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- StudyBlue
- North Carolina
- Carolina Day School
- Honors Geometry 1
- Codd
- Theorems for 5-13

Abigail W.

What is a parallelogram?

a quadrilateral with both pairs of opposite sides parallel

Opposite sides o a parallelogram are ______________

congruent

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Opposite angles of a parallelogram are ____________

congruent

Diagonals of a parallelogram __________ each other

bisect

If both pairs of opposite sides of a quadrilateral are congruent, the the quadrilateral is a _______

parallelogram

If one pair of opposite sides of a quadrilateral are both congruent and parallel the the quadrilateral is a ___________

parallelogram

If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a ___________

parallelogram

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a ___________.

parallelogram

What are the five ways to prove that a quadrilateral is a parallelogram?

1. Show that both pairs of opposite sides are parallel

2. Show that both pairs of opposite sides are congruent

3. Show that one pair of opposite sides are both congruent and parallel

4. Show that both pairs of opposite angles are congruent

5. Show that the diagonals bisect each other

If two lines are parallel, then all the points on one line are ___________ from the other line.

equidistant

If three parallel lines cut off congruent segments on one transversal, then cut off ___________ segments on every transversal.

congruent

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A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the ____________ of the third side.

midpoint

The segment that joins the midpoints of two sides of a triangle:

1.) is parallel to the third side

2.) is half as long as the third side

What is a rectangle?

a quadrilateral with four right angles, every rectangle is a parallelogram

What is a rhombus?

a quadrilateral with four congruent sides, every rhombus is a parallelogram

What is a square?

a quadrilateral with four right angles and four congruent sides. Every square is a rectangle, a rhombus, and a parallelogram

The diagonals of a rectangle are ___________

congruent

The diagonals of a _________ are perpendicular

rhombus

Each diagonal of a _________ bisects two angles of the __________

rhombus

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

midpoint

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 a trapezoid?

a quadrilateral with exactly one pair of parallel sides

What is an isosceles trapezoid?

a trapezoid with congruent legs

The base angles of an _________ __________ are congruent.

isosceles trapezoid

The median of a trapezoid:

1.) is parallel to the bases

2.) has a length equal to the average of the base length

The measure of an exterior angle of a triangle is greater than the measure of either __________ __________ ______

remote interior angle

Statement:

If p, then q

Inverse of a statement:

If not p, then nt q.

Contrapositive of a Statement

if not q, then not p

Converse of a statement:

If q then, p.

A statements and its _________ are logically equivalent.

contrapositive

A statement is not logically equivalent to its ______ or to its __________.

inverse, converse

How do you write an indirect proof?

1. Assume temporarily that the conclusion is not true

2. Reason logically until you reach a contradiction of a known fact

3. Point out that the temporarily assumption must be false, and that the conclusion must then be true.

If one side of a triangle is longer than a second, then the angle opposite the first side is _________ than the angle opposite the second side.

larger

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

longer

The perpendicular segment from a point to a line is the ____________ segment from the point to the line.

shortest

The perpendicular segment from a point to a plane is the ___________ segment from the point to the plane.

shortest

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

greater

If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is greater than that of the second triangle, then the third side of the first triangle is _________ than the third side of the second triangle.

longer

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

larger

What are the properties of proportion when a/b=c/d?

*ad=bc

*a/c=b/d

*b/a=d/c

*a+b/b=c+d/d

*If a/b=c/d=e/f.. Then a+c+e+/b+d+f...=a/b

What is the AA similarity postulate?

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

What is the SAS Similarity Theorem?

If an angle of one triangle is congruent to an angle of another triangles and their sides including those angles are in proportion, then the triangles are similar.

What is the SSS Similarity Theorem?

If the sides of two triangles are in proportion, then the triangles are similar.

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

proportionally

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

proportional

If three parallel lines intersect two transversals, then they divide the transversals ___________.

porportionally

What is the geometric mean of a and b?

the square root of a*b

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

similar

When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the ________ _____ between the segments of the hypotenuse.

geometric mean

When the altitude is drawn to the hypotenuse of a right triangle, each leg is the __________ ________ between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg.

geometric mean

What is the Pythagorean Theorem?

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

c squared= a squared + b squared

If the square of one side of a triangle is equal to the sum of the squares of another triangle, then the triangle is a ______ triangle.

right

If c2=a2+b2, then m<c=90 and ABC is a _____ triangle.

right

If c2<a2+b2, then m<C<90 and triangle ABC is ______.

acute

If c2>a2+b2, then m<C>90 and triangle ABC is ______.

obtuse

In a 45-45-90 triangle, the hypotenuse is ___ times as long as a leg.

radical 2

In a 30-60-90 triangle, the hypotenuse is ______ as long as the shorter leg and the longer leg is _____ times as long as the shorter leg.

twice, radical 3

What is the tangent ratio?

tangent of <A= leg opposite <A/ leg adjacent <A

*tan A= opposite/adjacent

What is the sine ratio?

sin A= leg opposite <A/ hypotenuse

What is the cosine ratio?

cos A= leg adjacent to <A/ hypotenuse

What is a chord?

a segment whose endpoints lie on a circle

What is a secant?

a line that contains a chord

What is a diameter?

a chord that contains the center of a circle

What is a tangent?

a line in the plane of a circle that intersects the circle in exactly one point

What is the point of tangency?

the point at which a tangent intersects the circle

If a line is tangent to a circle, then the line is ____________ to the radius drawn to the point of tangency.

perpendicular

Tangents to a circle from a point are ________.

congruent

If a line in the plane of a circle is perpendicular to a radius at its outer endpoint, then the line is ________ to the circle.

tangent

What is the arc addition postulate?

