Question

quadrilateral abcd is a rhombus. answers and information from one problem does not carry to the next problem.12. if ad = 13, then ab = .13. if m∠4 = 25°, then m∠5 = .14. if m∠dab = 130°, then m∠adc = .15. if m∠4 = 3x - 2 & m∠5 = 2x + 7, then x = .

Explanation:

Step1: Properties of a rhombus

In a rhombus, all sides are equal. So, \(AD = AB\). Given \(AD = 13\), then \(AB=13\).

Step2: Angles in a rhombus

The diagonals of a rhombus bisect the angles. In a rhombus \(ABCD\), \(\angle4=\angle5\) (diagonal \(BD\) bisects \(\angle ABC\)). Given \(m\angle4 = 25^{\circ}\), then \(m\angle5=25^{\circ}\).

Step3: Adjacent angles in a rhombus

In a rhombus, adjacent angles are supplementary (\(\angle DAB+\angle ADC = 180^{\circ}\)). Given \(m\angle DAB=130^{\circ}\), then \(m\angle ADC=180 - 130=50^{\circ}\).

Step4: Solving for \(x\)

Since \(\angle4=\angle5\) (diagonals of a rhombus bisect the angles), \(3x - 2=2x + 7\).
Subtract \(2x\) from both sides: \(3x-2x-2=2x - 2x+7\), which gives \(x-2 = 7\).
Add \(2\) to both sides: \(x=7 + 2=9\).

Answer:

1. \(13\)
2. \(25^{\circ}\)
3. \(50^{\circ}\)
4. \(9\)