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Curated practice problem set.. ​

Curated practice problem set.. ​-example-1
User Dave Rager
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a. Labeled diagram based on Priya and Han's conversation:

B 40

G 40

C(40)

b. No, there is nothing special about 40 degrees.

Theorem: If two lines are cut by a transversal such that the alternate interior angles are congruent, then the lines are parallel.

Proof:

Let lines l and m be cut by transversal t at points A and B, respectively.

Let ∠CAB≅∠ABM.

We want to prove that l∥m.

Draw line BX perpendicular to l at point B.

Since ∠ABX=90 ∘ , we have ∠CBX=180 ∘ −∠ABX−∠ABC=180 ∘ −90 ∘ −40 ∘ =50 ∘ .

Since ∠CBM=40 ∘ and ∠CBX=50 ∘ , we have ∠MBX=∠CBX−∠CBM=50 ∘ −40 ∘ =10 ∘ .

Since line BX is perpendicular to line l, we have ∠MBX+∠ABM=90 ∘ .

Substituting ∠MBX=10 ∘ into the equation above, we get 10 ∘ +∠ABM=90∘ .

Solving for ∠ABM, we get ∠ABM=80 ∘ .

Since ∠CAB≅∠ABM, we have ∠CAB=80 ∘ .

Since ∠CAB and ∠ABM are alternate interior angles, we have l∥m.

Therefore, if two lines are cut by a transversal such that the alternate interior angles are congruent, then the lines are parallel.

User Ray Zhou
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