Podcast
Questions and Answers
∠BEA ≅ ∠BEC is a true statement about the diagram.
∠BEA ≅ ∠BEC is a true statement about the diagram.
True
Segment AB is congruent to segment AB. This statement shows the _____ property.
Segment AB is congruent to segment AB. This statement shows the _____ property.
reflexive
Given that RT ≅ WX, which statement must be true?
Given that RT ≅ WX, which statement must be true?
RT + TW = WX + TW
What is a two-column proof?
What is a two-column proof?
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Given that D is the midpoint of AB and K is the midpoint of BC, which statement must be true?
Given that D is the midpoint of AB and K is the midpoint of BC, which statement must be true?
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What is the missing justification?
What is the missing justification?
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Given that ∠CEA is a right angle and EB bisects ∠CEA, which statement must be true?
Given that ∠CEA is a right angle and EB bisects ∠CEA, which statement must be true?
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Given that ∠ABC ≅ ∠DBE, which statement must be true?
Given that ∠ABC ≅ ∠DBE, which statement must be true?
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K is the midpoint of AB.
K is the midpoint of AB.
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Given that BA bisects ∠DBC, which statement must be true?
Given that BA bisects ∠DBC, which statement must be true?
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Name the three different types of proofs you've seen.
Name the three different types of proofs you've seen.
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Which property is shown? If m∠ABC = m∠CBD, then m∠CBD = m∠ABC.
Which property is shown? If m∠ABC = m∠CBD, then m∠CBD = m∠ABC.
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EB bisects ∠AEC. What statements are true regarding the given statement and diagram?
EB bisects ∠AEC. What statements are true regarding the given statement and diagram?
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Given: m∠ABC = m∠CBD. Prove: BC bisects ∠ABD. Justify each step in the flowchart proof.
Given: m∠ABC = m∠CBD. Prove: BC bisects ∠ABD. Justify each step in the flowchart proof.
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Describe the main parts of a proof.
Describe the main parts of a proof.
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Complete the paragraph proof. From the diagram, ∠CED is a right angle, which measures __° degrees.
Complete the paragraph proof. From the diagram, ∠CED is a right angle, which measures __° degrees.
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The measure of angle ______ must also be 90° by the _____.
The measure of angle ______ must also be 90° by the _____.
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Given: ∠ABC is a right angle and ∠DEF is a right angle. Prove: All right angles are congruent by showing that ∠ABC ≅ ∠DEF. What are the missing reasons in the steps of the proof?
Given: ∠ABC is a right angle and ∠DEF is a right angle. Prove: All right angles are congruent by showing that ∠ABC ≅ ∠DEF. What are the missing reasons in the steps of the proof?
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Identify the missing parts in the proof. Given: ∠ABC is a right angle. DB bisects ∠ABC. Prove: m∠CBD = 45°.
Identify the missing parts in the proof. Given: ∠ABC is a right angle. DB bisects ∠ABC. Prove: m∠CBD = 45°.
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Given: m∠A + m∠B = m∠B + m∠C. Prove: m∠C = m∠A. Write a paragraph proof to prove the statement.
Given: m∠A + m∠B = m∠B + m∠C. Prove: m∠C = m∠A. Write a paragraph proof to prove the statement.
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Study Notes
Proof Assignment and Quiz Flashcards Overview
- Understanding diagram relationships is crucial; for example, ∠BEA is congruent to ∠BEC.
- The reflexive property states that any segment is congruent to itself (e.g., segment AB ≅ segment AB).
- If segments RT and WX are congruent, their lengths can be summed with equal additional lengths, such as RT + TW = WX + TW.
- A two-column proof organizes statements and reasons in a logical format leading to a conclusion.
Midpoints and Segment Relationships
- If point D is the midpoint of segment AB and point K is the midpoint of segment BC, then the combined lengths of segments AK and BK equal segment AC (AK + BK = AC).
Properties of Angles
- The transitive property relates the equality of angles and segments through intermediary equalities.
- If ∠CEA is right and EB bisects it, then each of the resulting angles must be equal (m∠CEB = 45°).
- Congruent angles indicate that if ∠ABC ≅ ∠DBE, then ∠ABD must also be congruent to ∠CBE.
Bisectors and Angles
- K being the midpoint of AB is a critical assumption that leads to further deductions about lengths and congruencies in proofs.
- When a line bisects an angle, such as BA bisecting ∠DBC, the resulting angles are equal (m∠ABD = m∠ABC).
Types of Proofs
- There are three main types of proofs:
- Two-column proofs: List statements next to reasons
- Paragraph proofs: Written explanations of the relationships
- Flowchart proofs: Diagrams showing logical steps and conclusions.
Properties of Equality
- The symmetric property of equality indicates that if one angle equals another, then the second angle equals the first (if m∠ABC = m∠CBD, then m∠CBD = m∠ABC).
Angle Relationships
- If EB bisects ∠AEC, then several key angle relationships hold, including that ∠CED and ∠CEA are right angles, among others.
Constructing Proofs
- Comprehensive proofs consist of given information, steps using deductive reasoning, justification through theorems and definitions, ultimately concluding with a proven statement.
- To prove that m∠AEB = 45°, utilize known properties: EB bisects the angle, leading to the conclusion that each part measures half of the right angle.
Proving Angle Congruence
- To prove all right angles are congruent, one crucial reason includes the definition that states angles ABC and DEF are each equal to 90°.
Angle Measure Justifications
- To justify that m∠CBD = 45°, one can use the angle addition postulate within the setup of a right angle that has been bisected.
Proving Angle Equality
- A paragraph proof for angle equality utilizes the reflexive property and symmetric property, ensuring the relationship between the angles can be simplified down to their equality.
Studying That Suits You
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Description
Test your understanding of proof concepts with this quiz on congruences and properties. Each question challenges you to apply fundamental geometric principles and definitions. Perfect for students learning about two-column proofs and congruent segments.
