Unknown Angle Proofs — Writing Proofs

New York State Common Core Math Geometry, Module 1, Lesson 9

Worksheets for Geometry

Student Outcomes

• Students write unknown angle proofs, which does not require any new geometric facts. Rather, writing proofs requires students to string together facts they already know to reveal more information.

Unknown Angle Proofs — Writing Proofs

Classwork

Opening Exercise

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?

• Show Video for Whole Lesson

Discussion

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)?

PROOF:
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).

Exercises 1–6

1. You know that angles on a line sum to 180°.
   Prove that vertical angles are equal in measure.
   Make a plan:

• What do you know about ∠𝑤 and ∠𝑥? ∠𝑦 and ∠𝑥?
• What conclusion can you draw based on both pieces of knowledge?
• Write out your proof:

2. Given the diagram to the right, prove that 𝑚∠𝑤 + 𝑚∠𝑥 + 𝑚∠𝑧 = 180°.
   (Make a plan first. What do you know about ∠𝑥, ∠𝑦, and ∠𝑧?)
   Given the diagram to the right, prove that 𝑚∠𝑤 = 𝑚∠𝑦 + 𝑚∠𝑥
3. In the diagram to the right, prove that 𝑚∠𝑦 + 𝑚∠𝑧 = 𝑚∠𝑤 + 𝑚∠𝑥.
   (You need to write a label in the diagram that is not labeled yet for this proof.)
4. In the figure to the right, 𝐴𝐵 || 𝐶𝐷 and 𝐵𝐶 || 𝐷𝐸.
   Prove that 𝑚∠𝐴𝐵𝐶 = 𝑚∠𝐶𝐷𝐸.
5. In the figure to the right, prove that the sum of the angles marked by arrows is 900°.
   (You need to write several labels in the diagram for this proof.)
6. In the figure to the right, prove that 𝐷𝐶 ⊥ 𝐸𝐹.
   Draw in label 𝑍.

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