Math is Cool for Geometry Kids (Keep in mind that you have to write out the theorems. Word for word.)

What is the area equation for a sphere?

A = 4(pi)(r)^2

What is a function?

Correspondence between sets of numbers.

What are the 2 special right triangles we have learned?

The 45-45-90 triangle and the 30-60-90 triangle.

What is the theorem about two lines that are parallel to a third line?

Two lines parallel to a third line are parallel to each other.

What are parallel planes?

Two planes that do not intersect. If two parallel planes are cut by a third plane then the lines of intersection are parallel

What is the distributive property?

a (b + c) = ab + ac

What is the equation to find the area of a circle?

(pi)(r)^2

What is the equation of a circle with center (h,k) and radius r?

(x - h)^2 + (y - k)^2 = r^2 With circle equations, if the h and k are negative then when you graph, it is positive.

What is a secant?

A line that contains a chord.

What is the intersection of 2 lines?

A point.

What is a regular polygon?

A polygon that is both equiangular and equilateral.

What is an acute angle? What is an obtuse angle? Right angle? Straight angle?

Acute Angle: 0-89 Obtuse Angle: 91-180 Right Angle: 90 Straight angle: 180

What is a segment?

All points between any 2 endpoints.

What is the Right Angle Congruence theorem?

All right angles are congruent.

What is an inscribed angle?

An angle whose vertex is on a circle and whose sides contain chords of the circle. The measure of an inscribed angle is equal to half the measure of its intercepted arc? If two inscribed angles intercept the same arc, then the angles are congruent. An angle inscribed in a semicircle is a right angle. If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

What are some corollaries based off of the isosceles triangle information?

An equilateral triangle is equiangular. An equilateral triangle has three 60 degree angles. The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint.

What is an exterior angle?

An exterior angle is formed when one side of a triangle is extended. Since the exterior angle of a triangle is supplementary to the adjacent interior angle of the triangle, the other two angles of the triangle are called remote interior angles. Theorem: The measure of the exterior angle of a triangle equals the sum of the measures of the two remote interior angles.

What is a negation?

An opposite of the original conditional. Conditional Statement: The ball is blue. Negation: The ball is not blue.

What is congruence mapping?

Another name for an isometry.

What is the protractor postulate?

Given any line and a point on that line, all rays that can be drawn from O can be put into a one-to-one correspondence with real numbers from 0 to 180. The measure of an angle is the absolute value of the difference of the real numbers that the rays correspond with on the protractor. (Basically the ability to measure an angle with a protractor)

What is the definition of congruence?

If 2 angles are equal in measure, then they are congruent. If 2 angles are congruent, then their measures are equal.

What is the segment addition postulate?

If B is between A and C, then AB + BC = AC. Conversely, if AB + BC = AC, then B is between A and C.

What is the triangle proportionality theorem?

If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

What is the triangle angle-bisector theorem?

If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides.

What is sine, cosine, and tangent?

Methods to find the measurement of a leg or hypotenuse.

What does the word "congruent" mean?

Same size and same shape.

List all the forms linear equations can be written in.

Standard Form: Ax + By = C Slope-Intercept Form: y = mx + b Point-Slope Form: y - y1 = m(x - x1)

What are some properties of quadrilaterals?

The diagonals of a rectangle are congruent. If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle. If two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.

What is an angle?

The figure formed by two rays having the same endpoint. The two rays are called the sides of the angle and their common endpoint is the vertex of the angle.

What is the locus of points in a plane at are 1 cm from line m?

The locus of points in a plane 1 cm from line m is a pair of lines parallel to and 1 cm from line m.

What are major and minor arcs?

The major arc is the arc that has a larger central angle, and the minor arc has a lesser central angle. Three letters are used to name a semicircle or a major arc.

What is the measure of an angle formed by a chord and a tangent?

The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc.

What is the exterior angle inequality theorem?

