Question

1. ∠6 and ∠7 form a linear pair. twice the measure of... the measure of ∠7. find the measure of each angle.

refer to the figure at the right.

1. 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.
2. 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.

(diagram: lines intersecting at ( d ), with points ( a, b, c, e, f ) on the lines)

Explanation:

Problem 16:

Step1: Define linear pair property

A linear pair of angles sums to \(180^\circ\). Let \(m\angle7 = x\), then \(m\angle6 = 2x\) (assuming the missing part is "Twice the measure of \(\angle6\) is...", so \(2x + x=180^\circ\)).
\[2x + x = 180\]

Step2: Solve for \(x\)

Combine like terms: \(3x = 180\), then \(x=\frac{180}{3}=60\).
So \(m\angle7 = 60^\circ\), \(m\angle6 = 2\times60 = 120^\circ\).

Step1: Use right angle property

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

Step2: Simplify and solve for \(x\)

Combine like terms: \(10x + 20 = 90\). Subtract 20: \(10x = 70\), then \(x=\frac{70}{10}=7\).

Step1: Define supplementary angles

Supplementary angles sum to \(180^\circ\), so \(m\angle FDE + m\angle FDB = 180^\circ\), i.e., \((3x - 15)+(5x + 59)=180\).
\[3x - 15 + 5x + 59 = 180\]

Step2: Solve for \(x\)

Combine like terms: \(8x + 44 = 180\). Subtract 44: \(8x = 136\), then \(x=\frac{136}{8}=17\).

Answer:

\(m\angle6 = 120^\circ\), \(m\angle7 = 60^\circ\)

Problem 17: