Geometry Syllabus

List the standard triangle congruence criteria.

SSS (side-side-side), SAS (side-angle-side), ASA (angle-side-angle), AAS (angle-angle-side), and RHS/HL (right angle-hypotenuse-side) for right triangles.

Why is SSA (or AAA) generally not a valid congruence criterion?

AAA only guarantees similar (same shape) triangles, not congruent ones, since size can differ. SSA is ambiguous because two different triangles can often satisfy it (the ambiguous case).

What does it mean for two triangles to be similar, and what symbol is used?

Similar triangles have the same shape but not necessarily the same size: corresponding angles are equal and corresponding sides are in proportion. Notation: $\triangle ABC \sim \triangle DEF$.

List the criteria for triangle similarity.

AA (two pairs of equal angles), SAS (two sides in proportion with the included angle equal), and SSS (all three pairs of sides in proportion).

If $\triangle ABC \sim \triangle DEF$ with ratio of similarity $k$, how do their areas compare?

Corresponding sides are in ratio $k$, so the ratio of their areas is $k^{2}$: $\frac{[\triangle ABC]}{[\triangle DEF]}=k^{2}$.

State the Pythagorean Theorem.

In a right triangle with legs $a$ and $b$ and hypotenuse $c$ (opposite the right angle): $$a^{2}+b^{2}=c^{2}.$$

A right triangle has legs of length $6$ and $8$. Find the hypotenuse.

$c=\sqrt{6^{2}+8^{2}}=\sqrt{36+64}=\sqrt{100}=10$.

State the converse of the Pythagorean Theorem.

If the sides of a triangle satisfy $a^{2}+b^{2}=c^{2}$ (with $c$ the longest side), then the triangle is a right triangle with the right angle opposite side $c$.

What is a Pythagorean triple? Give two common examples.

A Pythagorean triple is a set of three positive integers $a$, $b$, $c$ satisfying $a^{2}+b^{2}=c^{2}$. Examples: $(3,4,5)$ and $(5,12,13)$.

What are the defining properties of an isosceles triangle (Base Angles Theorem)?

An isosceles triangle has two equal sides (legs) and the angles opposite those sides (base angles) are equal. Conversely, if two angles are equal, the sides opposite them are equal.

State the side and angle properties of an equilateral triangle.

An equilateral triangle has all three sides equal and all three interior angles equal to $60^\circ$. It is also equiangular.

What are the side ratios of a $45^\circ$-$45^\circ$-$90^\circ$ triangle?

The legs are equal and the hypotenuse is $\sqrt{2}$ times a leg: the ratio is $1:1:\sqrt{2}$.

What are the side ratios of a $30^\circ$-$60^\circ$-$90^\circ$ triangle?

Opposite $30^\circ:60^\circ:90^\circ$ the sides are in ratio $1:\sqrt{3}:2$ (short leg : long leg : hypotenuse).

Give the formula for the area of an equilateral triangle with side $s$.

$$A=\frac{\sqrt{3}}{4}s^{2}.$$

What is a quadrilateral, and what is the sum of its interior angles?

A quadrilateral is a four-sided polygon. The sum of its interior angles is always $360^\circ$.

Name the main special types of quadrilaterals.

Parallelogram, rectangle, square, rhombus, trapezoid (trapezium), and kite.

List the key properties of a parallelogram.

Both pairs of opposite sides are parallel and equal; opposite angles are equal; consecutive angles are supplementary; and the diagonals bisect each other.

How does a rectangle differ from a general parallelogram?

A rectangle is a parallelogram with four right angles ($90^\circ$). Additionally, its diagonals are equal in length (and still bisect each other).

How does a rhombus differ from a general parallelogram?

A rhombus is a parallelogram with all four sides equal. Its diagonals bisect each other at right angles ($90^\circ$) and bisect the interior angles.

What properties make a square special among quadrilaterals?

A square is both a rectangle and a rhombus: four equal sides, four right angles, and diagonals that are equal, bisect each other at right angles, and bisect the vertex angles.

What is a trapezoid (trapezium), and what makes it isosceles?

A trapezoid has exactly one pair of parallel sides (the bases). An isosceles trapezoid has its non-parallel legs equal, base angles equal, and equal diagonals.

What is a kite, and what are its diagonal properties?

A kite has two distinct pairs of consecutive (adjacent) equal sides. Its diagonals are perpendicular, and one diagonal bisects the other; one pair of opposite angles is equal.

Give the perimeter and area formulas for a rectangle with length $l$ and width $w$.

Perimeter $P=2(l+w)$ and area $A=l\times w$.

Give the perimeter and area formulas for a square with side $s$.

Perimeter $P=4s$ and area $A=s^{2}$.

Give the area formulas for a parallelogram and a rhombus.

Parallelogram: $A=b\times h$ (base times height). Rhombus: $A=\frac{1}{2}d_1 d_2$, where $d_1$ and $d_2$ are the diagonals.

State the area formula for a trapezoid with parallel sides $a$ and $b$ and height $h$.

$$A=\frac{1}{2}(a+b)\,h.$$

How are polygons classified by number of sides (name 3, 4, 5, 6, 8, 10 sides)?

3 = triangle, 4 = quadrilateral, 5 = pentagon, 6 = hexagon, 8 = octagon, 10 = decagon.

What is the difference between a convex and a concave polygon?

In a convex polygon every interior angle is less than $180^\circ$ and no vertex points inward. A concave polygon has at least one interior angle greater than $180^\circ$ (a reflex angle), pointing inward.

What is the formula for the sum of the interior angles of an $n$-sided polygon?

$$S=(n-2)\times 180^\circ.$$

What is the sum of the exterior angles of any convex polygon, and each exterior angle of a regular $n$-gon?

The exterior angles of any convex polygon sum to $360^\circ$. Each exterior angle of a regular $n$-gon is $\frac{360^\circ}{n}$, so each interior angle is $\frac{(n-2)\times 180^\circ}{n}$.