1.
Geometry Toolbox
You will need to use the definitions, postulates, algebraic properties
and theorems you have learned to justify your conclusions.
Click on the cards below to review each one as needed.
Postulates
Common Definitions
Algebraic Properties
Angle Addition Postulate
right triangles
congruent
reflexive property
Triangles
bisector
Angle Pairs and
Parallel Lines
vertical angles
Triangle Angle Sum Theorem
midpoint
perpendicular lines
Isosceles Triangles
perpendicular bisector
right angles
supplementary angles
alternate Interior angles
corresponding angles
Exterior Angles Theorem
isosceles triangles
complementary angles
Corresponding Parts of Congruent
Triangles are Congruent (CPCTC)
Equilateral Triangles
Line Segments in Triangles
same-side interior angles
Quadrilaterals
medians
Triangle Congruency
Criteria
SSS
SAS
ASA
AAS
angle bisector
parallelograms
altitudes
midsegments
rectangles
perpendicular bisector
rhombus
square
2.
Parallelogram
Properties of parallelograms:
Opposite sides are parallel
Opposite sides are congruent
Opposite angles are congruent
Consecutive angles are supplementary
Diagonals are bisect each other
Opposite sides are parallel
Opposite angles are congruent
Opposite sides are congruent
Consecutive angles are
supplementary
Diagonals bisect each other
3.
Rhombus
Properties of Rhombuses:
All properties of parallelograms apply to
rhombus:
Opposite sides are parallel
Opposite sides are congruent
Opposite angles are congruent
Consecutive angles are supplementary
Diagonals bisect each other
and
All sides are congruent
Diagonals bisect opposite angles
Diagonals are perpendicular
All sides are congruent
Diagonals are
perpendicular
Diagonals bisect
opposite angles
4.
Rectangle
Properties of Rectangles:
Diagonals are congruent
5.
Square
Properties of Squares:
All sides are congruent
Diagonals are
perpendicular
Diagonals are
congruent
Diagonals bisect
opposite angles
7.
Supplementary Angles
Supplementary angles are two angles whose
measures add up to 180°. Each angle is called
the supplement of the other. The angles may
or may not be adjacent to each other.
If m∠IJK=113° and the m∠KJL=67°, the sum is
180°. This means that ∠IJK and ∠KJL are
supplementary angles.
∠IJL is a straight angle.
8.
Vertical Angles
Two lines that intersect form four angles. The
angles that are opposite from each other are
vertical angles.
Vertical Angles Theorem:
Vertical angles are congruent.
9.
Angle Addition Postulate
Angle Addition Postulate
The sum of two adjacent angles is equal to the
measure of the larger angle that is created.
∠ABC+∠CBD=∠ABD
10.
Alternate Interior Angles
Alternate interior angles are in between two
parallel lines but on opposite sides of the
transversal (creates "Z" or backwards "Z")
Alternate Interior Angles Theorem
If two parallel lines are cut by a
transversal, then the alternate
interior angles are congruent.
Lines m and n are parallel
and are intersected by line t.
There are two pairs of
alternate interior angles:
∠4≅∠6
∠3≅∠5
11.
Corresponding Angles
Corresponding angles are the angles on
the same side of the parallel lines and
same side of the transversal.
Corresponding Angles Postulate
If two parallel lines are cut by a
transversal, then the corresponding
angles are congruent.
Lines m and n are parallel
and are intersected by line t.
There are four pairs of
corresponding angles:
∠1≅∠5
∠2≅∠6
∠4≅∠8
∠3≅∠7
12.
Same-Side Interior Angles
Same-Side Interior Angles are the angles
between the parallel lines and on the
same side of the transversal.
Same-Side Interior Angles
If two parallel lines are cut by a
transversal, then same-side interior
angles are supplementary.
Lines m and n are parallel
and are intersected by line t.
There are two pairs of
same-side interior
angles:
∠4+∠5=180°
∠3+∠6=180°
13.
Exterior Angles
An exterior angle is an angle that is outside of
a polygon.
The Triangle Exterior Angle Theorem
The measure of the exterior angle is
equal to the sum of the two remote
interior angles. The remote interior
angles are two interior angles of the
triangle that are not adjacent to the
exterior angle.
m∠A + m∠B = m∠BCD
14.
Right Triangles
A right triangle is a triangle with one angle
that is 90°. The side opposite the right angle is
called the hypotenuse and the two sides that
are not the hypotenuse are called legs.
Acute Angles of a Right Triangle Theorem
In a right triangle, the two acute angles are
complementary.
16.
Bisectors
The bisector of an angle divides an angle into
two congruent angles.
The bisector of a segment divides the segment
into two congruent segments (and goes
through the midpoint of the segment).
