Question

problem 10: given: \\( \angle b a c \cong \angle d c a, \overline{b a} \cong \overline{d c} \\)

problem 11: given: \\( \angle f \cong \angle i, \angle e \cong \angle h, \overline{e g} \cong \overline{h j} \\)

problem 12: given: \\( \angle o \cong \angle m, \overline{l o} \cong \overline{l m} \\)

problem 13: given: \\( \angle p \cong \angle r, \angle q s p \cong \angle s q r \\)

problem 14: given: \\( \overline{a c} \\| \overline{b d}, \angle b \cong \angle c \\)

choices for problems #1 - 4 (some will be used more than once):
aas
asa
alternate interior angles are \\( \cong \\)
given
reflexive property
sas
sss
vertical angles are \\( \cong \\)

Explanation:

Problem 10

Step1: Fill in the second statement

Given \( \overline{BA}\cong\overline{DC}\), so statement 2 is \( \overline{BA}\cong\overline{DC}\)

Step2: Determine the reason for statement 3

We have two triangles \( \triangle ABC\) and \( \triangle CDA\). We know \( \angle BAC\cong\angle DCA\), \( \overline{BA}\cong\overline{DC}\), and \( \overline{AC}\cong\overline{AC}\) (reflexive property). So the reason for statement 3 (if we consider the sides - angles - sides) is \(SAS\) (Side - Angle - Side) congruence criterion.

Step3: Determine the reason for statement 4

Since we have shown the two triangles \( \triangle ABC\) and \( \triangle CDA\) satisfy \(SAS\) (from steps 1 - 2), the reason for \( \triangle ABC\cong\triangle CDA\) is \(SAS\)

Problem 11

Step1: Fill in the second statement

Given \( \angle E\cong\angle H\), so statement 2 is \( \angle E\cong\angle H\)

Step2: Fill in the third statement

Given \( \overline{EG}\cong\overline{HJ}\), so statement 3 is \( \overline{EG}\cong\overline{HJ}\)

Step3: Determine the reason for statement 4

We have two angles (\( \angle F\cong\angle I\), \( \angle E\cong\angle H\)) and a non - included side (\( \overline{EG}\cong\overline{HJ}\)). The congruence criterion is \(AAS\) (Angle - Angle - Side)

Problem 12

Step1: Fill in the first blank

Given \( \angle O\cong\angle M\), so statement 1 is \( \angle O\cong\angle M\)

Step2: Fill in the second statement

Given \( \overline{LO}\cong\overline{LM}\), so statement 2 is \( \overline{LO}\cong\overline{LM}\)

Step3: Fill in the third blank

\( \angle KLO\) and \( \angle NLM\) are vertical angles. So \( \angle KLO\cong\angle NLM\) (vertical angles are congruent)

Step4: Determine the reason for statement 4

We have two angles (\( \angle O\cong\angle M\), \( \angle KLO\cong\angle NLM\)) and a side (\( \overline{LO}\cong\overline{LM}\)). The congruence criterion is \(ASA\) (Angle - Side - Angle)

Problem 13

Step1: Fill in the first blank

Given \( \angle P\cong\angle R\), so statement 1 is \( \angle P\cong\angle R\)

Step2: Fill in the second statement

Given \( \angle QSP\cong\angle SQR\), so statement 2 is \( \angle QSP\cong\angle SQR\)

Step3: Determine the reason for statement 4

We have two angles (\( \angle P\cong\angle R\), \( \angle QSP\cong\angle SQR\)) and a side (\( \overline{SQ}\cong\overline{SQ}\) - reflexive property). The congruence criterion is \(AAS\)

Problem 14

Step1: Fill in the second statement

Given \( \angle B\cong\angle C\), so statement 2 is \( \angle B\cong\angle C\)

Step2: Determine the reason for statement 3

Since \( \overline{AC}\parallel\overline{BD}\), by the alternate interior angles theorem, \( \angle CAD\cong\angle BDA\) (alternate interior angles are congruent)

Step3: Fill in the fourth statement

By reflexive property, \( \overline{AD}\cong\overline{DA}\)

Step4: Fill in the fifth blank

We have two angles (\( \angle CAD\cong\angle BDA\), \( \angle C\cong\angle B\)) and a side (\( \overline{AD}\cong\overline{DA}\)). The triangle \( \triangle ACD\cong\triangle DBA\) (using \(AAS\))

Answer:

• Problem 10:
• 2. \( \overline{BA}\cong\overline{DC}\)
• 3. \(SAS\)
• 4. \(SAS\)
• Problem 11:
• 2. \( \angle E\cong\angle H\)
• 3. \( \overline{EG}\cong\overline{HJ}\)
• 4. \(AAS\)
• Problem 12:
• 1. \(O\)
• 2. \( \overline{LO}\cong\overline{LM}\)
• 3. \(NLM\) (because vertical angles are congruent)
• 4. \(ASA\)
• Problem 13:
• 1. \(R\)
• 2. \( \angle QSP\cong\angle SQR\)
• 4. \(AAS\)
• Problem 14:
• 2. \( \angle B\cong\angle C\)
• 3. Alternate Interior Angles are \( \cong\)
• 4. \( \overline{AD}\cong\overline{DA}\)
• 5. \(DBA\) (by \(AAS\))