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Conjectures
Section
Name
2.5
Linear Pair Conjecture
2.5
Vertical Angles
Conjecture
2.6
Corresponding Angles
Conjecture OR
CA Conjecture
2.6
Alternate Interior
Angles Conjecture OR
AIA Conjecture
2.6
Alternate Exterior
Angles Conjecture OR
AEA Conjecture
Description
Chapter 2
Picture
If two angles form a linear pair,
then the measures of the angles
add up to 180°.
If two angles are vertical angles,
then they have equal measures
(are congruent).
If two parallel lines are cut by a
transversal, then corresponding
angles are congruent.
If two parallel lines are cut by a
transversal, then alternate
interior angles are congruent.
If two parallel lines are cut by a
transversal, then alternate
exterior angles are congruent.
If two parallel lines are cut by a
transversal, then corresponding
angles are congruent, alternate
interior angles are congruent, and
alternate exterior angles are
congruent.
2.6
Parallel Lines
Conjecture
2.6
Converse Parallel Lines If two lines are cut by a
Conjecture
transversal to form pairs of
congruent CA, AIA, and AEA, then
the lines are parallel.
1
Section
3.2
Name
Perpendicular Bisector
Conjecture
3.2
Converse
Perpendicular Bisector
Conjecture
3.3
Shortest Distance
Conjecture
3.4
Angle Bisector
Conjecture
3.7
Angle Bisector
Concurrency
Conjecture
3.7
3.7
3.7
Perpendicular Bisector
Concurrency
Conjecture
Altitude Concurrency
Conjecture
Circumcenter
Conjecture
Description
Chapter 3
Picture
If a point is on the perpendicular
bisector of a segment, then it is
equidistant from the endpoints.
If a point is equidistant from the
endpoints of a segment, then it is
on the perpendicular bisector of
the segment.
The shortest distance from a
point to a line is measured along
the perpendicular segment from
the point to the line.
If a point is on the bisector of an
angle, then it is equidistant from
the sides of the angle.
The three angle bisectors of a
triangle meet at point (are
concurrent).
The three perpendicular bisectors
of a triangle are concurrent.
Three altitudes (or the lines
containing the altitudes) of a
triangle are concurrent.
The circumcenter of a triangle is
equidistant from the vertices.
2
Section
Name
3.7
Incenter Conjecture
3.8
Median Concurrency
Conjecture
3.8
Centroid Conjecture
3.8
Center of Gravity
Conjecture
Description
Picture
The incenter of a triangle is
equidistant from the sides.
The three medians of a triangle
are concurrent.
The centroid of a triangle divides
each median into two parts so that
the distance from the centroid to
the vertex is twice the distance
from the centroid to the midpoint
of the opposite side.
The centroid of a triangle is the
center of gravity of the triangular
region.
Chapter 4
4.1
4.2
4.2
4.3
Triangle Sum
Conjecture
Isosceles Triangle
Conjecture
Converse Isosceles
Triangle Conjecture
Triangle Inequality
Conjecture
The sum of the measures of the
angles in every triangle is 180°.
If a triangle is isosceles, then its
base angles are congruent.
If a triangle has two congruent
angles, then it is an isosceles
triangle.
The sum of the lengths of any two
sides of a triangle is greater than
the length of the third side.
3
Section
Name
Description
In a triangle, if one side is longer
than another side, then the angle
opposite the longer side is larger
than the angle opposite the
shorter side.
The measure of an exterior angle
of a triangle is equal to the sum of
the measures of the remote
interior angles.
If the three sides of one triangle
are congruent to the three sides
of another triangle, then the
triangles are congruent.
If two sides and the included
angle of one triangle are congruent
to two sides and the included angle
of another triangle, then the
triangles are congruent.
4.3
Side-Angle Inequality
Conjecture
4.3
Triangle Exterior
Angle Conjecture
4.4
SSS Congruence
Conjecture
4.4
SAS Congruence
Conjecture
4.5
ASA Congruence
Conjecture
If two angles and the included
side of one triangle are congruent
to two angles and the included side
of another triangle, then the
triangles are congruent.
4.5
SAA Congruence
Conjecture
If two angles and a non-included
side of one triangle are congruent
to the corresponding two angles
and non-included side of another
triangle, then the triangles are
congruent.
4.8
Vertex Angle Bisector
Conjecture
4.8
Picture
In an isosceles triangle, the
bisector of the vertex angle is
also the altitude and the median to
the base.
Equilateral/Equiangular Every equilateral triangle is
Triangle Conjecture
equiangular, and, conversely, every
equiangular triangle is equilateral.
