Problem Regarding two congruent triangles $△ABC$ and $△A’B’C’$, when the two pairs of sides $AB$ and $A’B’$ and $AC$ and $A’C’$ are perpendicular to each other, the remaining pair $BC$ and $B’C’$ are also perpendicular to each other. However, assume that $∠A$ and $∠A’$ are not right angles. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$…

Problem There are $△ABC$ and $△A’B’C’$. $$AB=A’B’, \ BC=B’C’, \ and \ ∠A=∠A’.$$ Then, under which of the following conditions can we always have $△ABC≡△A’B’C’$? (1) Either $△ABC$ or $△A’B’C’$ is an obtuse triangle. (2) $∠C$ is a right angle. (3) $BC>AB$. (4) $AC$ and $A’C’$ are the maximum sides of $△ABC$ and $△A’B’C’$, respectively. (5) $AB>AC$…

Problem When the two sides of one of two triangles are equal to the two sides of the other, and the angles to one set of equal sides are also equal, the angles to the other set of equal sides are equal or supplementary angles. In the former case, the two triangles are congruent. $$…

Problem Two triangles which have two equal sides and equal median lines are congruent. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $AB=DE$, $AC=DF$ and $AP=DQ$. If we take points $A’$ and…

Problem Two triangles with equal bases, vertex angles, and heights are congruent. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution For $△ABC$ and $△A’B’C’$, suppose that $$BC=B’C’, \quad AD=A’D’ \quad and \quad ∠BAC=∠B’A’C’.$$…

Problem Two right triangles which have equal hypotenuses and the other equal sides respectively are congruent. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Suppose that $AC=DF$ and $AB=DE$ in $⊿ABC$ and $⊿DEF$…

Problem Two right triangles which have equal hypotenuses and equal acute angles are congruent. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let $∠B=∠E=∠R$, $AC=DF$ and $∠C=∠F$. Place $D$ on top of $A$,…

Problem Two triangles, in which the three corresponding sides are equal respectively, are congruent. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution In $△ABC$ and $△DEF$, $$AB=DE, \quad BC=EF \quad and \quad AC=DF.$$…

Problem Two triangles, in which a pair of corresponding sides and angles at both ends of them are respectively equal, are congruent. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Now, if we…

Problem Two triangles in which a pair of corresponding sides and the included angle are equal are congruent.   $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Now, if we put $A$ on…