A rhombus. Angle a is 100 degrees. Angle c is opposite to angle a. Angles b and d are opposite to each other. In rhombus abcd, the measure of angle a is 100°. Find the measure of the other angles.
The Correct Answer and Explanation is:
In a rhombus, opposite angles are equal, and the sum of adjacent angles is always 180° because a rhombus is a type of parallelogram. Here, we have rhombus ABCD, where angle A is given as 100°. Let’s analyze the measures of the other angles in this rhombus.
Solution:
- Since angle A is 100°, its opposite angle, angle C, must also be 100° because opposite angles in a rhombus are congruent (equal). So, angle C = 100°.
- In a rhombus, adjacent angles are supplementary, meaning their sum is 180°. Therefore, angle A and angle B are supplementary: [
\text{angle A} + \text{angle B} = 180°
]
[
100° + \text{angle B} = 180°
]
[
\text{angle B} = 180° – 100° = 80°
] - Since angle B is 80°, its opposite angle, angle D, is also 80°. So, angle D = 80°.
Summary of Angle Measures:
- Angle A = 100°
- Angle C = 100° (opposite to angle A)
- Angle B = 80°
- Angle D = 80° (opposite to angle B)
Explanation:
The properties of a rhombus provide a straightforward way to find unknown angles once one angle is known. A rhombus is a quadrilateral with all sides of equal length and opposite angles that are equal. Additionally, adjacent angles in a rhombus are supplementary, which means they add up to 180°. Knowing this, we determined that since angle A is 100°, angle C (opposite angle) is also 100°. Then, using the property of supplementary angles, we calculated angle B by subtracting angle A from 180°, giving us 80° for both angle B and its opposite, angle D. This consistency in angle measures reflects the symmetry of a rhombus, where opposite angles are equal, and adjacent angles always add to 180°.