The measure of an exterior angle of a triangle is greater than the measure of either remote interior angles.

What is the requirement for lines to be perpendicular on a graph?

The slope of one line has to be the negative recipricol of the other line.

What is the triangle inequality theorem?

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

What is the theorem concerning the measures of the angles in a triangle?

The sum of the measures of the angles of a triangle is 180 degrees.

What are some theorems involving parallel lines?

Theorem: If two lines are parallel, then all points on one line are equidistant from the other line. Theorem: If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. Theorem: A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of a third side.

What are adjacent angles?

Two angles in a place that have a common vertex and a common side, but no interior points that are common.

What are parallel lines?

Two lines that do not intersect and are coplanar.

What are skew lines?

Two lines that do not intersect and are not coplanar.

What are the properties of proportions?

a/b = c/d is equivalent to: ad = bc a/c = b/d b/a = d/c a + b / b = c + d / d a/b = c/d = e/f = ....., then a + c + e +..... / b + d + f +....... = a/b = ......

What is the equation to find the circumference of a circle?

(2)(pi)(r)

In space given plane Z and point B outside Z what is the locus of points in Z and are 3 cm from B?

0 points, 1 point, a circle.

What is the slope of a horizontal line?

0.

What is the Ruler Postulate?

1. All points on a number line can be paired with real numbers. 2. The distance between any two points is the absolute value of the difference of their coordinates.

How do you write an indirect proof?

1. Identify the statement you want to prove. 2. Assume temporarily that this statement is false by assuming that its opposite is true. 3. Reason logically until you reach a contradiction of a known fact (given). 4. Point out that the temporary assumption is false, and that the desired conclusion must be true. Indirect proofs are in paragraph form with statements and reasons still being part of the proof.

What is the equation to find the sum of the measures of the angles in a polygon with "n" sides?

180(n-2)

What is the sum of the measures of the exterior angles of a convex polygon with one angle at each vertex?

360 degrees.

What is the volume equation for a sphere?

4/3 * (pi)(r)^3

What is a prism?

A 3-D figure with two connected bases that are polygons.

What is an equation to find the area of a regular polygon?

A = 1/2 *ap where A is the area, a is apothem, and p is perimeter.

What is a central angle?

A central angle of a circle has the vertex at the center of the circle.

A diameter that is ________________ to a chord bisects the chord and its arc.

A diameter that is perpendicular to a chord bisects the chord and its arc.

What is Heron's formula?

A formula used for finding the area of a triangle when the lengths of the sides are known.

A line and a plane are _____________ if and only if they intersect and the line is _____________ to all of the lines that will pass through the point of intersection.

A line and a plane are perpendicular if and only if they intersect and the line is perpendicular to all of the lines that will pass through the point of intersection.

What are altitudes (triangles)?

A line from a vertex that meets the side opposite of it perpendicularly. The point of intersection for the altitudes is called the orthocenter. The right angles at the foot of an altitude are called orthogonal pairs. In the diagram, points F, D, and E are called the "feet" of the altitude.

What are medians (triangles)?

A line from the vertex of a triangle to the midpoint of the side it is opposite from. The point of intersection for all these medians is the centroid. The medians of a triangle also intersect in a point that is 2/3rds the distance from each vertex to the midpoint of the opposite side.

What is a line?

A line has one dimension and is represented by a line with two arrows but it extends without an end.

What is a transversal?

A line that intersects two or more coplanar lines in different points.

What is a common tangent?

A line that is tangent to each of two coplanar circles.

What is an auxiliary line?

A line, ray, or segment added to a diagram in a proof. It is denoted as a dashed line in the diagram.

What is the perpendicular bisector of a segment?

A line, segment, ray, or plane that intersects the segment at its midpoint and also forms four 90 degree angles.

What is the bisector of a segment?

A line, segment, ray, or plane that intersects the segment at its midpoint.

What is the intersection of 2 different planes?

A line.

What is a locus?

A locus is a figure that is the set of points that satisfy one or more conditions.

What is a one-to-one mapping?

A mapping from set A to set B where every member of set B has exactly one preimage in set A

What is a transformation?

A one to one mapping from a whole plane to another whole plane.

What is a plane?

A plane has 2 dimensions and is suggested by a floor or a wall. It has no length, width, or depth and is usually pictured by drawing a four-sided figure. It is often labeled with a capital letter.

What is a point?

A point has no dimension and is represented by a dot that may be named.

What is a reflection?

A point reflected over a line. If a point is reflected over the x-axis, the y-cordinate is multiplied by -1. If a point is reflected over the y-axis, the x-coordinate is multiplied by -1.

What is a polygon?

A polygon is formed by coplanar segments such that each segment intersects exactly two other segments, one at each endpoint, and no two segments with a common endpoint are collinear.

What is a convex polygon?

A polygon where no line containing a side of the polygon contains a point in the interior of the polygon.

What is a parallelogram?

A quadrilateral with both pairs of opposite sides parallel. Every parallelogram is a quadrilateral, but every quadrilateral is not a parallelogram.

What is a ray?

A ray consists of an endpoint and a line that extends beyond the arrow.

What is an angle bisector?

A ray that divides an angle into two congruent parts.

What is a rhombus?

A rhombus is a quadrilateral with four congruent sides. The diagonals of a rhombus are perpendicular. Each diagonal of a rhombus bisects a pair of opposite angles.

What is a diagonal in a polygon?

A segment joining two nonconsecutive vertices.

What is an counterexample?

A specific case that proves the conjecture false.

What is a corollary statement?

A statement that can be proved easily by applying a previously studied theorem. Corollaries can be used as reasons in a proof.

What is a bi-conditional?

A statement that contains the words "if and only if".

What is a tangent?

A tangent of a circle lies in the plane of the circle and intersects the circle in exactly one point called the point of tangency. If a line is tangent to a circle, then the line is perpendicular to the radius at the point of tangency.

What is a translation?

A transformation that glides all points of a plane in the same direction the same distance. A translation is an isometry.

What is an isometry?

A transformation that maps every segment to a congruent segment which means it preserves distance, angle measure, and area. The distance between 2 points in the preimage will be the same distance as the image of those 2 points.

What is a glide reflection?

A transformation where 1. Point P is mapped to P' then 2. A reflection of P' occurs to P''. A glide reflection is an isometry.

What is a trapezoid?

A trapezoid is a quad with exactly one pair of parallel sides. The parallel sides are bases, and the nonparallel sides are legs. A trapezoid has two pairs of base angles. A trapezoid with congruent legs is called an isosceles trapezoid.

What are the classifications for triangles?

A triangle can be classified by the number of congruent sides it has and by the measure of the angles. Scalene Triangle - No congruent sides Isosceles Triangle - At least two sides congruent Equilateral Triangle - All sides congruent Acute Triangle - Three acute angles Obtuse Triangle - One obtuse angle Right triangle - One right angle Equiangular Triangle - All angles congruent

What are the five theorems that prove triangle congruence?

AAS Theorem: If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. FOR RIGHT TRIANGLES ONLY (When using these theorems, you MUSt first state that you have right triangles): Hypotenuse-Leg Theorem (HL): If the hypotenuse and a leg of a right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent. Leg-Leg (LL) Theorem: If two legs of one right triangle are congruent to the two legs of another right triangle, then the triangles are congruent. Hypotenuse-Acute Angle (HA) Theorem: If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the triangles are congruent. Leg-Acute Angle (LA) Theorem: If a leg and an acute angle of one right triangle are congruent to the corresponding parts in another right triangle, then the triangles are congruent.

What are the properties of equality and what do they do?

