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The Encyclopedia of Geometry (0033)
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 $$BC=B’C’, \quad AD=A’D’ \quad and \quad ∠BAC=∠B’A’C’.$$…
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The Encyclopedia of Geometry (0032)
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 $⊿DEF$…
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The Encyclopedia of Geometry (0031)
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 $A$,…
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The Encyclopedia of Geometry (0030)
Problem Two triangles, in which the three corresponding sides are equal respectively, are congruent. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution In $△ABC$ and $△DEF$, $$AB=DE, \quad BC=EF \quad and \quad AC=DF.$$…
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The Encyclopedia of Geometry (0029)
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…
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The Encyclopedia of Geometry (0028)
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…
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The Encyclopedia of Geometry (0027)
Problem When two line segments $AB$ and $CD$ are parallel to each other in the same direction, if we take a point $P$ that is not between $AB$ and $CD$, then $$|∠ABP-∠CDP|=∠BPD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0026)
Problem Suppose that two line segments $AB$ and $CD$ are parallel to each other in the same direction. Then, if the point $P$ is between $AB$ and $CD$, $$∠ABP+∠CDP=∠BPD.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…
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The Encyclopedia of Geometry (0025)
Problem If the two sides $AB$ and $AC$ of $∠BAC$ are, respectively, parallel to the two sides $DE$ and $DF$ of $∠EDF$, then the bisector $AM$ of the former angle and the bisector $DN$ of the latter are parallel or perpendicular to each other. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$…
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The Encyclopedia of Geometry (0024)
Problem When a straight line $EF$ cuts across parallel lines $AB$ and $CD$, the bisectors of the alternate angles and the corresponding angles are parallel. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution…