| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | In a rhombus, the diagonals bisect each other at right angles, dividing the rhombus into four congruent right triangles. Given that m∠FJH = 82°, we can determine the measure of the angles in triangle FJH. Since FGHJ is a rhombus, all sides are equal, making triangle FJH an isosceles triangle. The base angles of an isosceles triangle are equal, so: m∠FJH = m∠HFJ = 82° The sum of the angles in any triangle is 180°, so we can find the measure of the third angle, ∠JFH: m∠JFH = 180° - m∠FJH - m∠HFJ m∠JFH = 180° - 82° - 82° m∠JFH = 16° Since the diagonals of the rhombus bisect each other at right angles, ∠JFH is split into two angles, each measuring half of 16°, or 8°. Therefore, m∠KHJ is one of these angles, plus the right angle formed by the diagonals: m∠KHJ = 8° + 90° m∠KHJ = 98° So, the measure of angle KHJ is 98°. |
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| deepseekr1 | Given a rhombus FGHJ with diagonals intersecting at point K, we need to find the measure of angle KHJ given that the measure of angle FJH is 82 degrees. 1. **Properties of a Rhombus**: - All sides are equal. - Diagonals bisect each other at right angles. - Diagonals bisect the vertex angles. 2. **Key Steps**: - Angle FJH is given as 82 degrees, which is the angle at vertex J of the rhombus. - In a rhombus, opposite angles are equal, and consecutive angles are supplementary. Therefore, the angle at vertex H is 180 - 82 = 98 degrees. - The diagonals of a rhombus bisect the vertex angles. Thus, the diagonal FH bisects the angle at vertex H (98 degrees) into two equal parts. - Each bisected angle at vertex H is 98 / 2 = 49 degrees. - Angle KHJ is the angle at vertex H between points K, H, and J. Since K lies on the diagonal FH, which bisects the angle at H, angle KHJ is 49 degrees. 3. **Verification**: - Considering triangle HKJ, which is right-angled at K (since diagonals intersect at right angles), we use the fact that the diagonal GJ bisects angle J into 41 degrees. In triangle HKJ, the remaining angle at H is 90 - 41 = 49 degrees. Thus, the measure of angle KHJ is \(\boxed{49}\). |
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| gpto1 | To solve the problem, we need to find the measure of angle \( \angle KHJ \) in rhombus \( FGHJ \) given that the diagonals intersect at point \( K \) and \( m\angle FJH = 82^\circ \). **Step 1: Understanding the Properties of a Rhombus** - In a rhombus, all sides are equal in length. - Opposite angles are equal: \( m\angle F = m\angle H \) and \( m\angle G = m\angle J \). - Adjacent angles are supplementary: \( m\angle F + m\angle G = 180^\circ \). - The diagonals bisect each other at right angles. - Each diagonal bisects a pair of opposite angles. **Step 2: Finding the Measure of Angle \( J \)** Since the diagonal \( GJ \) bisects angle \( J \), we have: \[ m\angle FJH = \frac{1}{2} m\angle J \] Given that \( m\angle FJH = 82^\circ \), we can calculate \( m\angle J \): \[ m\angle J = 2 \times 82^\circ = 164^\circ \] **Step 3: Finding the Measures of the Remaining Angles** Since opposite angles are equal in a rhombus: \[ m\angle G = m\angle J = 164^\circ \] Adjacent angles are supplementary: \[ m\angle F + m\angle G = 180^\circ \] \[ m\angle F + 164^\circ = 180^\circ \] \[ m\angle F = 16^\circ \] Since \( m\angle F = m\angle H \): \[ m\angle H = 16^\circ \] **Step 4: Finding \( m\angle KHJ \)** The diagonal \( FH \) bisects angle \( H \), so: \[ m\angle KHJ = \frac{1}{2} m\angle H \] \[ m\angle KHJ = \frac{1}{2} \times 16^\circ = 8^\circ \] **Answer:** \[ m\angle KHJ = 8^\circ \] \(\boxed{8^\circ}\) |
