Question

refer to the figure at the right. 17. if ( mangle adb = (6x - 4)^circ ) and ( mangle bdc = (4x + 24)^circ ), find the value of x such that ( angle adc ) is a right angle. 18. if ( mangle fde = (3x - 15)^circ ) and ( mangle fdb = (5x + 59)^circ ), find the value of x such that ( angle fde ) and ( angle fdb ) are supplementary.

Explanation:

Question 17

Step1: Recognize angle sum

Since \( \angle ADC \) is a right angle, \( m\angle ADC = 90^\circ \). Also, \( \angle ADB + \angle BDC=\angle ADC \), so \( (6x - 4)+(4x + 24)=90 \).

Step2: Simplify the equation

Combine like terms: \( 6x+4x - 4 + 24 = 90 \), which becomes \( 10x+20 = 90 \).

Step3: Solve for x

Subtract 20 from both sides: \( 10x=90 - 20=70 \). Then divide by 10: \( x = \frac{70}{10}=7 \).

Step1: Recall supplementary angles

Supplementary angles sum to \( 180^\circ \), so \( m\angle FDE + m\angle FDB = 180^\circ \). Substitute the expressions: \( (3x - 15)+(5x + 59)=180 \).

Step2: Simplify the equation

Combine like terms: \( 3x+5x - 15 + 59 = 180 \), which becomes \( 8x + 44 = 180 \).

Step3: Solve for x

Subtract 44 from both sides: \( 8x=180 - 44 = 136 \). Then divide by 8: \( x=\frac{136}{8}=17 \).

Answer:

\( x = 7 \)

Question 18