Question

1. triangles rst and vsu are shown below.

given: ∠r ≅ ∠v
rt ≅ vu
which additional condition is sufficient to prove that rs ≅ sv?
a. ts ≅ su
b. vs ⊥ ru
c. rs ≅ su
d. ∠vus ≅ ∠rst

1. use the proof to answer the question below.

given: ab ≅ bc; d is the midpoint of ac
prove: δabd ≅ δcbd

statement reason

1. ab ≅ bc; d is the midpoint of ac 1. given
2. ad ≅ cd 2. definition of midpoint
3. bd ≅ bd 3. reflexive property
4. δabd ≅ δcbd 4. ?

what reason can be used to prove that the triangles are congruent?
a. aas
b. asa
c. sas
d. sss

Explanation:

Question 4

Step1: Analyze given congruences

We know $\angle R \cong \angle V$ and $\overline{RT} \cong \overline{VU}$. We need to prove $\overline{RS} \cong \overline{SV}$, which requires $\triangle RST \cong \triangle VSU$.

Step2: Evaluate each option

• Option A: $\overline{TS} \cong \overline{SU}$ gives SSA, which does not prove triangle congruence.
• Option B: $\overline{VS} \perp \overline{RU}$ creates right angles, but does not provide enough congruent parts for the needed triangle congruence.
• Option C: $\overline{RS} \cong \overline{SU}$ is unrelated to linking the two triangles to prove $\overline{RS} \cong \overline{SV}$.
• Option D: $\angle VUS \cong \angle RST$ gives AAS congruence for $\triangle RST \cong \triangle VSU$ (two angles and a non-included side congruent). If the triangles are congruent, corresponding sides $\overline{RS} \cong \overline{SV}$.

Step1: List congruent sides

From the proof:

1. $\overline{AB} \cong \overline{BC}$ (Given)
2. $\overline{AD} \cong \overline{CD}$ (Definition of Midpoint)
3. $\overline{BD} \cong \overline{BD}$ (Reflexive Property)

Step2: Match to congruence rule

All three pairs of corresponding sides of $\triangle ABD$ and $\triangle CBD$ are congruent, which fits the SSS (Side-Side-Side) triangle congruence postulate.

Answer:

D. $\angle VUS \cong \angle RST$

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Question 5