if $\angle 4 \cong \angle 1$, then which statement(s) can be used to justify that $\angle 3 \cong \angle 1$? select all that apply.\
a) definition of adjacent angles\
b) vertical angles theorem\
c) reflexive property of congruence\
d) transitive property of congruence\
e) substitution property of congruence

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Accepted Answer

# Brief Explanations:
1. First, recall the Vertical Angles Theorem: Vertical angles are congruent. So, ∠1 ≅ ∠2? No, wait, ∠3 and ∠4 are vertical angles? Wait, no, looking at the diagram, ∠1 and ∠2 are adjacent, ∠1 and ∠3: Wait, actually, ∠3 and ∠4 are vertical angles? Wait, no, let's re-examine. The lines: two parallel lines? No, a transversal cutting two lines. Wait, ∠4 and ∠1: given ∠4 ≅ ∠1. Then, ∠3 and ∠4: are they vertical angles? Wait, no, ∠3 and ∠2: maybe. Wait, the Vertical Angles Theorem says vertical angles are congruent. Also, the Transitive Property of Congruence: if ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C. Wait, let's list the options:

- Option B: Vertical Angles Theorem. If ∠3 and ∠4 are vertical angles? Wait, no, maybe ∠1 and ∠3? Wait, no, the given is ∠4 ≅ ∠1. Then, if ∠3 ≅ ∠4 (by Vertical Angles Theorem), then by Transitive Property (if ∠4 ≅ ∠1 and ∠3 ≅ ∠4, then ∠3 ≅ ∠1). Wait, but the options:

Wait, the problem is: If ∠4 ≅ ∠1, then which statement(s) justify ∠3 ≅ ∠1.

Let's analyze each option:

- Option B: Vertical Angles Theorem. ∠3 and ∠4 are vertical angles, so ∠3 ≅ ∠4 (by Vertical Angles Theorem). Then, since ∠4 ≅ ∠1 (given), by Transitive Property (Option D), ∠3 ≅ ∠1. Wait, but the options:

Wait, maybe I misread. Let's check the options again:

A) Definition of adjacent angles: Adjacent angles share a common side and vertex, not related to congruence of ∠3 and ∠1. Eliminate A.

B) Vertical Angles Theorem: ∠3 and ∠4 are vertical angles, so ∠3 ≅ ∠4 (by Vertical Angles Theorem). Then, since ∠4 ≅ ∠1 (given), if we use Transitive Property (D), then ∠3 ≅ ∠1. But also, if ∠3 ≅ ∠4 (vertical angles) and ∠4 ≅ ∠1 (given), then by Transitive (D), ∠3 ≅ ∠1. Also, is there another way?

Wait, maybe ∠3 and ∠4 are vertical angles (so ∠3 ≅ ∠4 by Vertical Angles Theorem), and ∠4 ≅ ∠1 (given), so by Transitive Property (D), ∠3 ≅ ∠1. Also, is Substitution Property (E) applicable? Substitution Property: if a = b and b = c, then a = c (similar to transitive). Wait, Transitive and Substitution: sometimes they are considered, but in angle congruence, Transitive Property of Congruence states that if ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C. Substitution Property is more about replacing a quantity with its equal. But in this case, if ∠3 ≅ ∠4 (vertical angles) and ∠4 ≅ ∠1 (given), then by Transitive (D), ∠3 ≅ ∠1. Also, Vertical Angles Theorem (B) gives ∠3 ≅ ∠4, which is needed. Wait, but the problem is to justify ∠3 ≅ ∠1 given ∠4 ≅ ∠1.

Wait, maybe the correct options are B (Vertical Angles Theorem, to get ∠3 ≅ ∠4) and D (Transitive Property, to get ∠3 ≅ ∠1 from ∠3 ≅ ∠4 and ∠4 ≅ ∠1). Wait, but let's check the options again:

Wait, the options:

B) Vertical Angles Theorem: So ∠3 ≅ ∠4 (by Vertical Angles Theorem). Then, since ∠4 ≅ ∠1 (given), by Transitive Property (D), ∠3 ≅ ∠1. So both B and D? Wait, but the problem says "select all that apply".