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| GPT-4o | To solve the problem, we need to use the properties of a rhombus and its diagonals. 1. **Understanding the properties**: - A rhombus is a type of quadrilateral where all four sides have equal length. - The diagonals of a rhombus bisect each other at right angles (90 degrees). - The diagonals of a rhombus also bisect the angles of the rhombus. 2. **Given**: - FGHJ is a rhombus. - The diagonals intersect at point K. - \( m∠FJH = 82° \). 3. **Find**: - \( m∠KHJ \). 4. **Analyzing the angles**: - Since the diagonals bisect each other at right angles, they form four right triangles within the rhombus. - Given \( m∠FJH \) is 82°, we need to find \( m∠KHJ \). 5. **Relationship of angles**: - Since FGHJ is a rhombus, the diagonals bisect the angles of the rhombus. - Therefore, diagonal FJ bisects \( ∠FJH \) into two equal angles. - This means \( ∠FJH \) is composed of two equal parts, each being half of 82°. 6. **Calculation**: - \( ∠FJK \) and \( ∠HJK \) are each half of 82°. - \( m∠FJK = m∠HJK = \frac{82°}{2} = 41° \). 7. **Finding \( m∠KHJ \)**: - Since K is the intersection point of the diagonals and they bisect each other at right angles, \( m∠KHJ \) is one of the angles formed by the bisected diagonals. - \( m∠KHJ \) is the angle between one of the bisected angles \( ∠HJK \) and the right angle at K. 8. **Right triangle consideration**: - Consider the right triangle KJH. - \( m∠KJH = 90° \) (since the diagonals intersect at right angles). - \( m∠HJK = 41° \) (as calculated). 9. **Sum of angles in a triangle**: - The sum of angles in triangle KJH is 180°. - Therefore, \( m∠KHJ = 180° - m∠KJH - m∠HJK \). 10. **Calculation**: - \( m∠KHJ = 180° - 90° - 41° \). - \( m∠KHJ = 49° \). So, the measure of \( m∠KHJ \) is \( 49° \). |
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| llama3.1-70B-Instruct | Since the diagonals of a rhombus bisect each other at right angles, we can conclude that: m∠FKH = 90° (since the diagonals intersect at right angles) We are also given that m∠FJH = 82°. Since ∠FJH and ∠FKH are alternate interior angles, we can set up an equation: m∠FJH + m∠FKH = 180° (since they form a straight line) Substituting the values, we get: 82° + m∠FKH = 180° m∠FKH = 180° - 82° m∠FKH = 98° Now, since ∠FKH and ∠KHJ are vertical angles, we can conclude that: m∠KHJ = m∠FKH m∠KHJ = 98° Therefore, m∠KHJ = 98°. |
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| llama3.1-405B-Instruct | Since the diagonals of a rhombus bisect each other, we know that ∠FJH is bisected by diagonal HJ. Therefore, ∠KHJ is half the measure of ∠FJH. m∠KHJ = 1/2 × m∠FJH = 1/2 × 82 = 41 So, the measure of ∠KHJ is 41 degrees. |
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| DeepSeek-Coder-V2 | To solve the problem, we need to understand the properties of a rhombus and the relationships between its angles and diagonals. 1. **Properties of a Rhombus**: - All sides are congruent. - The diagonals bisect each other at right angles (90 degrees). - The diagonals bisect the angles of the rhombus. 2. **Given Information**: - The diagonals of rhombus FGHJ intersect at point K. - \( m\angle FJH = 82^\circ \). 3. **Objective**: - Find \( m\angle KHJ \). 4. **Step-by-Step Solution**: - Since the diagonals of a rhombus bisect the angles of the rhombus, \( \angle FJH \) is bisected by the diagonal FH. - Therefore, \( \angle FJH \) is divided into two equal angles: \( \angle FJH = 82^\circ \). - Each of these bisected angles is \( \frac{82^\circ}{2} = 41^\circ \). 5. **Conclusion**: - Since \( \angle KHJ \) is one of the bisected angles formed by the intersection of the diagonals, \( m\angle KHJ = 41^\circ \). Thus, the measure of \( \angle KHJ \) is \( \boxed{41} \). |
