Question

1. find the value of x. (6x + 7)° (6x - 17)°
2. find the value of x. (11x - 15)° (5x - 13)°
3. if bd ⊥ ac, m∠dbe=(2x - 1)°, and m∠cbe=(5x - 42)°, find the value of x.
4. find the value of x if qs bisects ∠pqr and m∠pqr = 82°. (10x + 1)°
5. find the values of x and y. (18y + 5)° (10x - 61)° (x + 10)°
6. find the values of x and y. (2y + 5)° (5x - 17)° (3x - 11)°

Explanation:

11.

Step1: Vertical - angles are equal

Since vertical angles are equal, we set up the equation \(6x + 7=6x - 17\). But this equation has no solution. There might be a mistake in the problem - setup. Assuming they are supplementary angles (if adjacent), \((6x + 7)+(6x - 17)=180\).
\[

$$\begin{align*} 6x+7 + 6x-17&=180\\ 12x-10&=180\\ 12x&=190\\ x&=\frac{95}{6}\approx15.83 \end{align*}$$

\]

12.

Step1: Supplementary - angles

The two angles \((11x - 15)\) and \((5x - 13)\) are supplementary, so \((11x - 15)+(5x - 13)=180\).

Step2: Simplify the equation

\[

$$\begin{align*} 11x-15 + 5x-13&=180\\ 16x-28&=180\\ 16x&=208\\ x& = 13 \end{align*}$$

\]

13.

Step1: Right - angle property

Since \(BD\perp AC\), \(\angle DBE+\angle CBE = 90^{\circ}\). So, \((2x - 1)+(5x - 42)=90\).

Step2: Simplify the equation

\[

$$\begin{align*} 2x-1+5x - 42&=90\\ 7x-43&=90\\ 7x&=133\\ x&=19 \end{align*}$$

\]

14.

Answer:

1. \(x=\frac{95}{6}\)
2. \(x = 13\)
3. \(x=19\)
4. \(x = 4\)
5. \(x=\frac{71}{9},y=\frac{58}{81}\)
6. \(x=\frac{59}{4},y=\frac{207}{8}\)