Question

proving congruence (triangle congruence: sss and sas)
homework

for exercises 37 – 39, consider $\triangle pfk$.

1. what angle is included between $\overline{pf}$ and $\overline{pk}$?
2. which sides include $\angle f$?
3. what angle is included between $\overline{fk}$ and $\overline{kp}$?

1. what additional information is needed to prove the two triangles congruent by sss triangle congruence?

for exercises 41 – 49, if the triangles can be proven congruent, state which rule applies and write a congruence statement. if not, write not enough information.

41.
42.
43.
44.
45.
46.
47.
48.
49.
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Explanation:

Step1: Identify included angle for 37

In $\triangle PFK$, the angle between $\overline{PF}$ and $\overline{PK}$ is the angle at vertex $P$.

Step2: Identify sides for 38

$\angle F$ is at vertex $F$, so the sides forming it are $\overline{PF}$ and $\overline{FK}$.

Step3: Identify included angle for 39

In $\triangle PFK$, the angle between $\overline{FK}$ and $\overline{KP}$ is the angle at vertex $K$.

Step4: Find SSS info for 40

SSS needs 3 pairs of congruent sides. Given $\overline{XZ}\cong\overline{RT}$, $\overline{YZ}\cong\overline{ST}$, so missing is $\overline{XY}\cong\overline{RS}$.

Step5: Analyze congruence for 41

Given $E$ is midpoint, so $\overline{WE}\cong\overline{YE}$; vertical angles $\angle QEW\cong\angle QEY$; $\overline{QE}\cong\overline{QE}$. Use SAS. Congruence: $\triangle QWE\cong\triangle QYE$.

Step6: Analyze congruence for 42

Given $\overline{NY}\cong\overline{MB}$, $\overline{YK}\cong\overline{BS}$, $\angle NYK\cong\angle MBS$. Use SAS. Congruence: $\triangle NYK\cong\triangle MBS$.

Step7: Analyze congruence for 43

Two pairs of congruent angles, one pair of non-included congruent sides. Use AAS. Congruence: $\triangle SRE\cong\triangle PJT$.

Step8: Analyze congruence for 44

$\overline{QP}\cong\overline{NO}$, $\overline{QH}\cong\overline{OH}$, vertical angles $\angle QHP\cong\angle OHN$. No congruence rule applies (SSA is invalid). Result: not enough information.

Step9: Analyze congruence for 45

$\angle J_1\cong\angle J_2$, $\overline{MJ}\cong\overline{KJ}$, $\overline{LJ}\cong\overline{IJ}$. No congruence rule applies (SSA is invalid). Result: not enough information.

Step10: Analyze congruence for 46

$\overline{AB}\cong\overline{DC}$, $\overline{BC}\cong\overline{AD}$, $\overline{BD}\cong\overline{DB}$. Use SSS. Congruence: $\triangle ABD\cong\triangle CDB$.

Step11: Analyze congruence for 47

$\overline{EG}\cong\overline{IH}$, $\overline{EF}\cong\overline{IF}$, vertical angles $\angle EFG\cong\angle IFH$. No congruence rule applies (SSA is invalid). Result: not enough information.

Step12: Analyze congruence for 48

$\overline{SB}\cong\overline{VB}$, $\overline{FB}\perp\overline{SV}$ so $\angle FBS\cong\angle FBV$, $\overline{FB}\cong\overline{FB}$. Use SAS. Congruence: $\triangle FSB\cong\triangle FVB$.

Step13: Analyze congruence for 49

In $\triangle IJK$ and $\triangle ILK$: $\overline{IJ}\cong\overline{IL}$, $\angle JIK\cong\angle LIK$, $\overline{IK}\cong\overline{IK}$. Use SAS. Congruence: $\triangle IJK\cong\triangle ILK$.

Answer:

1. $\boldsymbol{\angle P}$
2. $\boldsymbol{\overline{PF}}$ and $\boldsymbol{\overline{FK}}$
3. $\boldsymbol{\angle K}$
4. $\boldsymbol{\overline{XY} \cong \overline{RS}}$
5. SAS; $\boldsymbol{\triangle QWE \cong \triangle QYE}$
6. SAS; $\boldsymbol{\triangle NYK \cong \triangle MBS}$
7. AAS; $\boldsymbol{\triangle SRE \cong \triangle PJT}$
8. not enough information
9. not enough information
10. SSS; $\boldsymbol{\triangle ABD \cong \triangle CDB}$
11. not enough information
12. SAS; $\boldsymbol{\triangle FSB \cong \triangle FVB}$
13. SAS; $\boldsymbol{\triangle IJK \cong \triangle ILK}$