Question

multiple choice
$overleftrightarrow{ab}$ and $overleftrightarrow{cd}$ intersect at point $e$, $mangle aec = 6x + 20$, and $mangle deb = 10x$.
what is the value of $x$?
$4\frac{3}{8}$
$21\frac{1}{4}$
$10$
$5$
5 multiple choice 4 points
in the figure below, $overleftrightarrow{cd}$ intersects $overleftrightarrow{ab}$ at $f$, $mangle cfb = 50^{\circ}$, and $angle efa \cong angle afd$.
what is $mangle efc$?
options: $50^{\circ}$, $70^{\circ}$, $80^{\circ}$, $40^{\circ}$

Explanation:

First Question

Step1: Identify vertical angles

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

Step2: Set up equation

$6x + 20 = 10x$

Step3: Solve for x

Subtract $6x$ from both sides: $20 = 4x$
Divide by 4: $x = \frac{20}{4} = 5$

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Second Question

Step1: Find $\angle AFD$

$\angle CFB$ and $\angle AFD$ are vertical angles, so $m\angle AFD = 50^\circ$.

Step2: Find $\angle EFA$

$\angle EFA \cong \angle AFD$, so $m\angle EFA = 50^\circ$.

Step3: Calculate $\angle EFD$

$m\angle EFD = m\angle EFA + m\angle AFD = 50^\circ + 50^\circ = 100^\circ$

Step4: Find $\angle EFC$

$\angle EFD$ and $\angle EFC$ are supplementary (form a straight line):
$m\angle EFC = 180^\circ - m\angle EFD = 180^\circ - 100^\circ = 70^\circ$

Answer:

5