Question

A two-column proof

uses a visual representation of the logical flow of steps needed to reach a conclusion.
contains a table with a logical series of statements and reasons that reach a conclusion.
contains a set of sentences explaining the steps needed to reach a conclusion.
uses inductive reasoning to prove a statement.A paragraph proof

uses inductive reasoning to prove a statement.
contains a table with a logical series of statements and reasons.
uses a visual chart of the logical flow of steps needed to reach a conclusion.
contains a set of sentences explaining the steps needed to reach a conclusion. If m∠A = m∠B and m∠A + m∠C = m∠D, then
m∠B + m∠C = m∠D.

Which property is shown?

reflexive property
substitution property
symmetric property
transitive propertyA horizontal line has points D, B, C. A line extends vertically from point B to point A.
Given that Ray B A bisects ∠DBC, which statement must be true?

m∠ABD = m∠ABC
AB ≅ BC
B is the midpoint of DC.
m∠DBC = 90°Given: m∠AEB = 45°
∠AEC is a right angle.

Prove: Ray E B bisects ∠AEC.

3 lines extend from point E. A horizontal line extends to the left to point A and a vertical line extends up to point C. Another line extends halfway between the other 2 lines to point B. Lines A E and E C form a right angle.
Proof:
We are given that m∠AEB = 45° and ∠AEC is a right angle. The measure of ∠AEC is 90° by the definition of a right angle. Applying the
gives m∠AEB + m∠BEC = m∠AEC. Applying the substitution property gives 45° + m∠BEC = 90°. The subtraction property can be used to find m∠BEC = 45°, so ∠BEC ≅ ∠AEB because they have the same measure. Since Ray E B divides ∠AEC into two congruent angles, it is the angle bisector.4 lines extend from point E. A horizontal line extends to the left to point A and a vertical line extends up to point C. Another line extends halfway between the other 2 lines to point B. Lines A E and E C form a right angle.
Given that ∠CEA is a right angle and Ray E B bisects ∠CEA, which statement must be true?

∠BEA ≅ ∠CEA
∠CEB ≅ ∠CEA
m∠CEB = 45°
m∠CEA = 45°4 lines extend from point B. A line extends straight up from B to point A. Another line extends up and to the right to point C. Another line extends slight up and to the right to point D. The other line extends slightly down and to the right to point E.
Given that ∠ABC ≅ ∠DBE, which statement must be true?

∠ABC ≅ ∠ABD
∠ABD ≅ ∠CBE
∠CBD ≅ ∠DBE
∠CBD ≅ ∠ABCThe last line of a proof represents

the given information.
the argument.
the conclusion.
the assumptions.Segment AB is congruent to segment AB.

This statement shows the ______ property.

reflexive
symmetric
transitive
substitution4 lines are shown. One line contains points F, E, A. 3 lines come out of point E. One line goes to point D, another line goes to point B, and another line goes to point C. Angles B E C and B E A are congruent.
Which statement is true about the diagram?

∠DEF is a right angle.
m∠DEA = m∠FEC
∠BEA ≅ ∠BEC
Ray E B bisects ∠AEF.

Answer 1

1) Two-column proof
- Answer: contains a table with a logical series of statements and reasons that reach a conclusion.
- Reason: two-column proofs organize each statement with its corresponding reason in table form.

2) Paragraph proof
- Answer: contains a set of sentences explaining the steps needed to reach a conclusion.
- Reason: paragraph proofs present the logical steps in sentence form.

3) If m∠A = m∠B and m∠A + m∠C = m∠D, then m∠B + m∠C = m∠D.
- Answer: substitution property.
- Reason: m∠A is replaced by m∠B in the equation.

4) Diagram with D, B, C collinear and BA bisects ∠DBC.
- Answer: m∠ABD = m∠ABC.
- Reason: BA bisects ∠DBC, so the two adjacent angles formed with BA are congruent.

5) In the proof: "Applying the ___ gives m∠AEB + m∠BEC = m∠AEC."
- Answer: Angle Addition Postulate.
- Reason: the Angle Addition Postulate states a whole angle equals the sum of its two adjacent parts.

6) Given ∠CEA is a right angle and Ray EB bisects ∠CEA, which must be true?
- Answer: m∠CEB = 45°.
- Reason: a right angle measures 90°; a bisector divides it into two 45° angles.

7) From the B-ray diagram: Given ∠ABC ≅ ∠DBE, which must be true?
- Answer: ∠ABD ≅ ∠CBE.
- Reason: let ∠ABC = x and ∠CBD = y; then ∠ABD = x + y and ∠CBE = y + x, so they are equal.

8) The last line of a proof represents
- Answer: the conclusion.
- Reason: the final line states what was to be proven.

9) Segment AB ≅ segment AB.
- Answer: reflexive property.
- Reason: any segment (or angle) is congruent to itself.

10) Diagram with F–E–A collinear and ∠BEC ≅ ∠BEA.
- Answer: ∠BEA ≅ ∠BEC.
- Reason: this is exactly the given fact (the other choices are not supported by the given information).