Geometry

What people get wrong

1. Treating AAA (or just AA) as a congruence criterion — it only proves similarity, not congruence, because it says nothing about size. Only use SSS, SAS, ASA, AAS, or HL to prove triangles congruent.
2. Using SSA as if it were valid — two sides and a non-included angle can produce two different triangles (the 'ambiguous case'). If the given angle is not between the two given sides, look for HL (right triangle only) or find another pair of parts instead.
3. Mixing up rotation directions — writing $(x,y)\to(y,-x)$ when a 90° counterclockwise rotation was asked for. Memorize the two rules separately: CCW 90° is $(x,y)\to(-y,x)$, CW 90° is $(x,y)\to(y,-x)$, and check with a simple point like $(1,0)$ if unsure.
4. Writing proof steps without citing a definition, postulate, or theorem for each one — a correct-looking chain with missing justifications loses credit. State the reason (e.g., 'Alternate Interior Angles Theorem,' 'Reflexive Property') next to every step.

What people get wrong

1. Writing a similarity proportion with mismatched vertices (e.g., $AB/DE$ when the similarity statement is $\triangle ABC \sim \triangle DFE$). Instead, write the similarity statement first, then read the proportion directly from the vertex order.
2. Assuming SSA is a valid similarity or congruence shortcut. It is not, in general — only AA, SAS (with the angle between the two given sides), and SSS are valid; if a problem gives two sides and a non-included angle, look for another angle relationship instead.
3. Mislabeling opposite and adjacent sides in a trig ratio because the triangle is drawn in an unfamiliar orientation. Instead, always mark the reference angle first, then identify which leg touches it (adjacent) and which doesn't (opposite), regardless of how the triangle is rotated on the page.
4. In the converse of the Pythagorean theorem, squaring the wrong side as the 'hypotenuse.' Instead, always square the two shortest sides and compare their sum to the square of the longest side, never the other way around.

What people get wrong

1. Forgetting the factor of 1/3 for cones and pyramids — students often compute $Bh$ instead of $\frac{1}{3}Bh$. Always write the formula down first and check whether the solid comes to a point (cone/pyramid, needs the 1/3) or has two parallel congruent bases (prism/cylinder, does not).
2. Rotating about the wrong axis, which swaps which side becomes the radius and which becomes the height. Before computing, draw the axis of rotation as a line and label which dimension of the 2D shape lies along it and which sticks out perpendicular to it.
3. Assuming a plane cutting a cylinder or cone at any angle always gives the same shape as a perpendicular cut. A plane through a cylinder at an angle to the axis gives an ellipse, not a circle, and a plane through a cone at an angle can give a circle, ellipse, parabola, or hyperbola depending on the angle — for this course, focus on parallel-to-base (circle) versus through-the-axis (triangle) cuts.
4. Treating an oblique prism or cylinder as needing a different volume formula. By Cavalieri's Principle, an oblique solid has the same volume as a right solid with the same base area and the same perpendicular height — the slant does not change the volume.

What people get wrong

1. Proving a claim with one specific numerical figure instead of general coordinates. If the problem says 'prove for any rectangle,' place the vertices as $(0,0)$, $(a,0)$, $(a,b)$, $(0,b)$ with letters, not a single numeric example — a numeric case only proves that one case.
2. Taking the reciprocal of a slope without also flipping its sign when finding a perpendicular slope. Slope $-\tfrac{2}{3}$ becomes $\tfrac{3}{2}$, not $-\tfrac{3}{2}$ or $\tfrac{2}{3}$; always negate AND invert.
3. Concluding two segments are parallel just because their slopes match, without checking they aren't the same line. Two identical lines are not 'parallel sides' of a figure — confirm the points are actually distinct segments.
4. Applying the partition formula in the wrong direction. The formula $x_1+\tfrac{a}{a+b}(x_2-x_1)$ finds the point from $A$ toward $B$ in ratio $a:b$; swapping which endpoint is $A$ gives a different point on the segment.

What people get wrong

1. Using the central angle directly as the inscribed angle (or vice versa) instead of applying the factor of 2 — always ask whether the vertex of the angle is at the center or on the circle before using its measure.
2. Confusing the arc length formula with the sector area formula, since both start with angle/360 — remember arc length pairs with circumference ($2\pi r$, a length) and sector area pairs with area ($\pi r^2$, a squared quantity).
3. Making a sign error when completing the square, writing $(x+h)^2$ instead of $(x-h)^2$ — after completing the square, always rewrite the binomial so the sign matches subtraction of the center coordinate, e.g. $x^2-6x$ becomes $(x-3)^2-9$, giving center x-coordinate +3, not −3.
4. Treating the radius drawn to a point of tangency as anything other than perpendicular to the tangent — draw this radius explicitly and mark the right angle before setting up any Pythagorean or trigonometric relationship.

What people get wrong

1. Treating independent and mutually exclusive as the same idea. Mutually exclusive events (P(A and B)=0) are actually maximally dependent — knowing one occurred tells you the other definitely did not. Always check the correct definition for the situation before applying a rule.
2. Using the grand total as the denominator when the question asks for a conditional probability. If the question says 'given that' or 'among students who...', the denominator must be the row or column total for that condition, not the full sample size.
3. Forgetting to subtract the overlap in the addition rule when the events are not mutually exclusive, which double-counts outcomes in both A and B. Before adding P(A) + P(B), always ask whether an outcome could satisfy both events at once.
4. Reversing the order in conditional probability, computing P(B|A) when the question asks for P(A|B). Write P(A|B) = P(A and B)/P(B) explicitly and plug in the matching quantities rather than dividing whichever two numbers seem convenient.