will be: (6 - 2) × 180° = 4 × 180° = 720° From the image, we can see three interior angles are shown. Two of them have the measures of 120° and 135°, and the third angle is labeled as x. We also know that the remaining three interior angles are equal, which means each one has the same measure as x. Since the sum of all the interior angles is 720°, we can set up the following equation: 120° + 135° + x + x + x + x = 720° Now, we combine the known angles and the x's: 255° + 4x = 720° Subtracting 255° from both sides gives us: 4x = 720° - 255° 4x = 465° Finally, we solve for x by dividing both sides by 4: x = 465° ÷ 4 x = 116.25° So, each of the remaining three interior angles, including the one labeled as x in the diagram, measures 116.25°.
Finding Supplementary Angles in Leaf Veins
The question states that the veins in a leaf form a pair of supplementary angles. Supplementary angles are two angles whose measures add up to 180 degrees. Given the measures of the two angles as \( m\angle1 = (7x + 13)^\circ \) and \( m\angle2 = (25x + 7)^\circ \), we can set up an equation that reflects their supplementary nature: \[ (7x + 13) + (25x + 7) = 180 \] Simplify the equation: \[ 7x + 25x + 13 + 7 = 180 \] \[ 32x + 20 = 180 \] Now, solve for \( x \): \[ 32x = 180 - 20 \] \[ 32x = 160 \] \[ x = \frac{160}{32} \] \[ x = 5 \] Now that we have the value of \( x \), we can find the measure of each angle: \[ m\angle1 = 7x + 13 \] \[ m\angle1 = 7(5) + 13 \] \[ m\angle1 = 35 + 13 \] \[ m\angle1 = 48^\circ \] And for the other angle: \[ m\angle2 = 25x + 7 \] \[ m\angle2 = 25(5) + 7 \] \[ m\angle2 = 125 + 7 \] \[ m\angle2 = 132^\circ \] So the measures of the two angles are 48 degrees and 132 degrees, respectively.
Angle Measurement and Classification
The image displays a protractor placed over a series of angles. The angles are marked with points, and there are three parts to the question, asking to find the measure of each angle and classify them. a) \( \angle RQU \) To find the measure of \( \angle RQU \), trace the line from point Q (vertex of the angle) through point R until it meets the protractor. Then find where the other side of the angle (line from Q through U) intersects with the degree measures on the protractor. It appears that this angle measures 70 degrees based on the protractor. Angle classification is typically as follows: - Acute angle: less than 90 degrees - Right angle: exactly 90 degrees - Obtuse angle: more than 90 degrees but less than 180 degrees - Straight angle: exactly 180 degrees Since \( \angle RQU \) measures 70 degrees, it is an acute angle. b) \( \angle TQU \) Similarly, for \( \angle TQU \), we look at the measure where the ray starting at Q and passing through T intersects the protractor. The image resolution doesn't allow the exact value to be confidently determined, but we can provide an estimate. If the previous angle \( \angle RQU \) is 70 degrees and it looks like \( \angle TQU \) opens up wider, making an obtuse angle that is less than 180 degrees but larger than 90 degrees. Since I cannot precisely verify the measurement on the protractor, you will need to measure this angle with an actual protractor for an accurate result. Based on the image, it could be estimated to be between 110 and 130 degrees, but this is just an estimate. c) \( \angle UQS \) For \( \angle UQS \), follow the same process: trace the lines from Q through both U and S to see where they intersect the protractor. Again, based on the image, it is difficult to see the precise measurement, but we can determine the angle type. If the angle is wider than \( \angle TQU \) and appears to be more open, so it almost forms a straight angle without closing up. It is likely an obtuse angle that is less than 180 degrees. Again, precise measurement using the image is difficult, so an actual protractor should be used to find the exact measure. To summarize, based on what we can see: - \( \angle RQU \) is 70 degrees and acute. - \( \angle TQU \) appears to be an obtuse angle, but the exact measure needs to be taken with a protractor. - \( \angle UQS \) also appears to be an obtuse angle, and its measurement also requires use of a protractor for accuracy.
