Cumulative Assessment
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Identify linear pairs and compare angles.
90^(∘) Angles: ∠ MKJ, ∠ MKL, ∠ LKH, ∠ JKH, ∠ KHI, ∠ IHF, ∠ FHG, and ∠ GHK
133^(∘) Angles: ∠ MOR and ∠ POQ
47^(∘) Angles: ∠ MOP and ∠ QOR
55^(∘) Angles: ∠ KMO and ∠ DEC
125^(∘) Angles: ∠ AEC and ∠ NMK
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.
m∠ NMK =& 180^(∘)-55^(∘) = 125^(∘) m∠ POQ =& 180^(∘) - 47 ^(∘) = 133^(∘) m∠ MOR =& 180^(∘)-47^(∘) = 133^(∘)
Following along streets JL and GI, we find two more linear pairs. ∠ HKL &and ∠ HKJ ∠ GHK &and ∠ IHK Both ∠ HKL and ∠ GHK are right angles, so their supplements are also right angles. Let's mark the angles that we have found.
Now, along street MF, we can find another four linear pairs. ∠ MKL &and ∠ HKL ∠ KHI &and ∠ IHF ∠ FHG &and ∠ GHK ∠ HKJ &and ∠ JKM These angles are all 90^(∘) because the supplement to a right angle is also a right angle, similar to the previous two streets.
m∠ AEB= 37^(∘), m∠ CED= 55^(∘)
LHS-55^(∘)=RHS-55^(∘)
LHS-37^(∘)=RHS-37^(∘)
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 MF. Since they are all 90^(∘), they are all congruent. ∠ MKJ ≅ ∠ MKL≅ ∠ LKH ≅ ∠ JKH ≅ ∠ KHI ≅ ∠ IHF ≅ ∠ FHG ≅ ∠ GHK Next, let's look at the angles around point O.
Two angles measure 133^(∘) and two angles measure 47^(∘). Each pair of these angles are congruent. Therefore, ∠ MOR≅ ∠ POQ and ∠ MOP≅ ∠ QOR.
Looking to the north of the map, we can see that two angles measure 55^(∘), ∠ KMO and ∠ DEC. They are also congruent. There are no more congruent angles that are directly shown on our map. However, look at ∠ AEB and ∠ BEC. The sum of their measures is 37^(∘)+88^(∘) = 125^(∘). Therefore, ∠ AEC and ∠ NMK are congruent too!
