Unknown Angle Proofs — Proofs with Constructions

New York State Common Core Math Geometry, Module 1, Lesson 10

Worksheets for Geometry

Student Outcomes

• Students write unknown angle proofs involving auxiliary lines.

Unknown Angle Proofs — Proofs with Constructions

Classwork

Opening Exercise

In the figure on the right, 𝐴𝐵 ∥ 𝐷𝐸 and 𝐵𝐶 ∥ 𝐸𝐹. Prove that 𝑏 = 𝑒.
(Hint: Extend 𝐵𝐶 and 𝐸𝐷.)

Proof:

In the previous lesson, you used deductive reasoning with labeled diagrams to prove specific conjectures. What is different about the proof above? Drawing or extending segments, lines, or rays (referred to as auxiliary lines) is frequently useful in demonstrating steps in the deductive reasoning process. Once 𝐵𝐶 and 𝐸𝐷 were extended, it was relatively simple to prove the two angles congruent based on our knowledge of alternate interior angles. Sometimes there are several possible extensions or additional lines that would work equally well. For example, in this diagram, there are at least two possibilities for auxiliary lines. Can you spot them both?

Given: 𝐴𝐵 ∥ 𝐶𝐷. Prove: 𝑧 = 𝑥 + 𝑦

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Discussion

Here is one possibility:
Given: 𝐴𝐵 ∥ 𝐶𝐷.
Prove: 𝑧 = 𝑥 + 𝑦.
Extend the transversal as shown by the dotted line in the diagram.
Label angle measures 𝑣 and 𝑤, as shown.
What do you know about 𝑣 and 𝑥?
About 𝑤 and 𝑦? How does this help you?

Write a proof using the auxiliary segment drawn in the diagram to the right.

Another possibility appears here:
Given: 𝐴𝐵 ∥ 𝐶𝐷.
Prove: 𝑧 = 𝑥 + 𝑦.
Draw a segment parallel to 𝐴𝐵 through the vertex of the angle measuring 𝑧 degrees. This divides the angle into two parts as shown.
What do you know about 𝑣 and 𝑥?

About 𝑤 and 𝑦? How does this help you?

Write a proof using the auxiliary segment drawn in this diagram. Notice how this proof differs from the one above.

Examples

1. In the figure to the right, 𝐴𝐵 ∥ 𝐶𝐷 and 𝐵𝐶 ∥ 𝐷𝐸. Prove that 𝑚∠𝐴𝐵𝐶 = 𝑚∠𝐶𝐷𝐸. (Is an auxiliary segment necessary?)
2. In the figure to the right, 𝐴𝐵 ∥ 𝐶𝐷 and 𝐵𝐶 ∥ 𝐷𝐸. Prove that 𝑏 + 𝑑 = 180.
3. In the figure to the right, prove that 𝑑 = 𝑎 + 𝑏 + 𝑐.

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