In rhombus ABCD, shown below, angle A measures 116°.
If rhombus ABCD were reflected across the x-axis to create rhombus A'B'C'D', what would the measure of angle D' be?
When a figure is reflected across the x-axis, the y-coordinates of all the points are negated. Since a rhombus has equal opposite angles, we know that measures of angles A and C are equal, as are the measures of angles B and D. Angle A measures 116°, so angle C measures 116°. When a figure is reflected across the x-axis, angle C will occupy the same position as angle C'. In other words, angle C' has the same measure as angle C, which is 116°. Therefore, angle D' measure $\boxed{116\text{°}}$. [asy]
size(100);
pair A,B,C,D;
A=(0,0);
B=(2,2);
C=(4,0);
D=(2,-2);
draw(A--B--C--D--A);
draw(A--C);
label("A",A,SW);
label("B",B,N);
label("C",C,SE);
label("D",D,S);
[/asy]
When a figure is reflected across the x-axis, the angles remain the same. Therefore, the measure of angle D' would also be 116°.
To find the measure of angle D' in the reflected rhombus A'B'C'D', we need to understand the geometric properties of a rhombus and its reflection across the x-axis.
A rhombus is a quadrilateral with all sides equal in length. Opposite angles of a rhombus are congruent, meaning they have the same measure. When a figure is reflected across the x-axis, the x-coordinate of each point is multiplied by -1 while the y-coordinate remains the same. This reflection flips the figure over the x-axis.
In the given rhombus ABCD, angle A measures 116°. Since it is a rhombus, angle A is congruent to angle C and angle B is congruent to angle D.
When we reflect rhombus ABCD across the x-axis, the image will be rhombus A'B'C'D'. The x-coordinates of points A, B, C, and D will change signs since they will be multiplied by -1.
To find the measure of angle D' in rhombus A'B'C'D', we can first determine the measure of angle B' in the reflected rhombus. Since angle B is congruent to angle D, we can find the measure of angle D' by subtracting angle B' from 180°.
To find the measure of angle B' in the reflected rhombus, we need to consider the reflection of the points. Angle B is formed by lines AB and BC in rhombus ABCD. When we reflect rhombus ABCD across the x-axis, line AB will become line A'B'. Similarly, line BC will become line B'C'.
Since we know the measure of angle A, which is 116°, we can subtract this angle from 180° to find the measure of angle B in rhombus ABCD. Then, we can conclude that the measure of angle B' in rhombus A'B'C'D' is also equal to 116°.
Finally, to find the measure of angle D' in rhombus A'B'C'D', we subtract the measure of angle B' (116°) from 180°:
180° - 116° = 64°.
Therefore, the measure of angle D' in the reflected rhombus A'B'C'D' is 64°.