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SAS, ASA and AAS, a la Shmoop.

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Henry has always been a proficient wizard,

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but lately, it seems all of his potions have been turning out… a little funky.

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Henry is setting out to solve the problem logically.

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First, he double-checks his ingredients. They all seem to be in order.

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Next, his spell book. Looks fine. Finally, his two hourglasses.

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If the two halves of the hourglasses are differently shaped, or incongruent, the timing will be all off.

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Henry may have found the problem.

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Now the fastest way for Henry to determine congruency

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would be just to measure all the sides and angles.

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Unfortunately, Henry lost his measuring tools…

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…and he only has a few measurements originally provided by the hourglass manufacturers to work with…

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…as well as three very useful geometric rules.

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Each hourglass can be divided into two triangles, one standing upside down on the other.

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One rule for determining congruency between triangles is the Side-Angle-Side Rule, or

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SAS, which states…

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…“If two sides and the angle between them of one triangle are congruent to the

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corresponding parts of another triangle, the triangles are congruent.”

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For the smaller hourglass, Henry has the measurements of two of the sides;

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5 inches and 8 inches, and 5 inches and 8 inches.

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The measurements are equal, so the sides are congruent; we can use hash marks to keep track

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of congruent parts.

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As for the angle between them, Henry can use the Vertical Angle Theorem to conclude that

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the angles are also congruent.

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So by the SAS rule, the two triangular halves of this hourglass are congruent.

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Another rule for determining triangular congruency is the Angle-Side-Angle Rule,

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or ASA, which states…

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…“If two angles and the side between them of one triangle are congruent to the

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corresponding parts of another triangle, the triangles are congruent.”

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The triangles of Henry’s second hourglass have congruent corresponding angle measurements…

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…but that doesn’t prove congruency because the two halves might still be different sizes.

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So Henry measures the bases; and there you go they’re congruent!

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Since the bases lie between congruent, corresponding angles, the two halves of this hourglass are

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congruent by the ASA rule.

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Related to the ASA rule is the Angle-Angle-Side Theorem, or AAS, which states…

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…“If two angles and an adjacent side of one triangle are congruent to the corresponding

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parts of another triangle, the triangles are congruent.”

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This theorem is a bit of a shortcut because it helps save a few steps in any proof…

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…namely, if you know two angle measurements of any triangle, you can easily find the third.

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Looks like there isn’t really anything wrong with the shape of Henry’s hourglasses.

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Back to the drawing board.

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