QUESTION IMAGE
Question
- given: ( abcong de ), ( angle bcongangle e ), ( angle acongangle d ), prove: ( angle ccongangle f )
- given: ( angle pqscongangle rsq ), ( pqcong rs ), prove: ( angle qpscongangle srq )
- given: ( angle dcongangle j ), ( jlcong dl ), prove: ( \triangle jklcong\triangle del )
13.
Step1: Given information
$AB\cong DE$, $\angle B\cong\angle E$, $\angle A\cong\angle D$ are given.
Step2: Use ASA (Angle - Side - Angle) congruence criterion
Since two angles ($\angle A\cong\angle D$, $\angle B\cong\angle E$) and the included side ($AB\cong DE$) of $\triangle ABC$ and $\triangle DEF$ are congruent, by ASA, $\triangle ABC\cong\triangle DEF$.
Step3: Corresponding parts of congruent triangles
If $\triangle ABC\cong\triangle DEF$, then their corresponding angles are congruent. So, $\angle C\cong\angle F$.
| Statements | Reasons |
|---|---|
| $\angle B\cong\angle E$ | Given |
| $\angle A\cong\angle D$ | Given |
| $\triangle ABC\cong\triangle DEF$ | ASA (Angle - Side - Angle) |
| $\angle C\cong\angle F$ | Corresponding parts of congruent triangles (CPCTC) |
14.
Step1: Given information
$\angle PQS\cong\angle RSQ$, $PQ\cong RS$ are given.
Step2: Reflexive property
$QS\cong SQ$ (reflexive property of congruence, a segment is congruent to itself).
Step3: Use SAS (Side - Angle - Side) congruence criterion
Since $PQ\cong RS$, $\angle PQS\cong\angle RSQ$, and $QS\cong SQ$, by SAS, $\triangle PQS\cong\triangle RSQ$.
Step4: Corresponding parts of congruent triangles
If $\triangle PQS\cong\triangle RSQ$, then $\angle QPS\cong\angle SRQ$ (CPCTC).
| Statements | Reasons |
|---|---|
| $PQ\cong RS$ | Given |
| $QS\cong SQ$ | Reflexive property of congruence |
| $\triangle PQS\cong\triangle RSQ$ | SAS (Side - Angle - Side) |
| $\angle QPS\cong\angle SRQ$ | Corresponding parts of congruent triangles (CPCTC) |
15.
Step1: Given information
$\angle D\cong\angle J$, $JL\cong DL$ are given.
Step2: Reflexive property
$\angle L\cong\angle L$ (reflexive property of congruence, an angle is congruent to itself).
Step3: Use ASA (Angle - Side - Angle) congruence criterion
Since $\angle J\cong\angle D$, $JL\cong DL$, and $\angle L\cong\angle L$, by ASA, $\triangle JKL\cong\triangle DEL$.
| Statements | Reasons |
|---|---|
| $JL\cong DL$ | Given |
| $\angle L\cong\angle L$ | Reflexive property of congruence |
| $\triangle JKL\cong\triangle DEL$ | ASA (Angle - Side - Angle) |
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13.
Step1: Given information
$AB\cong DE$, $\angle B\cong\angle E$, $\angle A\cong\angle D$ are given.
Step2: Use ASA (Angle - Side - Angle) congruence criterion
Since two angles ($\angle A\cong\angle D$, $\angle B\cong\angle E$) and the included side ($AB\cong DE$) of $\triangle ABC$ and $\triangle DEF$ are congruent, by ASA, $\triangle ABC\cong\triangle DEF$.
Step3: Corresponding parts of congruent triangles
If $\triangle ABC\cong\triangle DEF$, then their corresponding angles are congruent. So, $\angle C\cong\angle F$.
| Statements | Reasons |
|---|---|
| $\angle B\cong\angle E$ | Given |
| $\angle A\cong\angle D$ | Given |
| $\triangle ABC\cong\triangle DEF$ | ASA (Angle - Side - Angle) |
| $\angle C\cong\angle F$ | Corresponding parts of congruent triangles (CPCTC) |
14.
Step1: Given information
$\angle PQS\cong\angle RSQ$, $PQ\cong RS$ are given.
Step2: Reflexive property
$QS\cong SQ$ (reflexive property of congruence, a segment is congruent to itself).
Step3: Use SAS (Side - Angle - Side) congruence criterion
Since $PQ\cong RS$, $\angle PQS\cong\angle RSQ$, and $QS\cong SQ$, by SAS, $\triangle PQS\cong\triangle RSQ$.
Step4: Corresponding parts of congruent triangles
If $\triangle PQS\cong\triangle RSQ$, then $\angle QPS\cong\angle SRQ$ (CPCTC).
| Statements | Reasons |
|---|---|
| $PQ\cong RS$ | Given |
| $QS\cong SQ$ | Reflexive property of congruence |
| $\triangle PQS\cong\triangle RSQ$ | SAS (Side - Angle - Side) |
| $\angle QPS\cong\angle SRQ$ | Corresponding parts of congruent triangles (CPCTC) |
15.
Step1: Given information
$\angle D\cong\angle J$, $JL\cong DL$ are given.
Step2: Reflexive property
$\angle L\cong\angle L$ (reflexive property of congruence, an angle is congruent to itself).
Step3: Use ASA (Angle - Side - Angle) congruence criterion
Since $\angle J\cong\angle D$, $JL\cong DL$, and $\angle L\cong\angle L$, by ASA, $\triangle JKL\cong\triangle DEL$.
| Statements | Reasons |
|---|---|
| $JL\cong DL$ | Given |
| $\angle L\cong\angle L$ | Reflexive property of congruence |
| $\triangle JKL\cong\triangle DEL$ | ASA (Angle - Side - Angle) |