6.2.1 Properties of Special Parallelograms (continued)
The properties of a rhombus are used here to determine the length of a side of a rhombus. The figure is given to be a rhombus with the lengths of two sides given as algebraic expressions. To find DC , begin by solving for x . By the definition of a rhombus, all sides are congruent, so AD = AB . Substitute the given algebraic expressions for AD and AB and solve for x . The solution yields x = 3. Again, by the definition of a rhombus all sides are congruent, so to find DC substitute the value for x into either of the expressions for the length of the two given sides. DC = AD = 4 x = 4(3) = 12. The properties of a rhombus are used here to determine the measure of an angle of a rhombus. The figure is given to be a rhombus with the measures of two angles given as algebraic expressions with the unknown y . To find m ∠ ADB begin by solving for y . By the properties of a rhombus the diagonals are perpendicular, so m ∠ BEC = (11 y + 2)° = 90°. Solve for y to yield y = 8. Substitute the value of y to find m ∠ DAC . m ∠ DAC = (7 y + 4)° = (7(8) + 4)° = 60° By the properties of a rhombus each diagonal bisects opposite angles, so m ∠ CAB = m ∠ DAC = 60°. Substitute to find m ∠ DAB . m ∠ DAB = m ∠ DAC + m ∠ CAB = 60° + 60° = 120° By the Same Side Interior Angles Theorem, m ∠ ADC + m ∠ DAB = m ∠ ADC + 120° = 180°. Therefore, m ∠ ADC = 60°. According to the properties of a rhombus each diagonal bisects opposite angles, so m ∠ ADB = (1/2)m ∠ ADC = (1/2)60° = 30°.
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