CamTutor

Example Question - triangle angles

Here are examples of questions we've helped users solve.

Solving for x in a Geometry Problem with Exterior Angle Theorem

This question involves a geometry problem with a triangle and an exterior angle. From the diagram, we see triangle ABC with an exterior angle DBC, which has a measure of (3x + 29)°. The interior angles opposite the exterior angle are angle BAC, which measures (6x + 14)°, and angle ABC, which appears to be a right angle (90°), as indicated by the little square box on angle ABC. According to the exterior angle theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles. Therefore, to solve for x, you would set up the equation based on the exterior angle theorem as follows: Exterior angle = Sum of opposite interior angles (3x + 29)° = (6x + 14)° + 90° Now let's solve for x. 3x + 29 = 6x + 14 + 90 3x + 29 = 6x + 104 Subtract 3x from both sides: 29 = 3x + 104 Subtract 104 from both sides: -75 = 3x Divide both sides by 3: -25 = x Therefore, x equals -25. However, if you are using this value of x to calculate angles in geometric problems, you should keep in mind that angle measures cannot be negative. There may be some information missing or a misunderstanding in interpreting the problem. Normally, you would expect to find a positive value for x in geometric problems involving angles. Please double-check to ensure there are no additional instructions or information provided that might affect the outcome of the solution.

Properties of a Triangle and Ratios in Geometric Configurations

Trong tam giác ABC với A = 30°, B = 70°, chúng ta có: A + B + C = 180° (tổng số đo các góc trong một tam giác) 30° + 70° + C = 180° 100° + C = 180° C = 80° Bây giờ, vì AB đối diện với góc C và AC đối diện với góc B, ta có thể so sánh các cặp cạnh-góc: - Cạnh AB đối diện với góc lớn nhất (C = 80°) nên AB là cạnh lớn nhất. - Cạnh AC đối diện với góc thứ hai về độ lớn (B = 70°) nên AC lớn hơn BC và nhỏ hơn AB. - Cạnh BC đối diện với góc nhỏ nhất (A = 30°) nên BC là cạnh nhỏ nhất. Vậy, ta có quan hệ giữa các cạnh như sau: BC < AC < AB. Nên phương án C là đúng. Đối với phần tỉ số giữa GD và BC, vì G là trọng tâm của tam giác, chia cạnh huyền thành ba phần bằng nhau, hai phần tính từ đỉnh đến trọng tâm, và một phần từ trọng tâm đến giữa cạnh đối diện. Vì vậy tỉ số của GD so với BC là 2:3. Điều này nghĩa là độ dài từ G đến D bằng hai phần trong tổng số năm phần bằng độ dài BC. Vậy phương án B đúng là GD = 2/3 BC.

Finding Angle of a Triangle

To find the value of \(x\) in the triangle depicted in the image, we can use the fact that the sum of the angles within a triangle always equals 180 degrees. The triangle in the image has one of its angles labeled 100 degrees and another labeled as \(30\) degrees. Let's call the third angle, which we are trying to find, \(x\). We can set up the equation: \[ 100^\circ + 30^\circ + x^\circ = 180^\circ \] Combining the known angles gives us: \[ 130^\circ + x^\circ = 180^\circ \] To solve for \(x\), we subtract 130 degrees from both sides of the equation: \[ x^\circ = 180^\circ - 130^\circ \] \[ x^\circ = 50^\circ \] So the value of \(x\) is 50 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°.

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°

Solving Triangle Angles with 'x' Degrees

The image shows a triangle with angles labeled in terms of 'x'. To solve for 'x', we'll use the fact that the sum of the interior angles of any triangle is always 180 degrees. The angles are labeled as follows: - One angle is x degrees. - The second angle is 4x + 50 degrees. - The third angle is 3x - 10 degrees. Let's write an equation that sums these angles to 180 degrees: x + (4x + 50) + (3x - 10) = 180 Now, we can simplify and solve for 'x': x + 4x + 50 + 3x - 10 = 180 8x + 40 = 180 8x = 180 - 40 8x = 140 x = 140 / 8 x = 17.5 Therefore, 'x' is 17.5 degrees.

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 \( 75^\circ \).

Geometry Problem: Angle Sum in Triangle and Quadrilateral

Unfortunately, the image is partially cropped, so the full text of the problem isn't visible. However, I can see the first two parts of the question which I can help you with: (i) For the triangle, show that \( 2a + 5b = 171 \) (ii) For the quadrilateral, show that \( 7a + 8b = 342 \) Let's solve each part step by step: **Part (i): For the triangle** In any triangle, the sum of the interior angles is always 180 degrees. The triangle shown in the image has angles \( 5b, a + b \), and \( a - 29 \). To show that \( 2a + 5b = 171 \), we'll add up the angles and set them equal to 180 degrees, which is the sum of all interior angles of a triangle. \[ 5b + (a + b) + (a - 29) = 180 \] \[ 5b + a + b + a - 29 = 180 \] \[ 2a + 6b - 29 = 180 \] We add \( 2a + 6b \) and subtract 29 to isolate \( 2a + 5b \) on one side: \[ (2a + 6b) - 29 = 180 \] \[ 2a + 6b = 180 + 29 \] \[ 2a + 6b = 209 \] However, to get \( 2a + 5b \), we need to subtract \( b \) from both sides: \[ (2a + 6b) - b = 209 - b \] \[ 2a + 5b = 209 - b \] Since our goal is to show \( 2a + 5b = 171 \), this means b must be \( 209 - 171 = 38 \). However, without additional information or an equation for b, we cannot verify this from the cropped image. There might be more information on the paper that provides a way to determine the value of b. **Part (ii): For the quadrilateral** The sum of the interior angles of a quadrilateral is \( 360 \) degrees. The quadrilateral shown in the image has angles \( 3a + b, a, 3a + 2b \), and \( 3b \). We'll add all angles and set them equal to 360 degrees: \[ (3a + b) + a + (3a + 2b) + 3b = 360 \] \[ 3a + b + a + 3a + 2b + 3b = 360 \] \[ 7a + 6b = 360 \] This is slightly different from the expression you were asked to show, \( 7a + 8b = 342 \). It seems there might be an error in the transcription of the problem or more information is needed that isn't visible due to the partial cropping of the image. In any case, if you have more details or can provide additional information or the rest of the image, I can assist you further.