Example Question - inscribed angles
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Circular Geometry Problem Involving Inscribed Angles
<p>We are given a circle with inscribed angles and we need to determine the value of angle \( x \).</p> <p>Firstly, we notice that angle \( KLN \) is an inscribed angle that subtends arc \( KM \). According to the inscribed angle theorem, an inscribed angle is half the measure of its subtended arc. Therefore, the measure of arc \( KM \) is \( 120^\circ \times 2 = 240^\circ \).</p> <p>Since angle \( LNM \) is also an inscribed angle subtending the arc \( LM \), and it measures \( 110^\circ \), the measure of arc \( LM \), by the inscribed angle theorem, is \( 110^\circ \times 2 = 220^\circ \).</p> <p>Now arc \( LM \) and arc \( KM \) overlap to form the entire circumference of the circle, which measures \( 360^\circ \). Therefore, the remaining arc, \( KL \), must have a measure of \( 360^\circ - 220^\circ = 140^\circ \).</p> <p>Since angle \( x \), which is \( \angle KMN \), is an inscribed angle that subtends arc \( KL \), by the inscribed angle theorem, \( x \) is half the measure of arc \( KL \), hence:</p> <p>\[ x = \frac{140^\circ}{2} = 70^\circ \]</p> <p>The answer is \( A: 70^\circ \).</p>
Cyclic Quadrilateral Angle Calculation
<p>In a cyclic quadrilateral, the sum of opposite angles is 180 degrees.</p> <p>Given angle K = 120 degrees, we find the opposite angle M, which is 110 degrees + x.</p> <p>120 degrees + (110 degrees + x) = 180 degrees</p> <p>230 degrees + x = 180 degrees</p> <p>x = 180 degrees - 230 degrees</p> <p>x = -50 degrees</p> <p>Since an angle cannot be negative in this context, we need to reconsider the equation for opposite angles.</p> <p>We use the property that the exterior angle is equal to the opposite interior angle.</p> <p>Thus the exterior angle at vertex L is 120 degrees which is equal to angle N + angle M.</p> <p>Angle N is 70 degrees because it is the base angle of the isosceles triangle KLN (angle K is 120 degrees).</p> <p>120 degrees = 70 degrees + angle M</p> <p>Angle M = 120 degrees - 70 degrees</p> <p>Angle M = 50 degrees</p> <p>Therefore, x = 50 degrees</p>
Geometry Problem Involving Circles and Triangles
<p>La imagen muestra cuatro subpreguntas marcadas como (a), (b), (c), y (d). Sin embargo, debido a la perspectiva y calidad de la imagen, no todos los detalles de las preguntas son totalmente claros. Para las partes que son legibles, puedo proporcionar las soluciones con los pasos matemáticos correspondientes.</p> <p>Para la pregunta (a), se puede resolver utilizando el teorema del ángulo inscrito que dice que el ángulo inscrito en una circunferencia es igual a la mitad del ángulo central que abarca el mismo arco. No obstante, el valor numérico específico y la ecuación completa no son claros en la imagen.</p> <p>Para la pregunta (b), se debe aplicar propiedades de ángulos inscritos y centrales, así como la relación entre los lados de los triángulos inscritos. Pero, de nuevo, no se pueden determinar valores específicos debido a la calidad de la imagen.</p> <p>Las preguntas (c) y (d) también parecen estar basadas en propiedades de las figuras dentro de los círculos y las relaciones entre ángulos y arcos, pero los detalles específicos son ilegibles.</p> <p>Por lo tanto, sólo puedo proporcionar orientación general y no una solución específica debido a la limitada legibilidad de la imagen. En un contexto ideal con información completa, se aplicarían teoremas de la geometría para proporcionar los pasos matemáticos detallados para resolver cada subpregunta.</p>
Geometry Problem on Inscribed Angles of a Circle
The image contains a geometry problem that states: "If 2 inscribed angles of a circle intercept the same arc, then the 2 angles are ____________. If m∠1 = 35°, then m∠2 = ________." The blank should be filled with "congruent," as inscribed angles that intercept the same arc are equal to each other. Thus, if m∠1 = 35°, then m∠2 also equals 35°, because they intercept the same arc.
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