Addition: If a = b, then a + c = b + c Subtraction: If a = b, then a - c = b - c Multiplication: If a = b, then ac = bc Division: If a = b and c ≠ 0, then a/c = b/c Reflexive: a = a Symmetric: If a = b, then b = a. Transitive: If a = b and b = c, then a = c. Substitution: If a = b, then a can be substituted for b in any equation or inequality. Substitution Property of Congruence does not exist.

What is a conditional statement?

An if-then statement, also known as the given information or hypothesis. If can also be written as: If p then q P implies q P only if q Q if p. Where p is the hypothesis and q is the conclusion.

What do the angle bisectors of a triangle form?

An incircle, which is a circle inside the triangle that is tangent to each side of the triangle. The point of intersection of the angle bisectors is also the center of the incircle and is called the incenter. The angle bisectors of a triangle intersect in a point that is equidistant from the three sides of the triangle.

What is a conjecture?

An unproven statement that is based on observations.

What are congruent angles?

Angles that have the same measure.

Between any ____ points there is exactly ___ line(s).

Between any 2 points, there is exactly 1 line.

What are concentric circles?

Circles that lie in the same plane and have the same center.

What are complementary and supplementary angles?

Complementary Angles - Two angles whose measures have the sum of 90 degrees. Supplementary Angles - Two angles whose measures have the sum of 180 degrees.

In the same circle or in congruent circles:

Congruent arcs have congruent chords. Congruent chords have congruent arcs too. Chords equally distant from the center or centers are congruent. Congruent chords are equally distant from the center or centers.

State the converse, inverse, and contrapositive of the following conditional: If a triangle is scalene, then it has no congruent sides.

Converse: If a triangle has no congruent sides, then it is scalene. Inverse: If a triangle is not scalene, then it has congruent sides. Contrapositive: If a triangle has congruent sides, then it is not scalene.

What is mapping?

Correspondence between sets of points.

What does it mean if two polygons are similar?

Corresponding angles are congruent and corresponding sides are in proportion. Corresponding vertices must be listed in the same order.

What is CPCTC?

Corresponding parts of congruent triangles.

What is inductive reasoning?

Finding a pattern for specific cases and then rewriting the conjecture for a general case.

What are the properties of inequality?

If a > b and d ≤ c, then a + c > b + d. If a > b and c > 0, then ac > bc and a/c > b/c. If a > b and c < 0, then ac < bc and a/c < b/c. If a > b and b > c, then a > c. If a = b + c and c > 0, then a > b..

List the equidistant theorems.

If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. If a point is equidistant from the sides of an angle, then the point lies on the angle bisector. If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the segment endpoints. If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment.

What is a theorem concerning isosceles trapezoids and base angles?

If a trapezoid is isosceles, then each pair of base angles is congruent.

What is geometric mean?

If a, b, and x are positive numbers and a/x = x/b then x is called the geometric mean between a and b.

What is the means-extremes property?

If a/b = c/d, then bc = ad. This is a property of proportions.

What is an inverse?

If not p, then not q. Where p is the hypothesis and q is the conclusion.

What is a contrapositive?

If not q, then not p. Where p is the hypothesis and q is the conclusion.

If one side of a triangle is __________ than a second side, then the angle ____________ the first side is larger than the angle ___________ the second side.

If one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side. The converse is also true.

What is the angle bisector theorem? How does this differ from the definition of angle bisector?

If ray BD is the bisector of ∠ABC, then m∠ABD = 1/2 m∠ABC and m∠ABC = 2m∠ABD Definition of Angle Bisector: An angle bisector splits the angle into 2 congruent angles.

If the _______ is drawn to the __________ of a right triangle, then the two triangles formed are ___________ to the original and to each other.

If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original and to each other.

Theorem: If two angles are _______________ to the same angle (or to congruent angles), then they are ______________.

If two angles are supplementary (Or Complementary) to the same angle (or to congruent angles), then they are congruent.

What is the converse of the isosceles triangle theorem?

If two angles of a triangle are congruent, then the sides opposite those angles are congruent. This, however, is not universal, so you have to write it out.

