Question

view submission calculator rhombus abcd has a diagonal ( overline{bd} ). ( mangle adb=(5x + 10)^{circ} ) ( mangle cdb=(4x + 19)^{circ} ) what is the ( mangle adc )? enter your answer in the box

Explanation:

Step1: Use the property of rhombus diagonals

In a rhombus, the diagonal bisects the angles. So, \(m\angle ADB=m\angle CDB\).

$$ 5x + 10=4x + 19 $$

Step2: Solve for \(x\)

Subtract \(4x\) from both sides: \(5x-4x+10 = 4x-4x + 19\), which gives \(x+10=19\).
Subtract \(10\) from both sides: \(x=19 - 10=9\).

Step3: Find \(m\angle ADB\) and \(m\angle CDB\)

Substitute \(x = 9\) into \(m\angle ADB=(5x + 10)^{\circ}\): \(m\angle ADB=(5\times9+10)^{\circ}=(45 + 10)^{\circ}=55^{\circ}\).
Substitute \(x = 9\) into \(m\angle CDB=(4x + 19)^{\circ}\): \(m\angle CDB=(4\times9+19)^{\circ}=(36+19)^{\circ}=55^{\circ}\).

Step4: Calculate \(m\angle ADC\)

Since \(m\angle ADC=m\angle ADB + m\angle CDB\), then \(m\angle ADC=55^{\circ}+55^{\circ}=110^{\circ}\).

Answer:

\(110^{\circ}\)