Download Geometry Lesson 31

Name: __________________________
Geometry Lesson 31
Date: ________________________
Objective: TSW use flow chart and paragraph proofs.
Period: ______________________
Flowchart Proof - A style of proof that uses boxes and arrows to show the structure of the proof.
A flowchart proof should be read from left to right or from top to bottom. Each part of the proof appears in a _____,
while the justification for each step is written __________ the
box. The arrows show the progression of the proof’s steps.
Example 1 Interpreting a Flowchart Proof
Use the given flowchart proof to write a two-column proof.
Given: ∠1 and ∠3 are congruent.
∠1 and ∠2 are supplementary.
Prove: ∠2 and ∠3 are supplementary.
SOLUTION
Statements
Reasons
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Flowchart proofs are useful when a proof has two different ________________ that could be performed at the same
time, rather than in sequence with one another. Whenever a proof does not proceed linearly from one step to another,
a flowchart proof should be considered.
Example 2 Writing a Flowchart Proof
Prove the Triangle Angle Sum Theorem: The sum of the
interior angles of a triangle is 180°.
Given: ∆𝐴𝐵𝐶
Prove: m∠1 + m∠2 + m∠3 = 180°
SOLUTION
1
2
Paragraph Proof - A style of proof in which statements and reasons are presented in paragraph form.
In a paragraph proof, every step of the proof must be explained by a sentence in the paragraph. Each sentence contains
a _____________________ and a _______________________________.
Example 3 Reading a Paragraph Proof
Use the given paragraph proof to write a two-column proof.
Given: ∠1 and ∠2 are complementary.
Prove: ∠3 and ∠4 are complementary.
∠1 and ∠2 are complementary, so m∠1 + m∠2 = 90° by the definition of complementary angles.
Angle 1 is congruent to ∠4, and ∠2 is congruent to ∠3, by the Vertical Angles Theorem. So m∠1 =
m∠4, and m∠2 = m∠3. By substitution, m∠4 + m∠3 = 90°. Therefore, ∠3 and ∠4 are
complementary by the definition of complementary angles.
SOLUTION
Statements
Reasons
1.
2.
3.
4.
5.
6.
A paragraph proof is good for ____________ proofs where each step follows logically from the one before. Paragraph
proofs are usually more compact than two-column proofs.
Example 4 Writing a Paragraph Proof
Prove Theorem 10-1: If two lines are cut by a transversal, then alternate interior angles
are congruent.
Given: Lines p and q are parallel.
Prove: ∠2 ≅ ∠3
SOLUTION
You Try!
b. Prove the Vertical Angles Theorem using a flowchart proof.
Given: a and b are intersecting lines.
Prove: ∠1 and ∠3 are congruent.
d. Prove Theorem 5-4: If two lines are perpendicular, then they form congruent adjacent
angles.
Given: Lines s and t are perpendicular.
Prove: Angles 1 and 2 are congruent adjacent angles.