4
Section
Name
Description
Chapter 5
Picture
5.1
Quadrilateral Sum
Conjecture
The sum of the measures of the
four interior angles of any
quadrilateral is 360°.
5.1
Pentagon Sum
Conjecture
The sum of the measures of the
five interior angles of any
pentagon is 540°.
5.1
Polygon Sum
Conjecture
5.2
5.2
Exterior Angle Sum
Conjecture
Equiangular Polygon
Conjecture
The sum of the measures of the n
interior angles of an n-gon is
180°(𝑛 − 2).
For any polygon, the sum of the
measures of a set of exterior
angles is 360°.
You can find the measure of each
interior angle of an equiangular ngon by using either of these
formulas: 180° −
5.3
360°
𝑛
or
180°(𝑛−2)
𝑛
Kite Angles Conjecture
The nonvertex angles of a kite are
congruent.
5.3
5.3
Kite Diagonals
Conjecture
Kite Diagonal Bisector
Conjecture
The diagonals of a kite are
perpendicular.
The diagonal connecting the
vertex angles of a kite is the
perpendicular bisector of the
other diagonal.
5
Section
5.3
Name
Kite Angle Bisector
Conjecture
Description
The vertex angles of a kite are
bisected by a diagonal.
5.3
Trapezoid Consecutive
Angles Conjecture
The consecutive angles between
the bases of a trapezoid are
supplementary.
5.3
Isosceles Trapezoid
Conjecture
5.3
5.4
5.4
5.4
5.5
Isosceles Trapezoid
Diagonals Conjecture
Three Midsegments
Conjecture
Triangle Midsegment
Conjecture
Trapezoid Midsegment
Conjecture
Parallelogram Opposite
Angles Conjecture
Picture
The base angles of an isosceles
trapezoid are congruent.
The diagonals of an isosceles
trapezoid are congruent.
The three midsegments of a
triangle divide it into four
congruent triangles.
A midsegment of a triangle is
parallel to the third side and half
the length of the third side.
The midsegment of a trapezoid is
parallel to the bases and is equal in
length to the average of the
lengths of the bases.
The opposite angles of a
parallelogram are congruent.
6
Section
5.5
5.5
5.5
5.6
5.6
5.6
5.6
5.6
Name
Parallelogram
Consecutive Angles
Conjecture
Parallelogram Opposite
Sides Conjecture
Parallelogram
Diagonals Conjecture
Double-Edged
Straightedge
Conjecture
Rhombus Diagonals
Conjecture
Rhombus Angles
Conjecture
Rectangle Diagonals
Conjecture
Square Diagonals
Conjecture
Description
Picture
The consecutive angles of a
parallelogram are supplementary.
The opposite sides of a
parallelogram are congruent.
The diagonals of a parallelogram
bisect each other.
If two parallel lines are
intersected by a second pair of
parallel lines that are the same
distance apart as the first pair,
then the parallelogram formed is a
rhombus.
The diagonals of a rhombus are
perpendicular, and they bisect
each other.
The diagonals of a rhombus bisect
the angles of the rhombus.
The diagonals of a rectangle are
congruent and bisect each other.
The diagonals of a square are
congruent, perpendicular, and
bisect each other.
7
Section
Name
7.1
Reflection Line
Conjecture
7.2
Coordinate
Transformations
Conjecture
7.2
Minimal Path
Conjecture
7.3
Reflections across
Parallel Lines
Conjecture
7.3
Reflections across
Intersecting Lines
Conjecture
Description
Chapter 7
Picture
The line of reflection is the
perpendicular of every segment
joining a point in the original
figure with its image.
(𝑥, 𝑦) → (−𝑥, 𝑦) is a reflection
across the y-axis
(𝑥, 𝑦) → (𝑥, −𝑦) is a reflection
across the x axis
(𝑥, 𝑦) → (−𝑥, −𝑦) is a rotation
about the origin
(𝑥, 𝑦) → (𝑦, 𝑥) is a reflection across
the line 𝑦 = 𝑥
If points A and B are on one side
of line 𝑙, then the minimal path
from point A to line 𝑙 to point B is
found by reflecting point B across
line 𝑙, drawing segment AB’, then
drawing segments AC and CB
where point C is the point of
intersection of segment AB’ and
line 𝑙.
A composition of two reflections
across two parallel lines is
equivalent to a single translation.
In addition, the distance from any
point to its second image under
the two reflections is twice the
distance between the parallel
lines.
A composition of two reflections
across a pair of intersecting lines
is equivalent to a single rotation.
The angle of rotation is twice the
acute angle between the pair of
intersecting reflection lines.
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