What are some corollary statements about triangles?

If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. Each angle of an equiangular triangle has a measure of 60 degrees. In a triangle, there can be at most one right angle or obtuse angle. The acute angles of a right triangle are complementary.

What is the AA Similarity Postulate?

If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

What is the area congruence postulate?

If two figures are congruent, then their areas are equal.

What is the SSS inequality theorem?

If two sides of a triangle are congruent to two sides of a second triangle, but 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.

What is the isosceles triangle theorem?

If two sides of a triangle are congruent, then the angles opposite those sides are congruent. This is a universal name, so you can just write "isosceles triangle theorem"

What is the SAS inequality theorem?

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

What are the ratios of areas concerning two triangles?

If two triangles have equal heights, then the ratio of their areas equals the ratio of their bases. If two triangles have equal bases, then the ratio of their areas equals the ratio of their heights. If two triangles are similar, then the ratio of their areas equals the square of their scale factor.

What is a scale factor?

In two similar polygons, the scale factor is the ratio of the lengths of two corresponding sides.

What is special about the 45-45-90 triangle?

It has a leg:leg:hypotenuse racial of 1:1:rt2

What is special about the 30-60-90 triangle?

It has a leg:leg:hypotenuse racial of 1:rt3:2. The "1" racial is always the leg opposite the 30-degree angle.

What is the LA, TA, and V equation for a cone?

LA = (pi)(r)(l) TA = LA + B V = 1/3 * Bh r --> radius l --> slant height B --> base area

What is the LA, TA, and V equation for a pyramid?

LA = 1/2 *pl TA = LA + B V = 1/3 * Bh l --> slant height p --> perimeter B --> base area h --> height

What is the LA, TA, and V equation for a prism?

LA = ph TA = LA + 2B V = Bh p --> perimeter h --> height B --> base area

What is the LA, TA, and V equation for a cylinder?

LA = ph TA = LA +2B V = Bh p --> perimeter h --> height B --> base area

What is a lateral face?

Lateral faces of a prism are parallelograms.

Say two lengths of a triangle are 5 and 7. What would the length of the third side be?

Let x = the third side of the triangle. x + 5 > 7 x > 2 5 + 7 > x 12 > x Tue length of the third side must be greater than 2 but less than 12.

What is the midpoint theorem? How does the midpoint theorem differ from the definition of midpoint?

Midpoint Theorem: If M is the midpoint of line AB then AM = 1/2 AB and AB = 2AM. Definition of Midpoint: A midpoint divides a segment into 2 congruent parts.

What are the names of all the shapes?

Number of Sides: 3 Name: Triangle Number of Sides: 4 Name: Quadrilateral Number of Sides: 5 Name: Pentagon Number of Sides: 6 Name: Hexagon Number of Sides: 8 Name: Octagon Number of Sides: 10 Name: Decagon Number of Sides: 12 Name: Dodecagon Number of Sides: n Name: n-gon Example of a n-gon Number of Sides: 9 Name: 9-gon

Opposite rays are ________.

Opposite rays are collinear.

What is preimage and image?

P' is the image of P and P is the preimage of P'.

What are collinear points?

Points that lie in the same line.

What are coplanar points?

Points that lie on the same plane.

How do you name a polygon?

Polygons are named using consecutive vertices.

What are the properties of congruence?

Reflexive: line DE ≅ line DE ∠A ≅ ∠A Symmetry: If line AB ≅ line CD, then line CD ≅ line AB. Transitive: If line AB ≅ line CD and line CD ≅ line EF, then line AB ≅ line EF If ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C.

What are the similar triangle theorems?

SAS Similarity Theorem: If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar. SSS Similarity Theorem: If the sides of two triangles are in proportion then the triangles are similar.

What are the three postulates that prove triangle congruence?

SSS Postulate: If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. SAS Postulate: If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. ASA Postulate: If two angles and the included side of one triangle are congruent to two angles and the included sides of a second triangle, then the two triangles are congruent.

