Given: \(\overline{KM} \perp \overline{JL}\); \(\angle J \cong \angle L\) | Prove: \(\triangle JMK \cong \triangle LMK\)
K J M L Right angles \(\angle KMJ \cong \angle KML\)
| # | Statements (Answer) | Reasons |
|---|---|---|
| 1 | \(\overline{KM} \perp \overline{JL}\); \(\angle J \cong \angle L\) | Given |
| 2 | \(\angle KMJ\) and \(\angle KML\) are right angles (or \(\angle JMK, \angle LMK\)) | Definition of Perpendicular |
| 3 | \(\angle KMJ \cong \angle KML\) (or \(\angle JMK \cong \angle LMK\)) | All right angles are congruent |
| 4 | \(\overline{KM} \cong \overline{KM}\) | Reflexive Property of Congruence |
| 5 | \(\triangle JMK \cong \triangle LMK\) (or \(\triangle KMJ \cong \triangle KML\)) | AAS |
Scoring Note Accept \(\angle JMK, \angle LMK\) naming. The proof establishes right angles via the definition of perpendicular lines before asserting congruence, concluding directly with triangle congruence via AAS.
Proof 4 ASA & CPCTC
Given: \(\overline{QS}\) bisects \(\angle PQR\); \(\angle PSQ \cong \angle RSQ\) | Prove: \(\overline{PQ} \cong \overline{RQ}\)
Q P S R \(\angle PQS \cong \angle RQS\) (bisector)
| # | Statements (Answer) | Reasons |
|---|---|---|
| 1 | \(\overline{QS}\) bisects \(\angle PQR\); \(\angle PSQ \cong \angle RSQ\) | Given |
| 2 | \(\angle PQS \cong \angle RQS\) | Definition of Angle Bisector |
| 3 | \(\overline{QS} \cong \overline{QS}\) | Reflexive Property of Congruence |
| 4 | \(\triangle PQS \cong \triangle RQS\) | ASA |
| 5 | \(\overline{PQ} \cong \overline{RQ}\) | CPCTC |
Scoring Note For step 2, accept \(\angle SQP \cong \angle SQR\). In step 4, triangle statement order must match: \(\triangle PSQ \cong \triangle RSQ\) or \(\triangle QPS \cong \triangle QRS\) are also acceptable.
Answer Key
Teacher Solutions & Scoring Guide • SSS, SAS, ASA, AAS & CPCTC
Page 3 Solutions
Proof 5 ASA & CPCTC
Given: \(\overline{AB} \parallel \overline{CD}\), \(\overline{AB} \cong \overline{CD}\) | Prove: \(\overline{AN} \cong \overline{CN}\)
A B N D C Alternate interior angles \(\angle A \cong \angle C\)
| # | Statements | Reasons |
|---|---|---|
| 1 | \(\overline{AB} \parallel \overline{CD}\), \(\overline{AB} \cong \overline{CD}\) | Given |
| 2 | \(\angle A \cong \angle C\) (or \(\angle BAN \cong \angle DCN\)) | Alternate Interior Angles |
| 3 | \(\angle B \cong \angle D\) (or \(\angle ABN \cong \angle CDN\)) | Alternate Interior Angles |
| 4 | \(\triangle ABN \cong \triangle CDN\) | ASA |
| 5 | \(\overline{AN} \cong \overline{CN}\) | CPCTC |
Scoring Note Accept alternate naming like \(\angle BAN \cong \angle DCN\) and \(\angle ABN \cong \angle CDN\). Students may also use vertical angles (\(\angle ANB \cong \angle CND\)) with AAS. Concludes with CPCTC.
Proof 6 SAS & CPCTC
Given: \(\angle WMY\) and \(\angle WMZ\) are right angles; \(M\) is midpoint of \(\overline{YZ}\) | Prove: \(\angle Y \cong \angle Z\)
W Y M Z \(\overline{YM} \cong \overline{ZM}\) & \(\overline{WM} \cong \overline{WM}\)
| # | Statements | Reasons |
|---|---|---|
| 1 | \(\angle WMY\) and \(\angle WMZ\) are right angles; \(M\) is midpoint of \(\overline{YZ}\) | Given |
| 2 | \(\overline{YM} \cong \overline{ZM}\) | Definition of Midpoint |
| 3 | \(\angle WMY \cong \angle WMZ\) | All right angles are congruent |
| 4 | \(\overline{WM} \cong \overline{WM}\) | Reflexive Property of Congruence |
| 5 | \(\triangle WMY \cong \triangle WMZ\) | SAS |
| 6 | \(\angle Y \cong \angle Z\) | CPCTC |
Scoring Note Step 1 is completely student-supplied. Step 3 statement can be written as \(\angle YMW \cong \angle ZMW\). Concludes with CPCTC.
Answer Key
Teacher Solutions & Scoring Guide • SSS, SAS, ASA, AAS & CPCTC
Page 4 Solutions
Proof 7 SSS & CPCTC
Given: \(\overline{AB} \cong \overline{CB}\), \(\overline{AD} \cong \overline{CD}\) | Prove: \(\angle A \cong \angle C\)
A B C D Shared diagonal \(\overline{BD}\)
| # | Statements (Answer) | Reasons (Answer) |
|---|---|---|
| 1 | \(\overline{AB} \cong \overline{CB}\), \(\overline{AD} \cong \overline{CD}\) | Given |
| 2 | \(\overline{BD} \cong \overline{BD}\) | Reflexive Property of Congruence |
| 3 | \(\triangle ABD \cong \triangle CBD\) | SSS Congruence Postulate |
| 4 | \(\angle A \cong \angle C\) (or \(\angle BAD \cong \angle BCD\)) | CPCTC |
Scoring Note Students may split given statements across two lines (1 & 2), making the proof 5 steps total. Give full credit if the logical chain from given \(\rightarrow\) reflexive \(\rightarrow\) SSS \(\rightarrow\) CPCTC is preserved.
Proof 8 AAS Postulate
Given: \(\angle A \cong \angle D\), \(\overline{AB} \cong \overline{DE}\) | Prove: \(\triangle ABC \cong \triangle DEC\)
A B C E D Vertical angles \(\angle ACB \cong \angle DCE\)
| # | Statements (Answer) | Reasons (Answer) |
|---|---|---|
| 1 | \(\angle A \cong \angle D\), \(\overline{AB} \cong \overline{DE}\) | Given |
| 2 | \(\angle ACB\) and \(\angle DCE\) are vertical angles | Definition of Vertical Angles |
| 3 | \(\angle ACB \cong \angle DCE\) | Vertical Angles Theorem |
| 4 | \(\triangle ABC \cong \triangle DEC\) | AAS Congruence Postulate |
Scoring Note Accept alternate naming like \(\angle BCA \cong \angle ECD\). The proof establishes vertical angles before asserting congruence, concluding directly with triangle congruence via AAS.