Determining Angles in a Regular Hexagon
Resimde verilen soruyu Türkçe olarak cevaplayacağım. Bu soruda altıgen ABCDEF'nin düzgün altıgen olduğu bilgisi verilmiş. Düzgün altıgenin iç açılarının her biri 120 derecedir. |PF| = |EF| olduğuna göre, FPE üçgeni eşkenar bir üçgen olup, E noktasındaki açı da 60 derece olacaktır. ∠APF = 35 derece verildiğine göre ve APF üçgenindeki diğer iki açı eşit olduğuna göre, bu iki açının toplamı (180 - 35) derece olur, yani 145 derece. Bu iki açı eşit oldukları için her biri 145/2 derecesidir ki bu da 72.5 derecedir. PEF üçgeninin bir dış açısı ise içte kalan iki açısının toplamı kadar olur. EPA açısı, PAF ve PAF'ın diğer eşit açısının toplamı kadar olur, yani 72.5 + 72.5 = 145 derece olur. EPA açısı ile EFP açısı bir doğru üzerinde komşu açılar olduğundan toplamları 180 derece verir. EPA + EFP = 180 145 + m(∠PEF) = 180 m(∠PEF) = 180 - 145 m(∠PEF) = 35 derece Sonuç olarak ∠PEF = 35 derecedir. Seçenekler arasında bu değeri gösteren şık yoktur, bu durumda soruda bir hata olabilir veya resimden eksik veya yanlış bir bilgi aktarılmış olabilir. Ancak verilen bilgiler doğrultusunda EFP açısının ölçüsü 35 derecedir.
Solving Triangles Using Angle Sum Property
En la imagen se presenta un triángulo con sus tres ángulos marcados. Para resolver la pregunta, es necesario recordar que la suma de los ángulos internos de un triángulo siempre es igual a 180 grados. El problema nos da tres expresiones para los ángulos del triángulo: x + 1, x y x + 2. Podemos establecer la siguiente ecuación sumando estas tres expresiones y estableciendo su suma igual a 180 grados: x + 1 + x + x + 2 = 180 Al simplificar la ecuación, combinamos los términos semejantes: 3x + 3 = 180 Ahora restamos 3 de ambos lados de la ecuación: 3x = 177 Y luego dividimos ambos lados entre 3 para despejar x: x = 177 / 3 x = 59 Con este valor, podemos encontrar la medida de cada ángulo del triángulo: Ángulo rojo (x + 1): 59 + 1 = 60 grados Ángulo verde claro (x): 59 grados Ángulo verde oscuro (x + 2): 59 + 2 = 61 grados Y con esto hemos resuelto el valor de x y también la medida de los ángulos del triángulo en la imagen.
Finding the Measure of an Angle in a Triangle
Veo una imagen de un triángulo en una hoja de papel cuadriculado con marcaciones de ángulos. El ángulo en la punta inferior derecha del triángulo está marcado con una "x", lo que indica que se debe encontrar su medida. El ángulo en la punta superior del triángulo está marcado como 16°. Se asume que el ángulo verde marcado en la punta inferior izquierda es un ángulo recto, que mide 90°, debido a la convención de colorear los ángulos rectos en verde y su aparente orientación en la cuadrícula. Para resolver por "x", utilizamos el hecho de que la suma de los ángulos internos de cualquier triángulo es igual a 180°. Entonces, sumamos los dos ángulos conocidos y restamos esta suma de 180° para hallar "x". \[ x + 16° + 90° = 180° \] Ahora, resolvamos para "x": \[ x + 106° = 180° \] \[ x = 180° - 106° \] \[ x = 74° \] Por lo tanto, el valor de "x" es 74°.
