Question

given lq = lr = os = ot and qr = st, select the true statement used to prove that ∠klm ≅ ∠nop.

show your work here

○ △klm ≅ △nop by the side - side - side criterion
○ $overleftrightarrow{kl} parallel overleftrightarrow{no}$ because lines with congruent alternate interior angles are parallel
○ $km = np$ because corresponding parts of congruent triangles are congruent
○ △qlr ≅ △sot by the side - side - side criterion

Explanation:

Step1: Analyze given information

We know \( LQ = LR = OS = OT \) and \( QR = ST \). We need to find which statement helps prove \( \angle KLM \cong \angle NOP \).

Step2: Check each option

• Option 1: \( \triangle KLM \cong \triangle NOP \) by SSS. But we don't have info on all three sides of these triangles, so this is incorrect.
• Option 2: \( \overleftrightarrow{KL} \parallel \overleftrightarrow{NO} \) - The goal is to prove angle congruence, not parallel lines, so this is irrelevant.
• Option 3: \( KM = NP \) - This is about segments, not angles, and we can't assume this yet without triangle congruence, so incorrect.
• Option 4: \( \triangle QLR \cong \triangle SOT \) by SSS. We have \( LQ = OS \), \( LR = OT \), and \( QR = ST \), so by SSS, \( \triangle QLR \cong \triangle SOT \). Then corresponding angles \( \angle KLM \) and \( \angle NOP \) (which are \( \angle QLR \) and \( \angle SOT \) or related) would be congruent. This makes sense.

Answer:

\( \triangle QLR \cong \triangle SOT \) by the side - side - side criterion