1. use the information given in the diagram. tell why each statement is true.
a) $overline{ab} parallel overline{dc}$
b) $angle cab cong angle acd$
c) $angle b cong angle d$
d) $angle bca cong angle dac$
e) $overline{ac} cong overline{ac}$
f) $overline{ab} cong overline{dc}, overline{bc} cong overline{ad}$
g) $ riangle abc cong riangle cda$
can you conclude the figures below are congruent? justify each answer.
2. $ riangle spq$ and $ riangle tuv$
3. $ riangle trk$ and $ riangle tuk$
#4 - 5: $ riangle abc cong riangle def$.
find the measures of the given angles or the lengths of the given sides.
4. $bc = 3z + 2, ef = z + 6$
5. $ac = 7a + 5, df = 5a + 9$

Question

1. use the information given in the diagram. tell why each statement is true.
a) $overline{ab} parallel overline{dc}$
 b) $angle cab cong angle acd$
 c) $angle b cong angle d$
 d) $angle bca cong angle dac$
 e) $overline{ac} cong overline{ac}$
 f) $overline{ab} cong overline{dc}, overline{bc} cong overline{ad}$
 g) $	riangle abc cong 	riangle cda$
can you conclude the figures below are congruent? justify each answer.
2. $	riangle spq$ and $	riangle tuv$
3. $	riangle trk$ and $	riangle tuk$
#4 - 5: $	riangle abc cong 	riangle def$.
find the measures of the given angles or the lengths of the given sides.
4. $bc = 3z + 2, ef = z + 6$
5. $ac = 7a + 5, df = 5a + 9$

Accepted Answer

### Problem 1 (a - g)
#### a) $\overline{AB} \parallel \overline{DC}$
# Explanation:
## Step1: Identify the figure
The diagram shows a quadrilateral with markings on sides and angles. From the markings (e.g., $\overline{BC} \cong \overline{AD}$ and $\overline{AB} \cong \overline{DC}$ implied by congruent triangles later, or alternate interior angles), we can see it's a parallelogram - like figure. In a parallelogram, opposite sides are parallel. Also, since $\angle CAB \cong \angle ACD$ (alternate interior angles), by the converse of alternate interior angles theorem, $\overline{AB} \parallel \overline{DC}$.
# Answer: By converse of alternate interior angles theorem (since $\angle CAB \cong \angle ACD$), $\overline{AB} \parallel \overline{DC}$

#### b) $\angle CAB \cong \angle ACD$
# Explanation:
## Step1: Identify the lines and angles
$\overline{AB} \parallel \overline{DC}$ (from part a) and $\overline{AC}$ is a transversal. When two parallel lines are cut by a transversal, alternate interior angles are congruent. So $\angle CAB$ and $\angle ACD$ are alternate interior angles, hence $\angle CAB \cong \angle ACD$.
# Answer: Alternate interior angles (since $\overline{AB} \parallel \overline{DC}$ and $\overline{AC}$ is transversal), so $\angle CAB \cong \angle ACD$

#### c) $\angle B \cong \angle D$
# Explanation:
## Step1: Recall triangle congruence
From part f, $\overline{AB} \cong \overline{DC}$, $\overline{BC} \cong \overline{AD}$ and $\overline{AC} \cong \overline{AC}$ (part e). So $	riangle ABC \cong 	riangle CDA$ (SSS). Corresponding parts of congruent triangles are congruent (CPCTC), so $\angle B \cong \angle D$.
# Answer: CPCTC (since $	riangle ABC \cong 	riangle CDA$), so $\angle B \cong \angle D$

#### d) $\angle BCA \cong \angle DAC$
# Explanation:
## Step1: Identify the lines and angles
$\overline{BC} \parallel \overline{AD}$ (since $\overline{BC} \cong \overline{AD}$ and in a quadrilateral with opposite sides congruent, it's a parallelogram, so $\overline{BC} \parallel \overline{AD}$) and $\overline{AC}$ is a transversal. Alternate interior angles are congruent. So $\angle BCA$ and $\angle DAC$ are alternate interior angles, hence $\angle BCA \cong \angle DAC$.
# Answer: Alternate interior angles (since $\overline{BC} \parallel \overline{AD}$ and $\overline{AC}$ is transversal), so $\angle BCA \cong \angle DAC$

#### e) $\overline{AC} \cong \overline{AC}$
# Explanation:
## Step1: Recall the reflexive property
The reflexive property of congruence states that any segment is congruent to itself. So $\overline{AC}$ is congruent to itself, hence $\overline{AC} \cong \overline{AC}$.
# Answer: Reflexive property of congruence (a segment is congruent to itself), so $\overline{AC} \cong \overline{AC}$

#### f) $\overline{AB} \cong \overline{DC}$, $\overline{BC} \cong \overline{AD}$
# Explanation:
## Step1: Identify the markings
From the diagram, there are tick marks on $\overline{AB}$ and $\overline{DC}$ (implied congruence) and on $\overline{BC}$ and $\overline{AD}$ (implied congruence). In a diagram, tick marks indicate congruent segments. So by the diagram's markings (or in the context of the quadrilateral, it's a parallelogram - like figure where opposite sides are congruent), $\overline{AB} \cong \overline{DC}$ and $\overline{BC} \cong \overline{AD}$.
# Answer: By diagram markings (tick marks) indicating congruent segments, so $\overline{AB} \cong \overline{DC}$ and $\overline{BC} \cong \overline{AD}$

