In triangle UVW, the measure of angle WUV is 50° and the measure of angle UVW is 31°. In triangle XYZ, the measure of angle ZXY is 50° and the measure of angle YZX is 99°. Determine how triangles UVW and XYZ can be shown to be similar.
A. Since WUV ZXY and VWU YZX, the triangles are similar by angle-angle.
B. Since VWU YZX and UW = YZ, the triangles are similar by angle-angle.
C. Since WUV ZXY and VWU XYZ, the triangles are similar by angle-angle.
D. Since WUV ZXY and UW = XY, the triangles are similar by angle-side.
A. Since WUV ≅ ZXY and VWU ≅ YZX, the triangles are similar by angle-angle.
To determine how triangles UVW and XYZ can be shown to be similar, we need to analyze the given information about the angle measurements.
In triangle UVW, we know that angle WUV measures 50° and angle UVW measures 31°. In triangle XYZ, angle ZXY measures 50° and angle YZX measures 99°.
To show that two triangles are similar, we can use the angle-angle similarity criterion, which states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
Now let's analyze the answer choices:
A. Since WUV = ZXY and VWU = YZX, the triangles are similar by angle-angle.
The given angles are WUV = 50° and ZXY = 50°, and VWU = 31° and YZX = 99°. Angle VWU is not congruent to angle YZX, so this option is not correct.
B. Since VWU = YZX and UW = YZ, the triangles are similar by angle-angle.
The given angles are VWU = 31° and YZX = 99°, which are congruent angles. However, the given lengths UW and YZ are not provided or indicated to be congruent. So, this option is also not correct.
C. Since WUV = ZXY and VWU = XYZ, the triangles are similar by angle-angle.
The given angles are WUV = 50° and ZXY = 50°, which are congruent angles. Additionally, the answer choice states that angle VWU is equal to angle XYZ, but this information is not given in the problem. Therefore, this option is not correct.
D. Since WUV = ZXY and UW = XY, the triangles are similar by angle-side.
The given angles are WUV = 50° and ZXY = 50°, which are congruent angles. Additionally, the given lengths UW = XY are provided and indicated as congruent. This option satisfies the angle-side similarity criterion, where two angles and the included side are congruent, so the triangles are similar.
Therefore, the correct answer is D. Since WUV = ZXY and UW = XY, the triangles UVW and XYZ can be shown to be similar by angle-side.
To determine if triangles UVW and XYZ are similar, we need to compare their angles and sides. In this problem, it is given that angle WUV in triangle UVW is 50°, and angle UVW is 31°. In triangle XYZ, it is given that angle ZXY is 50°, and angle YZX is 99°.
To show that two triangles are similar, we need either the angles in both triangles to be congruent, or the sides to be proportional.
Option A states that WUV is congruent to ZXY and VWU is congruent to YZX, so it claims the triangles are similar by angle-angle. However, the information given does not support this.
Option B states that VWU is congruent to YZX and UW is congruent to YZ. The information supports this statement, but it is not enough to conclude that the triangles are similar.
Option D states that WUV is congruent to ZXY and UW is congruent to XY. The information given also does not support this.
Option C states that WUV is congruent to ZXY and VWU is congruent to XYZ. According to the given information, WUV is indeed congruent to ZXY. However, there is no information given about VWU and XYZ. Therefore, we cannot conclude that VWU is congruent to XYZ.
Based on the information given, none of the options provide enough evidence to conclude that triangles UVW and XYZ are similar.
Thus, the correct answer is none of the provided options.