Segments and rays are collinear if they lie in the _________ _________.

Segments and rays are collinear if they lie in the same plane.

What are congruent segments?

Segments that have equal lengths.

What are the equations for sine, cosine, and tangent?

Sine = opposite leg/hypotenuse Cosine = adjacent leg/hypotenuse Tangent = opposite leg/adjacent leg

What is the slope equation?

Slope is represented by the letter m.

What is the requirement for lines to be parallel on a graph?

Slopes have to be equal.

What is a converse?

Switching the hypothesis and conclusion of a conditional statement around. (Basically the opposite) "If she is excited, he is happy" Turns into "If she is happy, she is excited." "If q, then p." Where p is the hypothesis and q is the conclusion.

Tangents to a circle from an exterior point are __________.

Tangents to a circle from an exterior point are congruent.

Write an indirect proof using the following. Given: Parallelogram ABCD; m∠A = 80 degrees Prove: Parallelogram ABCD is not a rectangle.

Temporarily assume that parallelogram ABCD is a rectangle. Because all angles in a rectangle are right, angle A is right. Since right angles are 90 degrees, the measure of angle A would be 90 degrees. This contradicts the given that the measure of angle A equals 80 degrees, so the assumption is false and parallelogram ABCD is not a rectangle.

What does it mean for a point to be equidistant to something?

That point is at an equal distance between two or more things.

What is the area addition postulate?

The area of a region is the sum of all areas of its non-overlapping parts.

What properties do pyramids have?

The base is a regular polygon. Lateral edges are congruent. Lateral faces are isosceles triangles. The height of a lateral face is called the slant height. The alittude meets the base at its center.

What is the measure of an angle formed by two chords that intersect inside a circle?

The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs.

What is the measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside a circle?

The measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs.

What is volume?

The measure of the amount of space inside a figure.

What is the arc addition postulate?

The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs.

What is the median of a trapezoid? How do you find the length of the medium?

The median of a trapezoid is the segment that joins the midpoints of the legs. The median of a trapezoid is parallel to the bases and has a length equal to half the sum of the bases.

The ________ of the hypotenuse of a right triangle is ___________ from the three vertices.

The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.

What is special about the relationship between isosceles triangles and altitudes, medians, angle bisectors, and perpendicular bisectors?

The perpendicular bisector, medians, angle bisector, and altitude are the same line from the vertex angle to the base of the isosceles triangle.

What is an apothem?

The perpendicular distance from the center of a polygon to its side.

What are opposite rays?

The picture says it all. 2 rays face the opposite direction and have the same endpoint.

What is the midpoint of a segment?

The point that divides the segment into two congruent segments.

What is a y-intercept?

The point where the line hits the y-axis. This is represented by the letter b.

What is a corollary about parallel lines and triangles?

The segment that joins the midpoints of two sides of a triangle is parallel to the third side and 1/2 the length of the third side.

What is an intersection?

The set of all points that figures have in common.

What is space?

The set of all points.

What is lateral area?

The sum of the areas of the lateral faces.

What is total area?

The total area is the sum of the areas of all the faces.

What are the theorems relating to a transversal perpendicular to parallel lines?

Theorem: If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. Theorem: In a plane, two lines perpendicular to the same line are parallel.

What are some theorems that proves a quadrilateral a parallelogram?

Theorem: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem: If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. Theorem: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

What are some theorems concerning parallelograms?

Theorem: Opposite sides of a parallelogram are congruent. Theorem: Opposite angles of a parallelogram are congruent. Theorem: Diagonals of a parallelogram bisect each other.

What are the two theorems concerning a point outside a line?

Theorem: Through a point outside a line, there is exactly one line parallel to the given line. Theorem: Through a point outside a line, there is exactly one line perpendicular to the given line.

What if a conditional and a converse statement are both true?

