Question

1. given: \\(\overline{ab} \cong \overline{ed}\\), \\(\overline{ab} \parallel \overline{de}\\), \\(c\\) is the midpoint of \\(\overline{ae}\\)\

prove: \\(\triangle abc \cong \triangle edc\\)\
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1. given: \\(\overline{yx} \cong \overline{zx}\\), \\(\overline{wx}\\) bisects \\(\angle yxz\\)\

prove: \\(\triangle wyx \cong \triangle wzx\\)\
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1. use the coordinates below to determine if \\(\triangle abc\\) and \\(\triangle def\\) are congruent.\

\\(\triangle abc\\): \\(a(2, -8)\\), \\(b(-5, -2)\\), \\(c(-7, 3)\\); \\(\triangle def\\): \\(d(-9, 7)\\), \\(e(-11, 12)\\), \\(f(-2, 1)\\)\
\\(ab = \\) \\(de = \\)\
\\(bc = \\) \\(ef = \\)\
\\(ac = \\) \\(df = \\)\
are the triangles congruent? if yes, explain your reasoning and write a congruency statement.

Explanation:

Problem 9: Prove $\triangle ABC \cong \triangle EDC$

Step 1: List Given Information

• $\overline{AB} \cong \overline{ED}$ (Given)
• $\overline{AB} \parallel \overline{DE}$ (Given)
• $C$ is the midpoint of $\overline{AE}$ (Given)

Step 2: Identify Vertical Angles

$\angle ACB \cong \angle ECD$ (Vertical angles are congruent)

Step 3: Identify Alternate Interior Angles

Since $\overline{AB} \parallel \overline{DE}$, $\angle A \cong \angle E$ (Alternate interior angles are congruent)

Step 4: Midpoint Implies Congruent Segments

$C$ is the midpoint of $\overline{AE}$, so $\overline{AC} \cong \overline{EC}$ (Definition of midpoint)

Step 5: Apply ASA Congruence

In $\triangle ABC$ and $\triangle EDC$:

• $\angle A \cong \angle E$ (Step 3)
• $\overline{AC} \cong \overline{EC}$ (Step 4)
• $\angle ACB \cong \angle ECD$ (Step 2)

Thus, $\triangle ABC \cong \triangle EDC$ (ASA Congruence Postulate)

Problem 10: Prove $\triangle WYX \cong \triangle WZX$

Step 1: List Given Information

• $\overline{YX} \cong \overline{ZX}$ (Given)
• $\overline{WX}$ bisects $\angle YXZ$ (Given)

Step 2: Angle Bisector Implies Congruent Angles

$\overline{WX}$ bisects $\angle YXZ$, so $\angle WXY \cong \angle WXZ$ (Definition of angle bisector)

Step 3: Identify Common Side

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

Step 4: Apply SAS Congruence

In $\triangle WYX$ and $\triangle WZX$:

• $\overline{YX} \cong \overline{ZX}$ (Given)
• $\angle WXY \cong \angle WXZ$ (Step 2)
• $\overline{WX} \cong \overline{WX}$ (Step 3)

Thus, $\triangle WYX \cong \triangle WZX$ (SAS Congruence Postulate)

Problem 11: Determine if $\triangle ABC$ and $\triangle DEF$ are Congruent

We use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Calculate $AB$ (for $\triangle ABC$):

$A(2, -8)$, $B(-5, -2)$
$$AB = \sqrt{(-5 - 2)^2 + (-2 - (-8))^2} = \sqrt{(-7)^2 + (6)^2} = \sqrt{49 + 36} = \sqrt{85}$$

Calculate $DE$ (for $\triangle DEF$):

$D(-9, 7)$, $E(-11, 12)$
$$DE = \sqrt{(-11 - (-9))^2 + (12 - 7)^2} = \sqrt{(-2)^2 + (5)^2} = \sqrt{4 + 25} = \sqrt{29}$$

Calculate $BC$ (for $\triangle ABC$):

$B(-5, -2)$, $C(-7, 3)$
$$BC = \sqrt{(-7 - (-5))^2 + (3 - (-2))^2} = \sqrt{(-2)^2 + (5)^2} = \sqrt{4 + 25} = \sqrt{29}$$

Calculate $EF$ (for $\triangle DEF$):

$E(-11, 12)$, $F(-2, 1)$
$$EF = \sqrt{(-2 - (-11))^2 + (1 - 12)^2} = \sqrt{(9)^2 + (-11)^2} = \sqrt{81 + 121} = \sqrt{202}$$

Calculate $AC$ (for $\triangle ABC$):

$A(2, -8)$, $C(-7, 3)$
$$AC = \sqrt{(-7 - 2)^2 + (3 - (-8))^2} = \sqrt{(-9)^2 + (11)^2} = \sqrt{81 + 121} = \sqrt{202}$$

Calculate $DF$ (for $\triangle DEF$):

$D(-9, 7)$, $F(-2, 1)$
$$DF = \sqrt{(-2 - (-9))^2 + (1 - 7)^2} = \sqrt{(7)^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85}$$

Compare Side Lengths:

• $AB = \sqrt{85}$, $DF = \sqrt{85}$
• $BC = \sqrt{29}$, $DE = \sqrt{29}$
• $AC = \sqrt{202}$, $EF = \sqrt{202}$

By SSS (Side-Side-Side) Congruence, $\triangle ABC \cong \triangle DFE$ (since $AB \cong DF$, $BC \cong DE$, $AC \cong EF$).

Answer:

s:

Problem 9 (Table):

Statements	Reasons
2. $\overline{AB} \parallel \overline{DE}$	Given
3. $C$ is the midpoint of $\overline{AE}$	Given
4. $\angle A \cong \angle E$	Alternate Interior Angles (from $AB \parallel DE$)
5. $\overline{AC} \cong \overline{EC}$	Definition of Midpoint
6. $\angle ACB \cong \angle ECD$	Vertical Angles Congruent
7. $\triangle ABC \cong \triangle EDC$	ASA Congruence

Problem 10 (Table):

Statements	Reasons
2. $\overline{WX}$ bisects $\angle YXZ$	Given
3. $\angle WXY \cong \angle WXZ$	Definition of Angle Bisector
4. $\overline{WX} \cong \overline{WX}$	Reflexive Property
5. $\triangle WYX \cong \triangle WZX$	SAS Congruence

Problem 11 (Table):

$AB = \boldsymbol{\sqrt{85}}$	$DE = \boldsymbol{\sqrt{29}}$
$BC = \boldsymbol{\sqrt{29}}$	$EF = \boldsymbol{\sqrt{202}}$
$AC = \boldsymbol{\sqrt{202}}$	$DF = \boldsymbol{\sqrt{85}}$

Congruence: Yes, $\triangle ABC \cong \triangle DFE$ (SSS Congruence, since all corresponding sides are congruent).