SSA

SSA is a method for determining whether two triangles are congruent by comparing two sides and a non-inclusive angle. However, unlike SAS, SSS, ASA, and SAA, this does not prove congruence in all cases.

Suppose we have two triangles, $\\triangle ABC$ and $\\triangle PQR$ . $\\triangle ABC\\cong?\\triangle PQR$ if $\\overline{AB}\\cong\\overline{PQ}$ , $\\overline{BC}\\cong\\overline{QR}$ , and either $\\angle BAC\\cong\\angle QPR$ or $\\angle BCA\\cong\\angle QRP$ .

Since this method does not prove congruence, it is more useful for disproving it. If the SSA method is attempted between $\\triangle ABC$ and $\\triangle PQR$ and fails for every $A B C$ , $B C A$ , and $C B A$ against every $P Q R$ , $Q R P$ , and $R P Q$ , then $\\triangle ABC\ot\\cong\\triangle PQR$ .

Suppose $\\triangle ABC$ and $\\triangle PQR$ the SSA test. The specific case where SSA fails, known as the ambiguous case, occurs if the congruent angles, $\\angle BAC$ and $\\angle QPR$ , are acute. Let us illustrate this.

Suppose we have a right triangle, $\\triangle XYZ$ , with right angle $\\angle XZY$ . Let $P$ and $Q$ be two points on $\\overleftrightarrow{XZ}$ equidistant from $Z$ such that $P$ is between $X$ and $Z$ and $Q$ is not. Since $\\angle XZY$ is right, this makes $\\angle PZY$ right, and $P$ , $Q$ are equidistant from $Z$ , thus $\\overleftrightarrow{YZ}$ bisects $P$ and $Q$ , and as such, every point on that line is equidistant from $P$ and $Q$ . From this, we know $Y$ is equidistant from $P$ and $Q$ , thus $\\overline{YP}\\cong\\overline{YQ}$ . Further, $\\angle YXP$ is in fact the same angle as $\\angle YXQ$ , thus $\\angle YXP\\cong\\angle YXQ$ . Since $\\overline{XY}\\cong\\overline{XY}$ , $\\triangle XYP$ and $\\triangle XYQ$ clearly meet the SSA test, and yet, just as clearly, are not congruent. This results from $\\angle YXZ$ being acute. This example also reveals the exception to the ambiguous case, namely $\\triangle XYZ$ . If $R$ is a point on $\\overleftrightarrow{XZ}$ such that $\\overline{YR}\\cong\\overline{YZ}$ , then $R\\cong Z$ . Proving this exception amounts to determining that $\\angle XZY$ is right, in which case the congruency could be proven instead with SAA.

However, if the congruent angles are not acute, i.e., they are either right or obtuse, then SSA is definitive.

单词	SSA
释义	SSA SSA is a method for determining whether two triangles are congruent by comparing two sides and a non-inclusive angle. However, unlike SAS, SSS, ASA, and SAA, this does not prove congruence in all cases. Suppose we have two triangles, $\\triangle ABC$ and $\\triangle PQR$ . $\\triangle ABC\\cong?\\triangle PQR$ if $\\overline{AB}\\cong\\overline{PQ}$ , $\\overline{BC}\\cong\\overline{QR}$ , and either $\\angle BAC\\cong\\angle QPR$ or $\\angle BCA\\cong\\angle QRP$ . Since this method does not prove congruence, it is more useful for disproving it. If the SSA method is attempted between $\\triangle ABC$ and $\\triangle PQR$ and fails for every $A B C$ , $B C A$ , and $C B A$ against every $P Q R$ , $Q R P$ , and $R P Q$ , then $\\triangle ABC\ot\\cong\\triangle PQR$ . Suppose $\\triangle ABC$ and $\\triangle PQR$ the SSA test. The specific case where SSA fails, known as the ambiguous case, occurs if the congruent angles, $\\angle BAC$ and $\\angle QPR$ , are acute. Let us illustrate this. Suppose we have a right triangle, $\\triangle XYZ$ , with right angle $\\angle XZY$ . Let $P$ and $Q$ be two points on $\\overleftrightarrow{XZ}$ equidistant from $Z$ such that $P$ is between $X$ and $Z$ and $Q$ is not. Since $\\angle XZY$ is right, this makes $\\angle PZY$ right, and $P$ , $Q$ are equidistant from $Z$ , thus $\\overleftrightarrow{YZ}$ bisects $P$ and $Q$ , and as such, every point on that line is equidistant from $P$ and $Q$ . From this, we know $Y$ is equidistant from $P$ and $Q$ , thus $\\overline{YP}\\cong\\overline{YQ}$ . Further, $\\angle YXP$ is in fact the same angle as $\\angle YXQ$ , thus $\\angle YXP\\cong\\angle YXQ$ . Since $\\overline{XY}\\cong\\overline{XY}$ , $\\triangle XYP$ and $\\triangle XYQ$ clearly meet the SSA test, and yet, just as clearly, are not congruent. This results from $\\angle YXZ$ being acute. This example also reveals the exception to the ambiguous case, namely $\\triangle XYZ$ . If $R$ is a point on $\\overleftrightarrow{XZ}$ such that $\\overline{YR}\\cong\\overline{YZ}$ , then $R\\cong Z$ . Proving this exception amounts to determining that $\\angle XZY$ is right, in which case the congruency could be proven instead with SAA. However, if the congruent angles are not acute, i.e., they are either right or obtuse, then SSA is definitive.
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