Unknown Angle Proofs — Writing Proofs
One of the main goals in studying geometry is to develop your ability to reason critically, to draw valid conclusions based upon observations and proven facts. Master detectives do this sort of thing all the time. Take a look as Sherlock Holmes uses seemingly insignificant observations to draw amazing conclusions.
Could you follow Sherlock Holmes’s reasoning as he described his thought process?
In geometry, we follow a similar deductive thought process (much like Holmes uses) to prove geometric claims. Let’s revisit an old friend—solving for unknown angles. Remember this one?
You needed to figure out the measure of 𝑎 and used the “fact” that an exterior angle of a triangle equals the sum of the measures of the opposite interior angles. The measure of ∠𝑎 must, therefore, be 36°.
Suppose that we rearrange the diagram just a little bit. Instead of using numbers, we use variables to represent angle measures.
Suppose further that we already know that the angles of a triangle sum to 180°. Given the labeled diagram to the right, can we prove that 𝑥 + 𝑦 = 𝑧 (or, in other words, that the exterior angle of a triangle equals the sum of the measures of the opposite interior angles)?
Label ∠𝑤, as shown in the diagram. 𝑚∠𝑥 + 𝑚∠𝑦 + 𝑚∠𝑤 = 180° The sum of the angle measures in a triangle is 180°. 𝑚∠𝑤 + 𝑚∠𝑧 = 180° Linear pairs form supplementary angles. 𝑚∠𝑥 + 𝑚∠𝑦 + 𝑚∠𝑤 = 𝑚∠𝑤 + 𝑚∠𝑧 Substitution property of equality ∴ 𝑚∠𝑥 + 𝑚∠𝑦 = 𝑚∠𝑧
Notice that each step in the proof was justified by a previously known or demonstrated fact. We end up with a newly proven fact (that an exterior angle of any triangle is the sum of the measures of the opposite interior angles of the triangle). This ability to identify the steps used to reach a conclusion based on known facts is deductive reasoning (i.e., the same type of reasoning that Sherlock Holmes used to accurately describe the doctor’s attacker in the video clip).
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