Question

jennifer is given the two triangles... can she prove the triangles are congruent, and why? a. yes, asa can be used to prove the triangles are congruent. b. yes, sas can be used to prove the triangles are congruent. c. no, the third side in each triangle is not given, so sss cannot be used. d. no, the congruent sides are not in corresponding locations relative to the given angles. this diagram shows two triangles. diagram of rts and xsw with right angles what additional information is... this figure shows lines m, n, and p. diagram of transversal p intersecting m and n, angles 1,2,3 on m; 6,7 on n which statement is sufficient to prove lines m and n are parallel? a. ∠1 ≅ ∠3 b. ∠1 ≅ ∠6 c. ∠3 ≅ ∠6 d. ∠6 ≅ ∠7

Explanation:

First Question (Triangles Congruence)

Step1: Recall Congruence Theorems

ASA (Angle - Side - Angle) requires two angles and the included side to be congruent. SAS (Side - Angle - Side) requires two sides and the included angle to be congruent. SSS (Side - Side - Side) requires all three sides to be congruent.

Step2: Analyze the Triangles

From the diagram (the first triangle problem), the congruent sides are not in the correct relative position to the given angles for SAS or ASA. For SAS, the side should be between the two angles, and for ASA, the side should be included between two angles. Here, the congruent side is not in the corresponding location relative to the given angles. So the answer is D.

Second Question (Parallel Lines)

Step1: Recall Parallel Line Theorems

To prove two lines are parallel, we can use corresponding angles, alternate interior angles, or consecutive interior angles. Corresponding angles: if corresponding angles are congruent, lines are parallel. Alternate interior angles: if alternate interior angles are congruent, lines are parallel.

Step2: Analyze Each Option

• Option A: $\angle1\cong\angle3$ are vertical angles, not related to parallel lines of $m$ and $n$.
• Option B: $\angle1\cong\angle6$: $\angle1$ and $\angle6$ are corresponding angles. If corresponding angles are congruent, then lines $m$ and $n$ are parallel (by the Corresponding Angles Postulate).
• Option C: $\angle3\cong\angle6$: $\angle3$ and $\angle6$ are not corresponding or alternate interior angles in a way that proves $m\parallel n$.
• Option D: $\angle6\cong\angle7$ are vertical angles, not related to parallel lines of $m$ and $n$. So the answer is B.

Third Question (Additional Information for Triangles, though not fully shown, assuming triangle congruence)

Assuming it's about proving $\triangle RTS\cong\triangle WXS$ (from the diagram with right angles at $T$ and $X$, and vertical angles at $S$). We already have $\angle T=\angle X = 90^{\circ}$, $\angle RST=\angle WXS$ (vertical angles). To prove congruence, we can use AAS (Angle - Angle - Side) or ASA. If we have $RT = WX$ (or $RS=WS$ or $TS = XS$), we can prove congruence. But since the question is about additional information, if we consider AAS, we need a pair of sides. For example, if $RT = WX$, then by AAS (two angles and a non - included side), the triangles are congruent. But since the question is not fully clear, but based on the diagram, if we assume we need to prove congruence, a possible additional information is a pair of corresponding sides (e.g., $RT = WX$ or $TS=XS$ or $RS = WS$).

Answer:

1. First Question: D. No, the congruent sides are not in corresponding locations relative to the given angles.
2. Second Question: B. $\angle1\cong\angle6$
3. Third Question (assuming triangle congruence): A possible additional information is $RT = WX$ (or other corresponding sides like $TS = XS$, $RS=WS$) to prove $\triangle RTS\cong\triangle WXS$ (using AAS or SAS or ASA depending on the side).