Wait, maybe I made a mistake. Let's re-express:

Given ∠4 ≅ ∠1. We need to show ∠3 ≅ ∠1.

If ∠3 ≅ ∠4 (by Vertical Angles Theorem, Option B), then by Transitive Property of Congruence (Option D: if ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C), so ∠3 ≅ ∠1 (since ∠3 ≅ ∠4 and ∠4 ≅ ∠1).

Also, is there another option? Option E: Substitution Property. Substitution Property is when you replace a variable with its equal, but in angle congruence, Transitive is more appropriate. Wait, maybe the Vertical Angles Theorem (B) and Transitive Property (D) are correct.

Wait, let's check the options again:

- Option B: Vertical Angles Theorem. ∠3 and ∠4 are vertical angles, so ∠3 ≅ ∠4. Correct.

- Option D: Transitive Property of Congruence. Since ∠3 ≅ ∠4 (from B) and ∠4 ≅ ∠1 (given), then ∠3 ≅ ∠1. Correct.

- Option E: Substitution Property. Substitution is like replacing ∠4 with ∠1 in ∠3 ≅ ∠4, giving ∠3 ≅ ∠1. Wait, maybe E is also correct? Wait, Transitive and Substitution: sometimes they are used interchangeably, but in formal terms, Transitive is a property of congruence (∠A ≅ ∠B, ∠B ≅ ∠C ⇒ ∠A ≅ ∠C), while Substitution is replacing a term with its equal (if x = y, then f(x) = f(y)). But in angle congruence, if ∠3 ≅ ∠4 and ∠4 ≅ ∠1, then by Transitive (D), ∠3 ≅ ∠1. Alternatively, using Substitution (E): since ∠4 ≅ ∠1, we can substitute ∠4 with ∠1 in ∠3 ≅ ∠4, getting ∠3 ≅ ∠1. So maybe B, D, E? Wait, no, let's check the Vertical Angles Theorem: ∠3 and ∠4 are vertical angles, so ∠3 ≅ ∠4 (B is correct). Then, given ∠4 ≅ ∠1, by Transitive (D) or Substitution (E), we get ∠3 ≅ ∠1.

Wait, but the initial analysis:

- A: Adjacent angles: no, eliminate.

- B: Vertical Angles Theorem: ∠3 ≅ ∠4 (correct, since they are vertical angles).

- C: Reflexive Property: ∠A ≅ ∠A, not applicable here. Eliminate C.

- D: Transitive Property: ∠3 ≅ ∠4 and ∠4 ≅ ∠1 ⇒ ∠3 ≅ ∠1 (correct).

- E: Substitution Property: Replace ∠4 with ∠1 in ∠3 ≅ ∠4 ⇒ ∠3 ≅ ∠1 (correct, since substitution allows replacing a congruent angle with another).

Wait, but maybe the intended answers are B and D, or B, D, E? Wait, let's check the problem again. The problem says "If ∠4 ≅ ∠1, then which statement(s) can be used to justify that ∠3 ≅ ∠1".

Let's confirm:

- Vertical Angles Theorem (B) gives ∠3 ≅ ∠4.

- Transitive Property (D) uses ∠3 ≅ ∠4 and ∠4 ≅ ∠1 to get ∠3 ≅ ∠1.

- Substitution Property (E) also works, as substituting ∠4 with ∠1 in ∠3 ≅ ∠4 gives ∠3 ≅ ∠1.

But maybe the question considers B and D as correct, or B, D, E. Wait, but let's check standard geometry:

- Vertical Angles Theorem: ∠3 ≅ ∠4 (B is correct).

- Transitive Property: ∠3 ≅ ∠4 and ∠4 ≅ ∠1 ⇒ ∠3 ≅ ∠1 (D is correct).

- Substitution Property: Since ∠4 ≅ ∠1, we can substitute ∠4 with ∠1 in the congruence ∠3 ≅ ∠4, resulting in ∠3 ≅ ∠1 (E is correct).

