Question

bottom what is pyramus? a lover ( quince a lover, that kills himself; for love. bottom that will ask some tears performing of it. if i do it, let the audie i will move storms, i will cor measure. to the east — yet my chief hu ould play rches rarely, or a p ake all split. the raging rocks and shivering sb hall break the l of prison gates. nd phibbus all shine fro d make anc foolish f as lofty. s. — th vein, ng. 3. given \\(\overline{gk} \cong \overline{ml}\\), \\(\angle gkm \cong \angle lmk\\) prove \\(\triangle gkm \cong \triangle lmk\\) statements 1. \\(\overline{gk} \cong \overline{ml}\\), \\(\angle gkm \cong \angle lmk\\) 2. \\(\overline{mk} \cong \overline{mk}\\) 3. \\(\triangle gkm \cong \triangle lmk\\) reasons 1. 2. 3. diagram of quadrilateral gmlk with diagonal mk 4. given \\(\angle s \cong \angle r\\) and \\(\overline{xt}\\) bisects \\(\angle sxr\\) prove: \\(\triangle sxt \cong \triangle rxt\\) statements 1. \\(\angle s \cong \angle r\\) and \\(\overline{xt}\\) bisects \\(\angle sxr\\) 2. \\(\angle sxt \cong \angle rxt\\) 3. \\(\overline{xt} \cong \overline{xt}\\) 4. \\(\triangle sxt \cong \triangle rxt\\) reasons 1. 2. 3. 4. diagram of triangle sxr with t on sr, xt as angle bisector 5. given \\(\overline{ft} \cong \overline{fr}\\) and \\(\overline{st} \cong \overline{sr}\\) prove \\(\triangle fts \cong \triangle frs\\) statements 1. \\(\overline{ft} \cong \overline{fr}\\) and \\(\overline{st} \cong \overline{sr}\\) 2. 3. reasons 1. 2. reflexive property 3. diagram of triangle ftr with s inside, fs as a segment

Explanation:

Problem 3 (Proving $\triangle GKM \cong \triangle LMK$)

Step 1: Identify Given Information

We are given $\overline{GK} \cong \overline{ML}$ and $\angle GKM \cong \angle LMK$. Also, $\overline{MK} \cong \overline{KM}$ (Reflexive Property of Congruence, as a segment is congruent to itself).

Step 2: Determine Congruence Criterion

Using the Side - Angle - Side (SAS) Congruence Criterion, which states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Here, $\overline{GK} \cong \overline{ML}$, $\angle GKM \cong \angle LMK$, and $\overline{MK} \cong \overline{KM}$. So, by SAS, $\triangle GKM \cong \triangle LMK$.

Reasons for Problem 3:

1. Given (for $\overline{GK} \cong \overline{ML}$, $\angle GKM \cong \angle LMK$)
2. Reflexive Property of Congruence (for $\overline{MK} \cong \overline{KM}$)
3. SAS (Side - Angle - Side) Congruence Criterion (for $\triangle GKM \cong \triangle LMK$)

Problem 4 (Proving $\triangle SXT \cong \triangle RXT$)

Step 1: Identify Given Information

We are given $\angle S \cong \angle R$ and $\overline{XT}$ bisects $\angle SXR$.

Step 2: Use Angle - Bisector Definition

Since $\overline{XT}$ bisects $\angle SXR$, by the definition of an angle bisector, $\angle SXT \cong \angle RXT$.

Step 3: Identify Common Side

$\overline{XT} \cong \overline{XT}$ (Reflexive Property of Congruence).

Step 4: Determine Congruence Criterion

Using the Angle - Angle - Side (AAS) Congruence Criterion, which states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the triangles are congruent. Here, $\angle S \cong \angle R$, $\angle SXT \cong \angle RXT$, and $\overline{XT} \cong \overline{XT}$. So, by AAS, $\triangle SXT \cong \triangle RXT$.

Reasons for Problem 4:

1. Given (for $\angle S \cong \angle R$ and $\overline{XT}$ bisects $\angle SXR$)
2. Definition of Angle Bisector (for $\angle SXT \cong \angle RXT$)
3. Reflexive Property of Congruence (for $\overline{XT} \cong \overline{XT}$)
4. AAS (Angle - Angle - Side) Congruence Criterion (for $\triangle SXT \cong \triangle RXT$)

Problem 5 (Proving $\triangle FTS \cong \triangle FRS$)

Step 1: Identify Given Information

We are given $\overline{FT} \cong \overline{FR}$ and $\overline{ST} \cong \overline{SR}$.

Step 2: Identify Common Side

$\overline{FS} \cong \overline{FS}$ (Reflexive Property of Congruence).

Step 3: Determine Congruence Criterion

Using the Side - Side - Side (SSS) Congruence Criterion, which states that if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. Here, $\overline{FT} \cong \overline{FR}$, $\overline{ST} \cong \overline{SR}$, and $\overline{FS} \cong \overline{FS}$. So, by SSS, $\triangle FTS \cong \triangle FRS$.

Statements and Reasons for Problem 5:

Statements:

1. $\overline{FT} \cong \overline{FR}$ and $\overline{ST} \cong \overline{SR}$
2. $\overline{FS} \cong \overline{FS}$
3. $\triangle FTS \cong \triangle FRS$

Reasons:

1. Given
2. Reflexive Property
3. SSS (Side - Side - Side) Congruence Criterion

Answer:

Step 1: Identify Given Information

We are given $\overline{FT} \cong \overline{FR}$ and $\overline{ST} \cong \overline{SR}$.

Step 2: Identify Common Side

$\overline{FS} \cong \overline{FS}$ (Reflexive Property of Congruence).

Step 3: Determine Congruence Criterion

Using the Side - Side - Side (SSS) Congruence Criterion, which states that if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. Here, $\overline{FT} \cong \overline{FR}$, $\overline{ST} \cong \overline{SR}$, and $\overline{FS} \cong \overline{FS}$. So, by SSS, $\triangle FTS \cong \triangle FRS$.

Statements and Reasons for Problem 5:

Statements:

1. $\overline{FT} \cong \overline{FR}$ and $\overline{ST} \cong \overline{SR}$
2. $\overline{FS} \cong \overline{FS}$
3. $\triangle FTS \cong \triangle FRS$

Reasons:

1. Given
2. Reflexive Property
3. SSS (Side - Side - Side) Congruence Criterion