Proof Assignment Quiz: Missing Parts & Angles

Questions and Answers

∠BEA ≅ ∠BEC is a true statement about the diagram.

True

Segment AB is congruent to segment AB. This statement shows the _____ property.

reflexive

Given that RT ≅ WX, which statement must be true?

RT + TW = WX + TW

What is a two-column proof?

It contains a table with a logical series of statements and reasons that reach a conclusion. Signup and view all the answers

Given that D is the midpoint of AB and K is the midpoint of BC, which statement must be true?

AK + BK = AC Signup and view all the answers

What is the missing justification?

transitive property Signup and view all the answers

Given that ∠CEA is a right angle and EB bisects ∠CEA, which statement must be true?

m∠CEB = 45° Signup and view all the answers

Given that ∠ABC ≅ ∠DBE, which statement must be true?

∠ABD ≅ ∠CBE Signup and view all the answers

K is the midpoint of AB.

True Signup and view all the answers

Given that BA bisects ∠DBC, which statement must be true?

m∠ABD = m∠ABC Signup and view all the answers

Name the three different types of proofs you've seen.

Two-column proof, paragraph proof, flowchart proof. Signup and view all the answers

Which property is shown? If m∠ABC = m∠CBD, then m∠CBD = m∠ABC.

symmetric property Signup and view all the answers

EB bisects ∠AEC. What statements are true regarding the given statement and diagram?

∠CED is a right angle, ∠CEA is a right angle, m∠CEB = m∠BEA, m∠DEB = 135° Signup and view all the answers

Given: m∠ABC = m∠CBD. Prove: BC bisects ∠ABD. Justify each step in the flowchart proof.

A: given, B: definition of congruent, C: definition of bisect Signup and view all the answers

Describe the main parts of a proof.

Proofs contain given information and a statement to be proven, using deductive reasoning and justification. Signup and view all the answers

Complete the paragraph proof. From the diagram, ∠CED is a right angle, which measures __° degrees.

90 Signup and view all the answers

The measure of angle __ must also be 90° by the _.

AEC, angle addition postulate Signup and view all the answers

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?

A: definition of right angle, B: substitution property, C: definition of congruent angles. Signup and view all the answers

Identify the missing parts in the proof. Given: ∠ABC is a right angle. DB bisects ∠ABC. Prove: m∠CBD = 45°.

A: given, B: measure of angle ABC = 90, C: angle addition postulate, D: 2 times the measure of angle CBD = 90. Signup and view all the answers

Given: m∠A + m∠B = m∠B + m∠C. Prove: m∠C = m∠A. Write a paragraph proof to prove the statement.

We are given that the sum of the measures of angles A and B is equal to the sum of the measures of angles B and C. The measure of angle B is equal to itself by the reflexive property, so you can subtract that measure from both sides of the equation. Now the measure of angle A equals the measure of angle C. By the symmetric property, this means the measure of angle C equals the measure of angle A. Signup and view all the answers

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.

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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.

Introduction to Proof Assignment Quiz