Question

consider the two triangles shown below.

note: the triangles are not drawn to scale.

are the two triangles congruent?

choose 1 answer:
a yes
b no
c there is not enough information to say.

Explanation:

Step1: Calculate the third angle of each triangle

For the first triangle, the sum of angles in a triangle is \(180^\circ\). So the third angle is \(180 - 75 - 46 = 59^\circ\)? Wait, no, wait. Wait the first triangle has angles \(75^\circ\), \(46^\circ\), so third angle: \(180 - 75 - 46 = 59^\circ\)? Wait no, the second triangle has angles \(76^\circ\), \(46^\circ\), so third angle: \(180 - 76 - 46 = 58^\circ\)? Wait, no, wait, maybe I misread. Wait the first triangle: angle \(75^\circ\), \(46^\circ\), side 8. The second triangle: angle \(76^\circ\), \(46^\circ\), side 8. Wait, no, wait the first triangle's angles: let's recalculate. Wait, sum of angles in a triangle is \(180^\circ\). First triangle: angles \(75^\circ\), \(46^\circ\), so third angle is \(180 - 75 - 46 = 59^\circ\). Second triangle: angles \(76^\circ\), \(46^\circ\), so third angle is \(180 - 76 - 46 = 58^\circ\). Wait, but that can't be. Wait, maybe I made a mistake. Wait, no, the first triangle: angle \(75^\circ\), \(46^\circ\), so third angle: \(180 - 75 - 46 = 59\). Second triangle: angle \(76^\circ\), \(46^\circ\), third angle: \(180 - 76 - 46 = 58\). Wait, but that would mean the angles are different. Wait, no, maybe the first triangle's angle is \(75\), second is \(76\)? Wait, the problem says "the two triangles". Wait, maybe I misread the angles. Wait, the first triangle: angle \(75^\circ\), \(46^\circ\), side 8. The second triangle: angle \(76^\circ\), \(46^\circ\), side 8. Wait, but that would mean the angles are not the same. Wait, no, maybe it's a typo, or maybe I misread. Wait, no, let's check again. Wait, the first triangle: angle \(75^\circ\), \(46^\circ\), so third angle is \(180 - 75 - 46 = 59\). Second triangle: angle \(76^\circ\), \(46^\circ\), third angle is \(180 - 76 - 46 = 58\). Wait, but that would mean the triangles have different angle measures, so they can't be congruent. But wait, maybe I made a mistake. Wait, no, the problem is about congruent triangles. Congruent triangles have all corresponding angles equal and all corresponding sides equal. Let's check the angles. First triangle: angles \(46^\circ\), \(75^\circ\), so third angle \(59^\circ\). Second triangle: angles \(46^\circ\), \(76^\circ\), so third angle \(58^\circ\). So the angles are not the same. Wait, but that can't be. Wait, maybe the first triangle's angle is \(76^\circ\)? Wait, the image: first triangle has \(75^\circ\), second has \(76^\circ\)? Wait, maybe it's a mistake, but according to the given, the first triangle has \(75^\circ\), \(46^\circ\), side 8. The second has \(76^\circ\), \(46^\circ\), side 8. So the angles are different, so the triangles are not congruent? But wait, no, maybe I misread the angles. Wait, maybe the first triangle's angle is \(76^\circ\)? Wait, the user's image: "75°" on first, "76°" on second. So that's a difference. So the angles are not equal, so the triangles can't be congruent. But wait, the answer options are Yes, No, or not enough. Wait, but if the angles are different, then No. But wait, maybe I made a mistake. Wait, let's recalculate the third angle for each. First triangle: \(180 - 46 - 75 = 59\). Second triangle: \(180 - 46 - 76 = 58\). So the third angles are different, so the triangles have different angle measures, so they can't be congruent. Therefore, the answer is No? But wait, maybe the side is included? Wait, the first triangle: side 8 is between \(46^\circ\) and \(75^\circ\)? The second triangle: side 8 is between \(46^\circ\) and \(76^\circ\)? Wait, no, the side 8 in the first triangle is adjacent to \(46^\circ\) and \(75^\circ\)? Wait, the first triangle: angle \(46^\circ\), side 8, angle \(75^\circ\). So that's an ASA (angle-side-angle) situation. The second triangle: angle \(46^\circ\), side 8, angle \(76^\circ\). So the angles adjacent to the side 8 are \(46^\circ\) and \(75^\circ\) in first, \(46^\circ\) and \(76^\circ\) in second. So the included angles with the side 8 are different (75 vs 76), so the triangles are not congruent. Therefore, the answer is No. But wait, the options are A Yes, B No, C not enough. So the answer is B No.

Step2: Determine congruence based on angles and side

Since the angles adjacent to the side of length 8 are different (75° vs 76°) in the two triangles, their corresponding angles are not equal. Congruent triangles require all corresponding angles and sides to be equal. Thus, the triangles are not congruent.

Answer:

B. No