Line Segment Bisector:
LK is a line segment that bisects HJ, point M is
the midpoint of HJ
Angle Bisector:
EG is a line segment that bisects ∠DGF
17.
Midpoint
The midpoint of a segment divides a segment
into two congruent segments.
If LK is a line segment that bisects HJ, point M
is the midpoint of HJ and LK is a line bisector of
HJ.
18.
Reflexive Property
(shared side or angle)
If two triangles share a side, the two sides are
congruent.
If two triangles share an angle, the two angles
are congruent.
19.
Perpendicular Lines
Perpendicular lines intersect to form 90° angles.
(right angles)
20.
Corresponding Parts
(CPCTC)
There are six statements that can be written about these triangles based on
their corresponding, congruent parts.
Corresponding Parts of Congruent Triangles
are Congruent
(CPCTC)
Corresponding parts can be proved congruent
using CPCTC if two triangles have already
been proved congruent by one of the triangle
congruence criteria (SSS, SAS, ASA, or AAS).
21.
SSS Postulate
If three sides of one triangle are congruent to three sides of
another triangle, then the triangles are congruent.
S
S
S
22.
SAS Postulate
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.
S
S
A
23.
ASA Postulate
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.
A
A
S
24.
AAS Postulate
If two angles and the non-included side of one triangle are
congruent to two angles and the non-included side of another
triangle, then the triangles are congruent.
A
A
S
25.
Congruent
Angles, segments or figures that are congruent
have exactly the same size and shape. This
means that the measures of the angles or
lengths of segments are equal.
26.
Isosceles Triangles
An isosceles triangle is a triangle with two
congruent sides.
The base of an isosceles triangle is the
side that is not a leg.
The base angles of an isosceles triangle
are the angles that are opposite the two
legs that are congruent.
The vertex angle is the angle that is not a
base angle (the angle that is opposite the
base of the isosceles triangle).
The altitude of the isosceles triangle is
the line segment that is drawn from the
vertex to the base of the isosceles
triangle. The altitude of a triangle is
always perpendicular to the base
27.
Isosceles Triangles
Altitude of an Isosceles Triangle Theorem
If a line segment is the angle bisector of the
vertex angle of an isosceles triangle, then it is
also the perpendicular bisector of the base.
Base Angles of Isosceles Triangles
Theorem
If a triangle is isosceles, the angles
that are opposite the two
congruent sides are also congruent.
28.
Equilateral Triangles
Equilateral triangles have all sides with the
same length.
An equiangular triangle is a triangle
whose angles all have the same
measure.
29.
Perpendicular Bisector
A perpendicular bisector is a line segment that
divides a segment into two congruent parts
and is perpendicular (creates a right angle)
with the segment it intersects.
30.
Right Angle
A right angle has a measure of 90°.
∠RST is a right angle.
The measure of ∠RST is 90°.
Segment RS is perpendicular to
segment ST. (RS⊥ST)
31.
Midsegments
The midsegment of a triangle is a segment
that joins the midpoints of two sides of a
triangle. The midpoint of a segment is the
point that divides the segment in half.
The Midsegment Theorem
The midsegment is parallel to its third side.
The midsegment is half of the length of the
third side.
The midsegment can be drawn from any two sides
of a triangle through the midpoints. The
midsegments do not intersect at one point.
32.
Median
The median of a triangle is a segment whose
endpoints are a vertex in a triangle and the
midpoint of the opposite side.
When all three of the medians of a
triangle are constructed, the medians
of a triangle meet at a point called the
centroid.
Another word for centroid is the
center of gravity, the point at which a
triangular shape will balance.
In this example, the medians intersect at point
G. Point G is the centroid of the triangle.
33.
Altitude
If all three altitudes are drawn in a
triangle, they meet at a point called the
orthocenter.
In this example, the three altitudes of
this triangle meet at point R, the
orthocenter.
34.
Angle Bisector
The angle bisector is a line segment that
divides an angle in half.
The angle bisectors of a triangle intersect
at a point called the incenter. The incenter
is the center of a circle that can be drawn
inside of the triangle (inscribed in the
triangle).
The angle bisectors of this triangle
intersect at point D, which is the
incenter. A circle with center at point D
can be inscribed inside ΔUVT.
35.
Perpendicular Bisectors
A perpendicular bisector is a line segment that is
perpendicular to a line segment and goes through the
midpoint of a line segment.
The perpendicular bisectors of the sides
of a triangle are concurrent at a point
called the circumcenter. This point is the
center of a circle that can be
circumscribed around the triangle.
The red lines represent the perpendicular
bisectors of the sides of ΔFEG. The
perpendicular bisectors intersect at point L,
the circumcenter. Point L is the center of the
circle that is circumscribed around ΔFEG.
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