Question

3
p
(x + 14)°
n
(5x + 10)°
m
r
o
x = ______
&∠
b/c
vocabulary term
∠onr = ______
4.
a
(3x + 16)°
c
(4x - 4)°
b
d
e
x = ______
&∠
b/c
vocabulary term
∠bce = ______

Explanation:

Problem 3

Step1: Identify vertical angles

$\angle PNM$ and $\angle ONR$ are vertical angles, so they are supplementary to adjacent angles. Alternatively, $\angle PNM=(x+14)^\circ$ and $\angle MNR=(5x+10)^\circ$ are supplementary (linear pair).
$ (x+14) + (5x+10) = 180 $

Step2: Solve for x

Combine like terms and isolate x.
$6x + 24 = 180$
$6x = 180 - 24$
$6x = 156$
$x = \frac{156}{6} = 26$

Step3: Calculate $\angle ONR$

$\angle ONR = \angle PNM = (x+14)^\circ$ (vertical angles).
$\angle ONR = 26 + 14 = 40$

Step1: Identify vertical angles

$\angle ACD$ and $\angle BCE$ are vertical angles, so they are equal.
$3x + 16 = 4x - 4$

Step2: Solve for x

Isolate x by rearranging terms.
$16 + 4 = 4x - 3x$
$x = 20$

Step3: Calculate $\angle BCE$

Substitute x into $\angle BCE = (4x-4)^\circ$.
$\angle BCE = 4(20) - 4 = 80 - 4 = 76$

Answer:

$\angle PNM$ & $\angle MNR$ b/c linear pair
$x = 26$
$\angle ONR = 40^\circ$

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Problem 4