Math 9G Geometry Flashcards

A comprehensive vocabulary review covering key definitions, postulates, corollaries, and theorems from the Math 9g High School Geometry module.

Trichotomy Property

A property of real numbers stating that for every $x$ and $y$, ONLY ONE of the following conditions holds: $x = y$, $x < y$, or $x > y$.

Theorem 7 – 1

If $a = b + c$ and $c > 0$, then $a > b$.

Exterior Angle of a Triangle

An angle formed by one side of a triangle and the extension of an adjacent side; specifically, if $C$ is between $A$ and $D$, then $\angle BCD$ is an exterior angle of $\Delta ABC$.

Remote Interior Angles

The interior angles of a triangle that are not adjacent to a given exterior angle; for exterior angle $\angle BCD$ of $\Delta ABC$, the remote interior angles are $\angle A$ and $\angle B$.

Theorem 7 – 2 (The Exterior Angle Theorem)

An exterior angle of a triangle is greater than each of its remote interior angles.

Corollary 7 – 2.1

If a triangle has one right angle, then its other angles are acute.

SAA Correspondence

A correspondence between two triangles where a pair of corresponding sides are congruent, and two pairs of corresponding angles are congruent.

Theorem 7 – 3 (The SAA Theorem)

Every SAA correspondence is a congruence.

Theorem 7 – 4 (The Hypotenuse-Leg Theorem)

Given a correspondence between two triangles, if the hypotenuse and one leg of one right triangle are congruent to the corresponding parts of the second right triangle, then the correspondence is a congruence.

Theorem 7 – 5

If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side.

Theorem 7 – 6

If angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle.

Theorem 7 – 7 (The First Minimum Theorem)

The shortest segment joining a point to a line is the perpendicular segment.

Theorem 7 – 8 (The Triangle Inequality)

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Theorem 7 – 9 (The Hinge Theorem)

If two sides of one triangle are congruent, respectively, to two sides of a second triangle, and the included angle of the first triangle is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second.

Theorem 7 – 10 (The Converse Hinge Theorem)

If two sides of one triangle are congruent, respectively, to two sides of a second triangle, and the third side of the first triangle is longer than the third side of the second, then the included angle of the first triangle is larger than the included angle of the second.

Skew Lines

Non-coplanar lines that do not intersect and are not parallel.

Theorem 9 – 1

Two parallel lines lie in exactly one plane.

Theorem 9 – 2

In a plane, two lines are parallel if they are both perpendicular to the same line.

Theorem 9 – 3 (Existence of Parallels)

Let $L$ be a line and $P$ be a point NOT on $L$. Then there is at least one line through $P$, parallel to $L$.

Transversal

A line that intersects two coplanar lines in two distinct points.

Alternate Interior Angles

Non-adjacent interior angles that lie on opposite sides of a transversal line.

Theorem 9 – 5 (The AIP Theorem)

Given two lines cut by a transversal, if a pair of alternate interior angles are congruent, then the lines are parallel.

Theorem 9 – 7 (The CAP Theorem)

Given two lines cut by a transversal, if a pair of corresponding angles are congruent, then the lines are parallel.

Theorem 9 – 8

Given two lines cut by a transversal, if a pair of interior angles on the same side of the transversal are supplementary, the lines are parallel.

Postulate 18 (The Parallel Postulate)

Through a given external point, there is only one line parallel to a given line.

Theorem 9 – 9 (The PAI Theorem)

If two parallel lines are cut by a transversal, then alternate interior angles are congruent.

Corollary 9 – 9.1 (The PCA Corollary)

If two parallel lines are cut by a transversal, each pair of corresponding angles are congruent.

Corollary 9 – 9.2

If two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary.

Theorem 9 – 13

For every triangle, the sum of the measures of the angles is $180$.

Corollary 9 – 13.2

The acute angles of a right triangle are complementary.

Corollary 9 – 13.3

For any triangle, the measure of an exterior angle is the sum of the measures of the remote interior angles.

Quadrilateral

The union of four segments $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$ formed by four coplanar points $A$, $B$, $C$, and $D$ (no three collinear) intersecting only at their endpoints.

Convex Quadrilateral

A quadrilateral in which no two of its vertices lie on opposite sides of a line containing any side of the quadrilateral.

Parallelogram

A quadrilateral in which both pairs of opposite sides are parallel.

