Question

sss proof #1
given: ( overline{pq} cong overline{st} ), ( overline{qr} cong overline{tr} ), ( r ) is the midpoint of ( overline{ps} )
prove: ( \triangle pqr cong \triangle str )
(diagram: triangles ( pqr ) and ( str ) with ( p, r, s ) colinear, ( q, t ) above ( pr, rs ) respectively)
statements | reasons
(blank table rows)
boxes: ( overline{pq} cong overline{st} ), ( \text{def. of midpoint} ), ( overline{qr} cong overline{tr} ), ( r ) is the midpoint of ( overline{ps} ), ( \text{given} ), ( overline{pr} cong overline{sr} ), ( \text{given} ), ( \text{sss} ), ( \text{given} ), ( \triangle pqr cong \triangle str )

sss proof #2
given: ( l ) is the midpoint of ( overline{jn} ), ( overline{jm} cong overline{nm} ),
prove: ( \triangle jlm cong \triangle nlm )
(diagram: triangle ( jnm ) with ( l ) midpoint of ( jn ), ( m ) above ( l ))
statements | reasons
(blank table rows)
boxes: ( overline{jl} cong overline{nl} ), ( l ) is the midpoint of ( overline{jn} ), ( overline{jm} cong overline{nm} ), ( \text{def. of midpoint} ), ( \triangle jlm cong \triangle nlm ), ( \text{given} ), ( \text{given} ), ( \text{sss} ), ( overline{lm} cong overline{lm} ), ( \text{reflexive property} )

Explanation:

SSS Proof #1

Step1: List Given Information

• \( \overline{PQ} \cong \overline{ST} \) (Given)
• \( \overline{QR} \cong \overline{TR} \) (Given)
• \( R \) is the midpoint of \( \overline{PS} \) (Given)

Step2: Use Midpoint Definition

Since \( R \) is the midpoint of \( \overline{PS} \), \( \overline{PR} \cong \overline{SR} \) (Def. of Midpoint)

Step3: Apply SSS Congruence

In \( \triangle PQR \) and \( \triangle STR \):

• \( \overline{PQ} \cong \overline{ST} \) (From Step 1)
• \( \overline{QR} \cong \overline{TR} \) (From Step 1)
• \( \overline{PR} \cong \overline{SR} \) (From Step 2)

By SSS (Side - Side - Side) Congruence Criterion, \( \triangle PQR \cong \triangle STR \)

SSS Proof #2

Step1: List Given Information

• \( L \) is the midpoint of \( \overline{JN} \) (Given)
• \( \overline{JM} \cong \overline{NM} \) (Given)

Step2: Use Midpoint Definition

Since \( L \) is the midpoint of \( \overline{JN} \), \( \overline{JL} \cong \overline{NL} \) (Def. of Midpoint)

Step3: Use Reflexive Property

\( \overline{LM} \cong \overline{LM} \) (Reflexive Property of Congruence)

Step4: Apply SSS Congruence

In \( \triangle JLM \) and \( \triangle NLM \):

• \( \overline{JL} \cong \overline{NL} \) (From Step 2)
• \( \overline{JM} \cong \overline{NM} \) (From Step 1)
• \( \overline{LM} \cong \overline{LM} \) (From Step 3)

By SSS (Side - Side - Side) Congruence Criterion, \( \triangle JLM \cong \triangle NLM \)

Filling the Tables

SSS Proof #1 Table

Statements	Reasons
\( \overline{QR} \cong \overline{TR} \)	Given
\( R \) is the midpoint of \( \overline{PS} \)	Given
\( \overline{PR} \cong \overline{SR} \)	Def. of Midpoint
\( \triangle PQR \cong \triangle STR \)	SSS

SSS Proof #2 Table

Statements	Reasons
\( \overline{JM} \cong \overline{NM} \)	Given
\( \overline{JL} \cong \overline{NL} \)	Def. of Midpoint
\( \overline{LM} \cong \overline{LM} \)	Reflexive Property
\( \triangle JLM \cong \triangle NLM \)	SSS

Answer:

Step1: List Given Information

• \( L \) is the midpoint of \( \overline{JN} \) (Given)
• \( \overline{JM} \cong \overline{NM} \) (Given)

Step2: Use Midpoint Definition

Since \( L \) is the midpoint of \( \overline{JN} \), \( \overline{JL} \cong \overline{NL} \) (Def. of Midpoint)

Step3: Use Reflexive Property

\( \overline{LM} \cong \overline{LM} \) (Reflexive Property of Congruence)

Step4: Apply SSS Congruence

In \( \triangle JLM \) and \( \triangle NLM \):

• \( \overline{JL} \cong \overline{NL} \) (From Step 2)
• \( \overline{JM} \cong \overline{NM} \) (From Step 1)
• \( \overline{LM} \cong \overline{LM} \) (From Step 3)

By SSS (Side - Side - Side) Congruence Criterion, \( \triangle JLM \cong \triangle NLM \)

Filling the Tables

SSS Proof #1 Table

Statements	Reasons
\( \overline{QR} \cong \overline{TR} \)	Given
\( R \) is the midpoint of \( \overline{PS} \)	Given
\( \overline{PR} \cong \overline{SR} \)	Def. of Midpoint
\( \triangle PQR \cong \triangle STR \)	SSS

SSS Proof #2 Table

Statements	Reasons
\( \overline{JM} \cong \overline{NM} \)	Given
\( \overline{JL} \cong \overline{NL} \)	Def. of Midpoint
\( \overline{LM} \cong \overline{LM} \)	Reflexive Property
\( \triangle JLM \cong \triangle NLM \)	SSS