Question

7 figure abcd is a parallelogram. ( m angle a=(6 x+15)^{circ} ) ( m angle c=(4 x+45)^{circ} ) select all of the measures of the angles in parallelogram ( a b c d ). a ( 75^{circ} ) b ( 87^{circ} ) c ( 93^{circ} ) d ( 105^{circ} )

Explanation:

Step1: Use the property of parallelogram

In a parallelogram, opposite angles are equal. So, \(m\angle A=m\angle C\).

$$6x + 15=4x + 45$$

Step2: Solve the equation for \(x\)

Subtract \(4x\) from both sides:

$$6x-4x + 15=4x-4x + 45$$

$$2x+15 = 45$$

Subtract \(15\) from both sides:

$$2x+15-15=45-15$$

$$2x=30$$

Divide both sides by \(2\):

$$x = 15$$

Step3: Find \(m\angle A\) and \(m\angle C\)

Substitute \(x = 15\) into \(m\angle A=(6x + 15)^{\circ}\):

$$m\angle A=(6\times15 + 15)^{\circ}=(90+15)^{\circ}=105^{\circ}$$

Since \(m\angle A=m\angle C\), \(m\angle C = 105^{\circ}\)

Step4: Find \(m\angle B\) and \(m\angle D\)

In a parallelogram, consecutive angles are supplementary (\(m\angle A+m\angle B = 180^{\circ}\)).
Let \(m\angle B=y\), then \(105^{\circ}+y=180^{\circ}\), \(y = 180^{\circ}-105^{\circ}=75^{\circ}\)
Since \(m\angle B=m\angle D\), \(m\angle D = 75^{\circ}\)

Answer:

A. \(75^{\circ}\), D. \(105^{\circ}\)