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| Qwen2-72B-Instruct | In a rhombus, the diagonals bisect each other at right angles. This means they divide the rhombus into four congruent right triangles. Given that m∠FJH = 82°, we can determine the measure of the angles in triangle FJH. Since FGHJ is a rhombus, all sides are equal, making triangle FJH an isosceles triangle. The base angles of an isosceles triangle are equal, so: m∠FJH = m∠HFJ = 82° The sum of the angles in any triangle is 180°, so we can find the measure of the third angle, ∠JFH: m∠JFH = 180° - m∠FJH - m∠HFJ m∠JFH = 180° - 82° - 82° m∠JFH = 16° Now, since the diagonals of the rhombus bisect each other at right angles, ∠JFH is split into two angles, each measuring half of 16°, or 8°. Therefore, m∠KHJ is one of these angles, plus the right angle formed by the diagonals: m∠KHJ = 8° + 90° m∠KHJ = 98° So, the measure of angle KHJ is 98°. |
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| baichuan4 | In a rhombus, the diagonals are perpendicular bisectors of each other. This means that the diagonals intersect at right angles (90 degrees). When the diagonals intersect, they form four triangles within the rhombus. Each of these triangles is a right triangle because the diagonals are perpendicular. Given that ∠FJH = 82 degrees, we can see that ∠FJH is one of the angles in the right triangle FJH. Since the sum of the angles in a triangle is always 180 degrees, and we know one angle is 82 degrees, the other non-right angle in triangle FJH must be: 180 - 90 - 82 = 90 - 82 = 8 degrees Now, since the diagonal FH is a bisector of angle KHJ, this means that ∠KHJ is also divided into two congruent angles by FH. Therefore, each of these angles will be 8 degrees. So, the measure of ∠KHJ is 8 degrees. |
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| doubao | In a rhombus, the diagonals are perpendicular bisectors of each other. This means that ∠KHJ is a right angle. So m∠KHJ = 90°. | 49 |
| minimax | In a rhombus, all sides are equal in length, and the diagonals bisect each other at right angles. This means that each diagonal divides the rhombus into two congruent right triangles. Since the diagonals intersect at right angles, we can use this information to find the measure of the angles at the vertices where the diagonals intersect. Given that m∠FJH = 82 degrees, we know that this angle is part of one of the right triangles created by the diagonals. Since the diagonals bisect each other at K, ∠FJK and ∠HJK are congruent right angles (each being half of 90 degrees). Therefore, ∠FJK and ∠HJK are both 45 degrees. Now, we can find m∠KHJ by subtracting the measure of ∠HJK from 180 degrees (since the sum of the angles in any triangle is 180 degrees): m∠KHJ = 180° - m∠HJK m∠KHJ = 180° - 45° m∠KHJ = 135° So, the measure of angle KHJ is 135 degrees. |
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| kimi | In a rhombus, the diagonals are perpendicular bisectors of each other. This means that they intersect at a right angle (90 degrees) at their point of intersection, which is point K in this case. Given that m∠FJH = 82 degrees, we can use the fact that the diagonals of a rhombus bisect the angles at the vertices. Therefore, the angle m∠FJH is bisected by the diagonal JH, so the two angles formed by the bisecting diagonal are equal. Let's denote the two angles formed by the diagonal JH as m∠FJH and m∠FJH'. Since the diagonals bisect the angles, we have: m∠FJH = m∠FJH' = 82 degrees Now, since the diagonals intersect at right angles, the angle m∠KHJ is the complement of m∠FJH' (because they form a right angle together). Therefore: m∠KHJ = 90 degrees - m∠FJH' = 90 degrees - 82 degrees = 8 degrees So, the measure of angle m∠KHJ is 8 degrees. |