They can be written as one statement using "if and only if".

What do the perpendicular bisectors of a triangle form?

They form a circle that intersects each vertex of the triangle, with the point of intersection of the perpendicular bisectors being the circumcenter. The circumcenter of an acute triangle is inside the triangle, the circumcenter of an obtuse triangle is outside the triangle, and the circumcenter of a right triangle is on the midpoint of the hypotenuse. The perpendicular bisectors of the sides of a triangle intersect in a point that is equidistant from the three vertices of a triangle.

Through any __ points not on the _______ line, there is exactly 1 _____.

Through any 3 points not on the same line, there is exactly 1 plane.

What are corresponding angles?

Two angles in corresponding positions relative to the two lines. Theorem: If two lines are parallel and a transversal cuts it, corresponding angles are congruent. Theorem: If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.

What are vertical angles?

Two angles such that the sides of one angle are opposite rays to the sides of the other angle. When the two lines intersect, two pairs of vertical angles are formed. Vertical angles are congruent.

What are congruent figures?

Two figures that have the same size and the same shape. Two polygons are congruent if and only if their vertices can be matched up so that their corresponding parts are congruent.

What are same side interior angles?

Two interior angles on the same side of a transversal. Theorem: If two lines are parallel and a transversal cuts it, same side interior angles are supplementary. Theorem: If two lines are cut by a transversal and same side interior angles are supplementary, then the lines are parallel.

What are perpendicular lines? What are the theorems concerning perpendicular lines?

Two lines that intersect to form right angles. Perpendicular Lines Theorems: If two lines are perpendicular, then they form congruent adjacent angles. OR If two lines form congruent adjacent angles, then the lines are perpendicular. OR If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary. Keep in mind that you have to write out the actual theorem, you can NOT write Perpendicular Lines Theorem.

What are alternate exterior angles?

Two nonadjacent exterior angles on opposite sides of a transversal. Theorem: If two lines are parallel and a transversal cuts it, alternate exterior angles are congruent. Theorem: If two lines are cut by a transversal and alternate exterior angles are congruent, then the lines are parallel.

What are alternate interior angles?

Two nonadjacent interior angles on opposite sides of a tranversal. Theorem: If two lines are parallel and a transversal cuts it, alternate interior angles are congruent. Theorem: If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.

What is the slope of a vertical line?

Undefined.

What is the product of a secant segment and a tangent segment drawn to a circle from an external point?

When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment is equal to the square of the tangent segment.

What are the ways geometric mean can be used with right triangles?

When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse. When the altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and segment of the hypotenuse that is adjacent to that leg.

When two chords intersect inside a circle, the product of the segments of one chord __________ the product of the segments of the other chord.

When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord.

What are concurrent lines?

When two or more lines intersect in one point, the lines are concurrent.

What is the product of one secant segment and another secant segment drawn to a circle from an external point?

When two secant segments are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment.

If the scale factor of two similar solids is a : b, then what is the ratio of a) corresponding perimeters b) base areas, lateral areas, and total areas c) volumes

a) a : b b) a^2 : b^2 c) a^3 : b^3

G maps each point (x,y) to the point (2x,y-1). a) Express this using function notation. b) Find the images of (3,0) and (1,4).

a) g: (x,y) --> (2x,y-1) b) g: (3,0) --> (6,-1) g: (1,4) --> (2,3)

What is the angle addition postulate?

m∠AOB + m∠BOC =m∠AOC

What are some key facts to remember about rotations?

o In a rotation, a counterclockwise rotation is considered positive o In a rotation, a clockwise rotation is considered negative o A rotation 90 degrees counterclockwise would be written with fancy R: R0, 90 o A rotation is an isometry, meaning it preserves angle measurement, etc. o Half-turn: a rotation about point O through 180 degrees; usually denoted by HO o A half-turn about the origin can be written HO: (x, y) ---> (-x, -y) Credit to Phoebe.