#### g) $	riangle ABC \cong 	riangle CDA$
# Explanation:
## Step1: Identify the congruent sides
We have $\overline{AB} \cong \overline{DC}$ (from part f), $\overline{BC} \cong \overline{AD}$ (from part f) and $\overline{AC} \cong \overline{AC}$ (from part e). By the Side - Side - Side (SSS) congruence criterion, if three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. So $	riangle ABC \cong 	riangle CDA$ by SSS.
# Answer: By SSS congruence criterion ( $\overline{AB} \cong \overline{DC}$, $\overline{BC} \cong \overline{AD}$, $\overline{AC} \cong \overline{AC}$), so $	riangle ABC \cong 	riangle CDA$

### Problem 2: $	riangle SPQ$ and $	riangle TUV$
# Explanation:
## Step1: List the side lengths
For $	riangle SPQ$: $SQ = 5$, $QP = 6$, $SP = 7$
For $	riangle TUV$: $TV = 4$, $TU = 7$, $UV = 8$
## Step2: Check congruence criteria (SSS, SAS, ASA, AAS)
SSS: We need all three sides to be congruent. $SQ = 5
eq TV = 4$, $QP = 6
eq UV = 8$, $SP = 7 = TU = 7$. Since not all sides are congruent, SSS fails.
SAS: We need two sides and included angle. We don't have angle information, and sides don't match for SAS.
ASA/AAS: No angle information given. So the triangles are not congruent.
# Answer: No, $	riangle SPQ$ and $	riangle TUV$ are not congruent. Because the side lengths are $5,6,7$ for $	riangle SPQ$ and $4,7,8$ for $	riangle TUV$; not all corresponding sides are congruent (SSS fails), and no angle - side correspondence matches other criteria.

### Problem 3: $	riangle TRK$ and $	riangle TUK$
# Explanation:
## Step1: Identify congruent parts
From the diagram: $\overline{TR} \cong \overline{TU}$ (tick marks), $\overline{RK} \cong \overline{UK}$ (tick marks), and $\overline{TK}$ is common to both triangles. Also, $\angle RTK \cong \angle UTK$ (marked as congruent angles).
## Step2: Apply congruence criterion
By Side - Angle - Side (SAS) congruence criterion: $\overline{TR} \cong \overline{TU}$, $\angle RTK \cong \angle UTK$, $\overline{TK} \cong \overline{TK}$ (reflexive). So $	riangle TRK \cong 	riangle TUK$ by SAS.
# Answer: Yes, $	riangle TRK \cong 	riangle TUK$ by SAS congruence criterion ( $\overline{TR} \cong \overline{TU}$, $\angle RTK \cong \angle UTK$, $\overline{TK} \cong \overline{TK}$)

### Problem 4: $	riangle ABC \cong 	riangle DEF$, $BC = 3z + 2$, $EF = z + 6$
# Explanation:
## Step1: Recall CPCTC for sides
Since $	riangle ABC \cong 	riangle DEF$, corresponding sides $BC$ and $EF$ are congruent. So $BC = EF$.
## Step2: Set up the equation
Substitute the expressions: $3z + 2 = z + 6$
## Step3: Solve for $z$
Subtract $z$ from both sides: $3z - z+ 2 = z - z + 6\Rightarrow 2z + 2 = 6$
Subtract 2 from both sides: $2z+2 - 2=6 - 2\Rightarrow 2z = 4$
Divide by 2: $z=\frac{4}{2}=2$
## Step4: Find $BC$ and $EF$ (optional, but to check)
$BC = 3(2)+2 = 8$, $EF = 2 + 6 = 8$. They are equal, so $z = 2$.
# Answer: $z = 2$

### Problem 5: $	riangle ABC \cong 	riangle DEF$, $AC = 7a + 5$, $DF = 5a + 9$
# Explanation:
## Step1: Recall CPCTC for sides
Since $	riangle ABC \cong 	riangle DEF$, corresponding sides $AC$ and $DF$ are congruent. So $AC = DF$.
## Step2: Set up the equation
Substitute the expressions: $7a + 5 = 5a + 9$
## Step3: Solve for $a$
Subtract $5a$ from both sides: $7a - 5a+ 5 = 5a - 5a + 9\Rightarrow 2a + 5 = 9$
Subtract 5 from both sides: $2a+5 - 5=9 - 5\Rightarrow 2a = 4$
Divide by 2: $a=\frac{4}{2}=2$
## Step4: Find $AC$ and $DF$ (optional, but to check)
$AC = 7(2)+5 = 19$, $DF = 5(2)+9 = 19$. They are equal, so $a = 2$.
# Answer: $a = 2$

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QUESTION IMAGE

Question

Explanation:

Problem 1 (a - g)

a) $\overline{AB} \parallel \overline{DC}$

Step1: Identify the figure

Step1: Identify the lines and angles

Step1: Recall triangle congruence

Answer:

b) $\angle CAB \cong \angle ACD$