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| wenxinyiyan | 首先，我们知道在菱形中，对角线互相垂直并且平分对方。因此，我们可以得出∠KHJ是直角三角形FKH中的一个锐角。 其次，由于菱形的相邻角是互补的，所以我们可以得出∠FJH + ∠KHJ = 180°。 现在，我们已知∠FJH = 82°，所以我们可以将这个值代入上述等式中，得到： 82° + ∠KHJ = 180° 然后，我们可以通过简单的代数运算解出∠KHJ的值： ∠KHJ = 180° - 82° ∠KHJ = 98° 所以，我们可以得出∠KHJ的度数为98°。 |
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| zhipu | To solve for the measure of angle KHJ in the rhombus FGHJ, we need to use the properties of a rhombus and the given information about the angles. A rhombus has the following properties that are relevant to this problem: 1. All sides are congruent. 2. The diagonals are perpendicular bisectors of each other. 3. Each diagonal bisects a pair of opposite angles. Given that m∠FJH = 82°, we know that this is one of the angles at the vertices of the rhombus. Since the diagonals of a rhombus bisect the vertex angles, the angle FJH is split into two equal angles where the diagonals intersect. Thus, the angle FJK (where K is the intersection of the diagonals) is half of angle FJH: m∠FJK = m∠FJH / 2 m∠FJK = 82° / 2 m∠FJK = 41° Now, because the diagonals of a rhombus are perpendicular, we know that: m∠FKH = 90° Angle KHJ is part of the right angle FKH, along with angle FJK. Since they are complementary angles (they add up to 90°), we can find m∠KHJ by subtracting m∠FJK from 90°: m∠KHJ = 90° - m∠FJK m∠KHJ = 90° - 41° m∠KHJ = 49° Therefore, the measure of angle KHJ is 49 degrees. |
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| gpto1 | To solve the problem, we need to find the measure of angle \( \angle KHJ \) in rhombus \( FGHJ \) given that the diagonals intersect at point \( K \) and \( m\angle FJH = 82^\circ \). **Step 1: Understanding the Properties of a Rhombus** - In a rhombus, all sides are equal in length. - Opposite angles are equal: \( m\angle F = m\angle H \) and \( m\angle G = m\angle J \). - Adjacent angles are supplementary: \( m\angle F + m\angle G = 180^\circ \). - The diagonals bisect each other at right angles. - Each diagonal bisects a pair of opposite angles. **Step 2: Finding the Measure of Angle \( J \)** Since the diagonal \( GJ \) bisects angle \( J \), we have: \[ m\angle FJH = \frac{1}{2} m\angle J \] Given that \( m\angle FJH = 82^\circ \), we can calculate \( m\angle J \): \[ m\angle J = 2 \times 82^\circ = 164^\circ \] **Step 3: Finding the Measures of the Remaining Angles** Since opposite angles are equal in a rhombus: \[ m\angle G = m\angle J = 164^\circ \] Adjacent angles are supplementary: \[ m\angle F + m\angle G = 180^\circ \] \[ m\angle F + 164^\circ = 180^\circ \] \[ m\angle F = 16^\circ \] Since \( m\angle F = m\angle H \): \[ m\angle H = 16^\circ \] **Step 4: Finding \( m\angle KHJ \)** The diagonal \( FH \) bisects angle \( H \), so: \[ m\angle KHJ = \frac{1}{2} m\angle H \] \[ m\angle KHJ = \frac{1}{2} \times 16^\circ = 8^\circ \] **Answer:** \[ m\angle KHJ = 8^\circ \] \(\boxed{8^\circ}\) |
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| deepseekv3 | server error: Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-1ff2e1ff-de38-9871-ab23-8a7e535c98ee', 'request_id': '1ff2e1ff-de38-9871-ab23-8a7e535c98ee'} | 49 |