But maybe the answer is B, D, E? Wait, no, maybe I made a mistake. Let's check the diagram again. The angles: ∠1 and ∠2 are adjacent, ∠1 and ∠3: maybe ∠1 and ∠3 are corresponding angles? No, the given is ∠4 ≅ ∠1. So, ∠3 and ∠4 are vertical angles (so ∠3 ≅ ∠4 by B), then ∠4 ≅ ∠1 (given), so by D (Transitive) or E (Substitution), ∠3 ≅ ∠1.

So the correct options are B, D, E? Wait, but let's check the options again:

A: No.

B: Yes (Vertical Angles Theorem for ∠3 ≅ ∠4).

C: No (Reflexive is self-congruence).

D: Yes (Transitive: ∠3 ≅ ∠4 and ∠4 ≅ ∠1 ⇒ ∠3 ≅ ∠1).

E: Yes (Substitution: replace ∠4 with ∠1 in ∠3 ≅ ∠4).

But maybe the intended answer is B and D, or B, D, E. However, in some textbooks, Transitive and Substitution are similar, but Transitive is a property of congruence (three angles), while Substitution is replacing a term. But in this case, since ∠3 ≅ ∠4 (B) and ∠4 ≅ ∠1 (given), Transitive (D) applies. Also, Substitution (E) applies. And Vertical Angles (B) is needed to get ∠3 ≅ ∠4.

Wait, but maybe the answer is B, D, and E? Or B and D? Let's check the problem again. The user says "select all that apply".

After re-evaluating:

- B: Vertical Angles Theorem (∠3 ≅ ∠4) – correct.

- D: Transitive Property (∠3 ≅ ∠4 and ∠4 ≅ ∠1 ⇒ ∠3 ≅ ∠1) – correct.

- E: Substitution Property (replace ∠4 with ∠1 in ∠3 ≅ ∠4 ⇒ ∠3 ≅ ∠1) – correct.

But maybe the initial problem's diagram has ∠3 and ∠4 as vertical angles, so B is correct. Then, with ∠4 ≅ ∠1, Transitive (D) or Substitution (E) works.

However, in many standard problems, the Vertical Angles Theorem (B) and Transitive Property (D) are the main ones, but Substitution (E) is also valid.

But let's check the options again:

- A: Adjacent angles: no.

- B: Vertical Angles Theorem: yes, ∠3 and ∠4 are vertical angles, so ∠3 ≅ ∠4.

- C: Reflexive: no.

- D: Transitive: yes, ∠3 ≅ ∠4 and ∠4 ≅ ∠1 ⇒ ∠3 ≅ ∠1.

- E: Substitution: yes, substitute ∠4 with ∠1 in ∠3 ≅ ∠4.

So the correct options are B, D, E? Wait, but maybe the answer is B and D. Let's see, in some contexts, Substitution is considered part of Transitive, but no, they are distinct.

Alternatively, maybe the problem is designed so that B (Vertical Angles) and D (Transitive) are correct, and E is also correct. But I think the intended answers are B, D, and E? Wait, no, let's check with a standard example.

Suppose ∠3 ≅ ∠4 (vertical angles, B), ∠4 ≅ ∠1 (given). Then by Transitive (D), ∠3 ≅ ∠1. Substitution (E) is also valid because you can substitute ∠4 with ∠1 in the congruence statement ∠3 ≅ ∠4.

So the correct options are B, D, and E? Wait, but the initial check:

- B: Vertical Angles Theorem – correct.

- D: Transitive Property – correct.

- E: Substitution Property – correct.

But maybe the answer is B, D, E. However, I need to confirm.

Wait, the Reflexive Property (C) is ∠A ≅ ∠A, so not applicable. Adjacent angles (A) are not about congruence here. So the correct options are B, D, E.

But maybe the problem's diagram shows ∠3 and ∠4 as vertical angles, so B is correct. Then, with ∠4 ≅ ∠1, Transitive (D) or Substitution (E) gives ∠3 ≅ ∠1.

So the answer is B, D, E. But let's check the options again:

A) Definition of adjacent angles – no.

B) Vertical Angles Theorem – yes.

C) Reflexive Property – no.

D) Transitive Property – yes.

E) Substitution Property – yes.

So the correct options are B, D, E.
# Answer:
B. Vertical Angles Theorem, D. Transitive Property of Congruence, E. Substitution Property of Congruence

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