Question

in the accompanying diagram, (overrightarrow{bd} perp overleftrightarrow{abc}) at (b) and (overrightarrow{be} perp overrightarrow{bf}) at (b). if (mangle fbc = 20), what is (mangle ebd)?

(\bigcirc) (110)
(\bigcirc) (20)
(\bigcirc) (70)
(\bigcirc) (90)

2 multiple choice 4 points
(overleftrightarrow{ab}) and (overleftrightarrow{cd}) intersect at point (e), (mangle aec = 6x + 20), and (mangle deb = 10x).
what is the value of (x)?
(\bigcirc) (21\frac{1}{4})
(\bigcirc) (5)
(\bigcirc) (10)
(\bigcirc) (4\frac{3}{8})

Explanation:

First Question

Step1: Find $m\angle DBC$

Since $\overrightarrow{BD} \perp \overleftrightarrow{ABC}$, $m\angle DBC = 90^\circ$

Step2: Calculate $m\angle DBF$

$m\angle DBF = m\angle DBC - m\angle FBC = 90^\circ - 20^\circ = 70^\circ$

Step3: Calculate $m\angle EBD$

Since $\overrightarrow{BE} \perp \overrightarrow{BF}$, $m\angle EBF = 90^\circ$. So $m\angle EBD = 90^\circ + m\angle DBF = 90^\circ + 20^\circ = 110^\circ$

---

Second Question

Step1: Set angles equal (vertical angles)

$\angle AEC$ and $\angle DEB$ are vertical angles, so $m\angle AEC = m\angle DEB$

Step2: Substitute given expressions

$6x + 20 = 10x$

Step3: Solve for $x$

$20 = 10x - 6x$
$20 = 4x$
$x = \frac{20}{4} = 5$

Answer:

110