Exercise 1 Page 436

Let's begin by making a rough sketch of the map, highlighting the red streets.

Since all of the streets are straight angles, we can identify some linear pairs. Let's look for linear pairs one street at a time.

Street $\overline{NQ}$

Following street $\overline{NQ},$ we can find three pairs. Linear pairs are supplementary angles, so the sum of each pair will be $180^{\circ }.$ We can subtract the known values from $180$ to find the missing values.

Streets $\overline{JL}$ and $\overline{GI}$

Following along streets $\overline{JL}$ and $\overline{GI},$ we find two more linear pairs. Both $\angle HKL$ and $\angle GHK$ are right angles, so their supplements are also right angles. Let's mark the angles that we have found.

Street $\overline{MF}$

Now, along street $\overline{MF},$ we can find another four linear pairs. These angles are all $90^{\circ }$ because the supplement to a right angle is also a right angle, similar to the previous two streets.

Street $\overline{AD}$

Street $\overline{AD}$ is a straight angle, so the sum of $m\angle AEB,$ $m\angle BEC,$ and $m\angle CED$ is $180^{\circ }.$

Finding the Congruent Angles

Now that we found the measures of the angles in the map, we can figure out which ones are congruent. There's a cluster of right angles along $\overline{MF}.$ Since they are all $90^{\circ },$ they are all congruent. Next, let's look at the angles around point $O.$

Two angles measure $133^{\circ }$ and two angles measure $47^{\circ }.$ Each pair of these angles are congruent. Therefore, $\angle MOR\cong \angle POQ$ and $\angle MOP\cong \angle QOR.$

Looking to the north of the map, we can see that two angles measure $55^{\circ },$ $\angle KMO$ and $\angle DEC.$ They are also congruent. There are no more congruent angles that are directly shown on our map. However, look at $\angle AEB$ and $\angle BEC.$ The sum of their measures is $37^{\circ }+88^{\circ }=125^{\circ }.$ Therefore, $\angle AEC$ and $\angle NMK$ are congruent too!