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

In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are ___________.

congruent

In the same circle or in congruent circles:

(1) congruent arcs have congruent chords

(2) congruent chords have congruent arcs

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

bisects

In the same circle or in congruent circles:

(1) chords equally distant from the center (or centers) are congruent

(2) congruent chords are equally distant from the center (or centers)

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 __________ of its intercepted arc

measure

If two inscribed angles intercept the same arc, then the angles are _________.

congruent

An angle inscribed in a semi-circle is a _____ angle.

right

If a quadrilateral is inscribed in a circle, then its opposite angles are _____________.

supplementary

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

equal

The measure of an angle formed by two chord that intersect inside a circle is equal to half the _____ of the measures of the intercepted arc.

sum

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 _____________ of the measures of the intercepted arcs.

difference

m<1= 1/2(x-y)

m<2= 1/2(x-y)

m<3= 1/2(x-y)

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

product

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 ____________ of the other secant segment and its external segment.

product

*ab=cd

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 ________ to the square of the tangent segment.

equal

The area of a square is the square of the length of a side, what is the formula?

A=s2

If two figures are congruent, then they have the same _______

area

The area of a region the sum of the _______ of its non-overlapping parts.

areas

The area of a rectangle equals the product of its ________ and _________

base, height

*A=bh

The area of a parallelgoram equals the product of a _______ and the _________ to that _________.

base, height

*A=bh

The area of a triangle equals half the product of a ________ and the ________ to that ________.

base, height

*A=1/2bh

The area of a rhombus equals half the product of its ___________.

diagonals

*A=1/2(d1*d2)

The area of a trapezoid equals half the product of the ________ and the sum of the ________.

height, bases

*A=1/2h(b1+b2)

The area of a regular polygon is equal to half the product of the __________ and the _____________.

A=1/2ap

Formula for the circumference of a circle:

C=2*pi*r

Formula for area of a circle:

A=pi*r2

Length of AB= x/360*2pir

Area of sector AOB= x/360*2pi*r2

Comparing the areas of triangles:

1. If two triangles have equal heights, then the ratio of their areas equals the ratios of their bases

2. If two triangles have equal bases, then the ratio of their areas equals the ratio of their heights

3. If two triangles are similar, then the ratio of their areas equals the square of their scale factor

If the scale factor of two similar figures is a:b, then:

(1) the ratio of the perimeters is a:b

(2) the ratio of the areas is a2:b2

What is the altitude of a prism?

a segment joining the two base planes and is perpendicular to both

What is the formula for lateral area?

LA= ph

*p=perimeter

*h=height

What is the formula for total area?

TA= LA+2B

*LA=latereal area= ph

*B= area of the base

What is the formula for volume?

V=Bh

*B=area of the base

*h=height

The lateral area of a right prism equals the ___________ of a base times the _________ of the prism.

perimeter, height

The volume of a right prism equals the _______ of a base times the ________ of a prism.

area, height

What is the formula for the lateral area of a pyramid?

LA= 1/2pl

*p=perimeter of the base

*l= slant height

The lateral area of a regular pyramid equals half the ____________ of the base times the ________ __________.

perimeter, slant height

What is the formula for the volume of a pyramid?

V= 1/3Bh

*B= area of the base

*h=height

The volume of a pyramid equals one third the _______ of a base times the _________ of the pyramid.

area, height

What is the formula for the lateral area of a cylinder?

LA= 2*pi*r*h

*r= radius

*h=height

The lateral area of a cylinder equals the ________ of a base times the ________ of a cylinder.

circumference, height

What is the formula for the volume of a cylinder?

V= pi*r2*h

The volume of a cylinder equals the _______ of a base times the ________ of the cylinder.

area, height

What is the formula for the lateral area of a cone?

LA= 1/2*2pir*l

or

LA=pi*r*l

*l=slant height

The lateral of a cone equals half the ________ of the base times the _______ ________.

circumference, slant height

What is the formula for the volume of a cone?

V=1/3*pi*r2*h

The volume of a cone equals one third the ______ of the base times the ________ of the cone.

area, height

What is the formula for the area of a sphere?

A=4pir2

The area of a sphere equals ____ times the square of the radius.

4pi

What is the formula for the volume of a sphere?

V=4/3pir3

The volume of a sphere equals _____ times the cube of the radius.

4/3pi

If the scale factor of two similar solids is a:b, then:

(1) the ratio of corresponding perimeters is a:b

(2) the ratio of the base areas, of the lateral areas, and of the total areas is a2:b2

(3) the ratio of the volumes is a3:b3

What is the distance formula?

the square root of x2-x1 squared puls y2-y1 sqaured

What is the formula for the center of a circle with a radius of r?

(x-a)2+(y-b)2=r2

What is the formula for the slope of a line?

change in y over the change in x

Two nonvertical lines are parallel if and only if their slopes are ________

equal

Two nonvertical line are perpendicular if and only if the product of their slope is _____

-1

How do you find the magnitude of a vector?

find the distance btw the 2 points

What is the midpoint formula?

(x1+x2/2, y1+y2/2)

What is standard form?

Ax+By=C

What is slope intercept form?

y=mx+b

What is point slope form?

y-y1= m(x-x1)

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