Example Question - sum of angles
Here are examples of questions we've helped users solve.
Finding the Value of an Angle in a Kite-like Figure
\begin{align*} m\angle EAD + m\angle ADC + m\angle DAB &= 180^\circ \quad \text{(Sum of interior angles of $\triangle ADB$)} \\ x + 118^\circ + 30^\circ &= 180^\circ \\ x + 148^\circ &= 180^\circ \\ x &= 180^\circ - 148^\circ \\ x &= 32^\circ \end{align*}
Determining the Value of an Angle in a Quadrilateral
Given: - Quadrilateral \( ABCD \) - \( AB = BC = CD = 4 \) - \( \angle C = 32^\circ \) - \( \angle A = 115^\circ \) - \( BD \) is a diagonal To find: - \( \angle x \) in \( \triangle BCD \) Step 1: Identify that \( ABCD \) is a kite since two pairs of adjacent sides are equal. Step 2: Use the fact that in a kite, one diagonal is the perpendicular bisector of the other. Step 3: This makes \( \triangle BCD \) and \( \triangle ABD \) right triangles. Therefore, \( \angle BDC = \angle ABD = 90^\circ \). Step 4: Calculate the remaining angle in \( \triangle BCD \) (angle \( x \)) using the fact that the sum of angles in a triangle is \( 180^\circ \). \[ x + 32^\circ + 90^\circ = 180^\circ \] \[ x + 122^\circ = 180^\circ \] \[ x = 180^\circ - 122^\circ \] \[ x = 58^\circ \]
Calculation of Tangent Angle with Given Secant Value
La imagen muestra un problema matemático que dice lo siguiente: "Si \( \pi < x < \frac{3\pi}{2} \) y \( \sec(x) = -\frac{13}{5} \), ¿cuánto es el valor de \( \tan \left(\frac{\pi}{4} + x \right) \)?" Para resolver este problema, primero debemos recordar que la secante de un ángulo \( x \) es igual al inverso del coseno de \( x \): \( \sec(x) = \frac{1}{\cos(x)} \). Dado que nos han proporcionado que \( \sec(x) = -\frac{13}{5} \), podemos calcular el coseno de \( x \): \[ \cos(x) = -\frac{5}{13} \] Ahora, necesitamos encontrar el seno de \( x \). Ya que sabemos que el ángulo está en el tercer cuadrante (donde seno y coseno son negativos), y que la identidad pitagórica \( \sin^2(x) + \cos^2(x) = 1 \) se mantiene siempre, tenemos: \[ \sin^2(x) = 1 - \cos^2(x) \] \[ \sin^2(x) = 1 - \left(-\frac{5}{13}\right)^2 \] \[ \sin^2(x) = 1 - \frac{25}{169} \] \[ \sin^2(x) = \frac{169}{169} - \frac{25}{169} \] \[ \sin^2(x) = \frac{144}{169} \] Tomando la raíz cuadrada para encontrar el seno y recordando que debe ser negativo (pues estamos en el tercer cuadrante), obtenemos: \[ \sin(x) = -\frac{12}{13} \] Ahora, vamos a calcular la tangente de \( \frac{\pi}{4} + x \) usando la fórmula de la tangente de la suma de ángulos: \[ \tan\left(\frac{\pi}{4} + x\right) = \frac{\tan\left(\frac{\pi}{4}\right) + \tan(x)}{1 - \tan\left(\frac{\pi}{4}\right)\tan(x)} \] Sabiendo que \( \tan\left(\frac{\pi}{4}\right) = 1 \) y que \( \tan(x) = \frac{\sin(x)}{\cos(x)} = \frac{-12/13}{-5/13} = \frac{12}{5} \), tenemos: \[ \tan\left(\frac{\pi}{4} + x\right) = \frac{1 + \frac{12}{5}}{1 - 1 \cdot \frac{12}{5}} \] \[ \tan\left(\frac{\pi}{4} + x\right) = \frac{\frac{5}{5} + \frac{12}{5}}{1 - \frac{12}{5}} \] \[ \tan\left(\frac{\pi}{4} + x\right) = \frac{\frac{17}{5}}{1 - \frac{12}{5}} \] \[ \tan\left(\frac{\pi}{4} + x\right) = \frac{\frac{17}{5}}{\frac{5}{5} - \frac{12}{5}} \] \[ \tan\left(\frac{\pi}{4} + x\right) = \frac{17}{5} \cdot \frac{5}{-7} \] \[ \tan\left(\frac{\pi}{4} + x\right) = \frac{17}{-7} \] \[ \tan\left(\frac{\pi}{4} + x\right) = -\frac{17}{7} \] Por lo tanto, el valor de \( \tan\left(\frac{\pi}{4} + x\right) \) es \( -\frac{17}{7} \).
Properties of an Equilateral Triangle
The statement in the image is indeed true. An equilateral triangle has three sides of equal length and, consequently, three angles that are equal in measure. By definition, the sum of the angles in any triangle is 180 degrees. Since all angles are equal in an equilateral triangle, each angle must be one third of 180 degrees, which is 60 degrees. Therefore, an equilateral triangle has three angles that are each 60 degrees.
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°.
Deduction of Angles in a Star-shaped Figure
The image shows two geometric diagrams. However, the second diagram with the question "Find the sum of angle a and angle b" is the relevant part here. The image is a bit blurry, but I will attempt to help you with the information visible. From the diagram, it appears to be a star-shaped figure, suggesting that angle a and angle b are interior angles of a pentagon inscribed in the star. When multiple triangles are radiating from a central point, this implies the sum of the angles at the central point is 360 degrees due to the circular arrangement. It isn't evident from the diagram whether there are any special properties, such as equal sides or angles, that we could use directly. Yet, if we consider a regular pentagon (since it appears all sides may be equal), then the sum of the interior angles of any polygon can be calculated using the formula: Sum of interior angles = (n - 2) * 180 degrees Where n is the number of sides. For a pentagon (n=5), the sum of its interior angles is: Sum = (5-2) * 180 = 3 * 180 = 540 degrees Since a regular pentagon has all angles equal, each angle is: Each angle = 540 degrees / 5 = 108 degrees However, the sum of angles a and b, as exhibited in the star figure, appears to be less straightforward than simply adding two angles of a regular pentagon because these angles are formed by the intersection of the extensions of the sides of the pentagon. Angle a seems to be an external angle of the pentagon, and since the external angles of a polygon add up to 360 degrees, angle a would be Angle a = 360 degrees / 5 = 72 degrees (if the pentagon is regular). Angle b is an internal angle of the overlapping triangles. Without additional information on the relationships between the angles in the figure, we cannot definitively calculate the value of angle b. To accurately determine the sum of angle a and angle b, I would need further details about the properties of the figure and clear visibility of any marked angles or sides. If any other angles or side lengths are given, they might help solve for angles a and b. Please provide additional information, such as whether the star is regular, and if there are any given angles or relationships stated in the problem.
Calculating Sum of Angles in a Pentagonal Pyramid
The image shows a pentagonal pyramid with its base ABCDE and with the apex labeled as V. The question asks to find the sum of angle C and angle B. There are handwritten notes on the image providing values for angles at base vertices C and B, which are as follows: \( \angle C = 108^\circ - 36^\circ = 72^\circ \) \( \angle B = 108^\circ - 36^\circ = 72^\circ \) The sum of \( \angle C \) and \( \angle B \) would be: \( 72^\circ + 72^\circ = 144^\circ \) So the sum of angles C and B is \( 144^\circ \).
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