Trapezoid

A quadrilateral in which one and only one pair of opposite sides are parallel.

Isosceles Trapezoid

A trapezoid whose non-parallel sides (legs) are congruent.

Theorem 9 – 14

Each diagonal separates a parallelogram into two congruent triangles.

Theorem 9 – 15

In a parallelogram, any two opposite sides are congruent.

Theorem 9 – 16

In a parallelogram, any two opposite angles are congruent.

Theorem 9 – 17

In a parallelogram, any two consecutive angles are supplementary.

Theorem 9 – 18

The diagonals of a parallelogram bisect each other.

Theorem 9 – 22 (The Midline Theorem)

The segment between the midpoints of two sides of a triangle is parallel to the third side and half as long.

Rhombus

A parallelogram all of whose sides are congruent.

Rectangle

A parallelogram all of whose angles are congruent.

Square

A rectangle all of whose sides are congruent.

Theorem 9 – 26

The median to the hypotenuse of a right triangle is half as long as the hypotenuse.

Theorem 9 – 27 (The 30–60–90 Triangle Theorem)

If an acute angle of a right triangle has measure $30$, then the opposite side is half as long as the hypotenuse.

Triangular Region

The union of a triangle and its interior.

Polygonal Region

The union of a finite number of triangular regions in a plane such that if two intersect, their intersection is either a point or a segment.

Postulate 19 (The Area Postulate)

To every polygonal region there corresponds a unique positive real number.

Postulate 21 (The Area Addition Postulate)

If two polygonal regions intersect only in edges and vertices (or do not intersect at all), then the area of their union is the sum of their areas.

Theorem 11 – 1 (Area of a Rectangle)

The area of a rectangle is the product of its base and its altitude.

Theorem 11 – 2 (Area of a Right Triangle)

The area of a right triangle is half the product of its legs.

Theorem 11 – 3 (Area of a Triangle)

The area of a triangle is half the product of any base and the corresponding altitude.

Theorem 11 – 4 (Area of a Trapezoid)

The area of a trapezoid is half the product of its altitude and the sum of its bases.

Theorem 11 – 5 (Area of a Parallelogram)

The area of a parallelogram is the product of any base and the corresponding altitude.

Theorem 11 – 8 (The Pythagorean Theorem)

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

Pythagorean Triple

A set of positive numbers $a$, $b$, and $c$ that satisfies $c^2 = a^2 + b^2$.

Theorem 11 – 10 (The Isosceles Right Triangle Theorem)

In an isosceles right triangle, the hypotenuse is $\sqrt{2}$ times as long as each of the legs.

Theorem 11 – 12

In a 30-60-90 triangle, the longer leg (the leg opposite the acute angle with measure $60$) is $\frac{1}{2}\sqrt{3}$ times as long as the hypotenuse.

Geometric Mean

For positive numbers $a$, $b$, and $c$, if $\frac{a}{b} = \frac{b}{c}$, then $b = \sqrt{a \cdot c}$ is called the geometric mean between $a$ and $c$.

Arithmetic Mean

For any two positive numbers $a$ and $c$, the average value given by $\frac{1}{2}(a + c)$.

Theorem 12 – 2 (The Basic Proportionality Theorem)

If a line parallel to one side of a triangle intersects the two other sides in distinct points, then it cuts off segments which are proportional to these sides.

Theorem 12 – 4 (The AAA Similarity Theorem)

Given a correspondence between two triangles, if corresponding angles are congruent, then the correspondence is a similarity.

Corollary 12 – 4.1 (The AA Corollary)

Given a correspondence between two triangles, if two pairs of corresponding angles are congruent, then the correspondence is a similarity.

Theorem 12 – 6 (The SAS Similarity Theorem)

Given a correspondence between two triangles, if two pairs of corresponding sides are proportional and the included angles are congruent, then the correspondence is a similarity.

Theorem 12 – 7 (The SSS Similarity Theorem)

Given a correspondence between two triangles, if corresponding sides are proportional, then the correspondence is a similarity.

Theorem 12 – 8

In any right triangle, the altitude to the hypotenuse separates the triangle into two triangles which are similar to each other and to the original triangle.

Theorem 12 – 10

If two triangles are similar, then the ratio of their areas is the square of the ratio of any two corresponding sides.