Finding Angle Measurement in Shifted Triangle
In the given figure, there is a triangle APB with a 45° angle at A. It's stated that point A is shifted to point B along the ray PX such that PB = 2PA. We're asked to find the measure of ∠BPY. When point A is moved to point B, since PB is twice PA, triangle PBY will be an isosceles triangle where PB = PY. Therefore, the angles opposite the equal sides (PB and PY) will also be equal. Let's denote the measure of these angles as ∠PBY and ∠BPY, respectively. Because the sum of angles in a triangle is always equal to 180°, we can use this rule to find the measure of ∠BPY. For triangle PBY: ∠PBY + ∠BPY + ∠B = 180° Since ∠PBY and ∠BPY are equal (isosceles triangle), we can say: 2∠BPY + 45° = 180° Now, we solve for ∠BPY: 2∠BPY = 180° - 45° 2∠BPY = 135° Dividing both sides by 2 gives us: ∠BPY = 135° / 2 ∠BPY = 67.5° So, the measure of ∠BPY is 67.5°, which is greater than 45°. The correct answer is: a) Greater than 45°
Understanding Obtuse Angles
An obtuse angle is one that is greater than 90 degrees but less than 180 degrees. Looking at Fig. 2.9, we see several angles, but we are only interested in those greater than 90 degrees: - The angle labeled 45 degrees is acute, not obtuse. - The angle labeled 65 degrees is acute, not obtuse. - The angle labeled 20 degrees is acute, not obtuse. - The angle labeled 30 degrees is acute, not obtuse. However, there are angles not labeled, but we can deduce their measure by looking at the image carefully: - The angle adjacent to the 65-degree angle, between the blue arrow pointing up and the blue arrow pointing to the bottom right, is obtuse because if we subtract 65 from 180 we get 115 (180 - 65 = 115), which is obtuse. - The angle adjacent to the 45-degree angle, between the blue arrow pointing up and the blue arrow pointing to the bottom left, is obtuse by the same logic: 180 - 45 = 135 degrees, which is obtuse. Therefore, from the labeled angles and considering the structure of the lines, we find two obtuse angles. The correct answer is: a) 2
Angle Calculation Result
The image shows two straight lines that are designated as line m and line n, and there is an angle of 140° formed between these lines. Since the lines m and n are given to be parallel (as indicated by m || n), the angle corresponding to the given 140° angle on the parallel line below must be the same, due to the concept of alternate interior angles between parallel lines. So, the value of x is the same as the given angle because it is an alternate interior angle to the 140° angle. Therefore, the value of x is 140°.
Solving Unknown Angles in a Triangle
The image shows a diagram of a triangle with one of its angles labeled as 26 degrees. It seems the problem is asking you to find the unknown angles x, y, and z of the triangle. To solve for the unknown angles, we would use the fact that the sum of the angles in any triangle is 180 degrees. Given that one angle is 26 degrees, we can find the sum of the other two angles: 180 degrees (sum of all angles in a triangle) - 26 degrees (given angle) = 154 degrees Now we have an equation for the combined measure of angles x and y: x + y = 154 degrees However, we can't find the exact measures of x and y without additional information. For angle z, it's not clear from the diagram whether z is supposed to be an angle on the inside of the triangle or an external angle, nor is there an indication of the relationship between z and other angles. If z is an external angle adjacent to the 26-degree angle, then: z = 180 degrees - 26 degrees (since the sum of a straight line is 180 degrees) z = 154 degrees But if z is an angle within the triangle, we would need to know its relationship to the other angles to solve for it. Please check if there is more information provided or if there is a specific question you need to answer.
Calculation of Angles in Triangles
To find the value of \( x \), we need to look at the triangle \( ABC \). Since we know two angles in the triangle, \( 15^\circ \) (angle \( A \)) and \( 45^\circ \) (angle \( B \)), we can find angle \( C \) because the sum of the angles in a triangle is \( 180^\circ \). Let's calculate angle \( C \): \( C = 180^\circ - A - B = 180^\circ - 15^\circ - 45^\circ = 120^\circ \) Now we can focus on triangle \( BCD \), it appears to be an isosceles triangle since \( BD = DC \). Therefore, angle \( B \) and angle \( C \) in this triangle should be equal. However, were also given that \( DE \) is congruent to \( BD \), which means triangle \( BDE \) is also isosceles, and angles \( B \) and \( E \) must be equal. Since the large angle \( C \) of the big triangle \( ABC \) is split into two angles \( C \) and \( E \) within the triangle \( BCD \) and \( BDE \) respectively, we must have: \( C \) (angle within triangle \( BCD \)) \( + E = 120^\circ \) (angle C of \( ABC \)) Since \( C \) (within \( BCD \)) is equal to \( B = 45^\circ \), we get: \( 45^\circ + E = 120^\circ \) Now we can solve for \( E \): \( E = 120^\circ - 45^\circ = 75^\circ \) This is the value of angle \( E \), and since \( DE \) is congruent to \( BD \), angle \( D \) in triangle \( BDE \) is also \( 75^\circ \). Our goal is to find the angle \( x \) which is the angle \( D \) in triangle \( ADE \). Since \( ADE \) is an isosceles triangle with \( AE = DE \), the angles at \( A \) and \( D \) are equal. Therefore, angle \( x \) (which is angle \( D \) in \( ADE \)) equals \( 75^\circ \).
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