Question

use the diagram at the right. solve for x. find the angle measures to check your work.
$m\angle aob = 4x - 2$, $m\angle boc = 5x - 10$, $m\angle cod = 2x + 14$
understanding the problem

1. the diagram shows that $\angle aob$ and ________ are congruent.

<ocd
<cob
<dco
<boc
<bod
<abo
<cod

Explanation:

Step1: Identify congruent angles

By observing the diagram (assuming standard angle - congruence markings), if $\angle AOB$ and $\angle COD$ are congruent (since no other information is given about congruence and we need to solve for $x$ using the angle - measures), we set up the equation based on the equality of their angle - measures.
$4x - 2=2x + 14$

Step2: Solve the equation for $x$

Subtract $2x$ from both sides:
$4x-2x - 2=2x-2x + 14$
$2x-2 = 14$
Add 2 to both sides:
$2x-2 + 2=14 + 2$
$2x=16$
Divide both sides by 2:
$x = 8$

Step3: Check the work

Find the measure of $\angle AOB$: $m\angle AOB=4x - 2=4\times8 - 2=32 - 2=30$
Find the measure of $\angle COD$: $m\angle COD=2x + 14=2\times8+14=16 + 14=30$
For the multiple - choice part:
Since $\angle AOB$ and $\angle COD$ are congruent, the answer to "The diagram shows that $\angle AOB$ and ____ are congruent" is $\angle COD$.

Answer:

The value of $x$ is 8.
The answer for the multiple - choice question is $\angle COD$.