Midterm Study Guide
Unit 1
Section 9-1: Translations
Pre-image- the starting figure in a transformation
Image- the finishing figure in a transformation
Rigid Motion- preserves size and angle measures; also called isometries.
 Translations
 Reflections
 Rotations


List all pairs of corresponding sides and angles in the correct order
o Corresponding angles: ÐN and ÐS , ÐI and ÐU , ÐD and ÐP .
o Corresponding sides: NI and S U , I D and U P , ND and S P .
∆NID  ∆SUP
Translations: “slide”
 T<1.4>: translate the figure one unit to the right and four units up [add 1 to the x-values
and 4 to the y-values  (x+1, y+4)]
Section 9-2: Reflections
Reflections: “flip”
 Rx-axis(P): reflect the coordinate P across the x-axis (The distance from P to the line of
reflection should be the same as the distance from P’ to the line of reflection)
Section 9-3: Rotations
Rotations: “turn”

r(90°,O)(ABCD): rotate quadrilateral ABCD 90° about the origin.
90˚: switch the position of x and y and change the sign of the new x
180˚: change the sign of x and y
270˚: switch the position of x and y and change the sign of the new y
360˚: back to original
Section 9-6: Dilations
Dilations: increases or decreases the size of a figure

The scale factor of a dilation is the ratio of the length of any side in the image to the
length of its corresponding side in the pre-image.

The center of dilation is a fixed point in the plane about which all points are expanded or
contracted.

If the image is smaller than the original figure, then the dilation is a reduction. (Scale
factor is between 0 and 1)

If the image is larger than the original figure, then the dilation is an enlargement. (Scale
factor is greater than 1)
Section 1-2: Points, Lines, and Planes

Point: dot on the coordinate plane; indicates a location; has no size

Line: two points connected and go in both directions forever

Plane: flat surface that extends in all directions

Ray: part of a line that has one endpoint and extends forever in the other direction

Line segment: part of a line with two endpoints

Collinear Points: points that lie in the same line

Coplanar Points: points that lie in the same plane

Two lines always intersect at a point.

Two planes always intersect at a line.
Section 1-3: Measuring Segments

Segment Addition Postulate: If three points A, B, and C are collinear and B is between A
and C, then AB + BC = AC
Section 1-7: Midpoint & Distance in the Coordinate Plane

The midpoint of a segment is the point that divides the segment into two congruent parts.
o
Midpoint formula: (
x1 + x 2 y1 + y 2
,
)
2
2

Tick marks indicate congruence. Therefore, this means the lengths are equal.
o KL = L M

The distance between two points can be found by using the distance formula:
d = (x 2 - x1 )2 + (y 2 - y1 )2
Unit 2
Sections 1-4 & 1-5: Measuring Angles & Exploring Angle Pairs
Bisect: to divide into two congruent parts
Segment Bisector: segment, ray, line, or plane that
intersects a segment at its midpoint
Angle Bisector: a ray that divides and angle into two congruent angles
Complementary Angles: two angles whose measures have a sum of 90°
Supplementary Angles: two angles whose measures have a sum of 180°
Vertical Angles: nonadjacent angles formed by 2 intersecting lines
- vertical angles are congruent
Example:
Ð1& Ð3 are vertical angles
Ð2 & Ð4 are vertical angles
Linear pair (supplementary angles): two adjacent angles on the same line
- add up to 180°
Example:
Ð1& Ð2 are linear pairs
Adjacent Angles: two angles with a common vertex and side but no common interior points
Example:
ÐABC & ÐCBD are adjacent angles
Acute Angle: measures between 0° and 90°
Right Angle: equal to 90°
Obtuse Angle: measures between 90° and 180°
Straight Angle: equal to 180°
Section 1-8 Review: Classifying Polygons
Polygon: a flat shape consisting of straight lines that are joined to form a closed chain or circuit;
no curved lines

Convex: has no diagonal with points outside the polygon

Concave: has at least one diagonal with points outside the polygon
Section 1-8: Perimeter, Circumference, & Area
Section 3-1: Lines & Angles
Definition
Parallel lines:
 Coplanar lines
 Do not intersect

Symbol: ||
“is parallel to”
Skew lines:
 noncoplanar
 not parallel
 do not intersect
Parallel planes:
 planes that do not
intersect
Symbols
Diagram

AE || BF

AD || BC

AB and CG are skew.
plane ABCD || plane EFGH
Section 3-2 Properties of Parallel Lines
Ð1& Ð5
Corresponding angles:
Ð2 & Ð6
Ð3 & Ð7
Ð4 & Ð8
- corresponding angles are equal
Alternate interior angles:
Ð4 & Ð5
Ð2 & Ð7
- alternate interior angles are equal
Alternate exterior angles:
Ð3 & Ð6
Ð1& Ð8
- alternate exterior angles are equal
Same side interior angles:
Ð2 & Ð5
Ð4 & Ð7
- same side interior angles are supplementary
Section 3-3 Proving Lines Parallel
Showing lines are parallel:
-
If two lines are cut by a transversal so that the corresponding angles are congruent,
then the lines are parallel.
If Ð1@ Ð5, then r s
-
If two lines are cut by a transversal so that alternate interior angles are congruent,
then the lines are parallel.
If Ð4 @ Ð5, then r s
-
If two lines are cut by a transversal so that alternate exterior angles are congruent,
the lines are parallel.
If Ð1@ Ð8, then r s
-
If two lines are cut by a transversal so that the same-side interior angles are
supplementary, then the lines are parallel.
If
mÐ3+ mÐ5 =180°, then r s
Section 3-7: Equations of Lines on the Coordinate Plane

Slope formula: m =
y 2 - y1
x 2 - x1
 Types of slopes:
Positive
Negative
Zero
y=4
Undefined
x=4

Slope intercept form: y mxb
o m = slope
o b = y-intercept
o Use 
when
 given slope and y-intercept
o When you are given two points: First find the slope, then use slope-intercept form

Rules for graphing in slope intercept form:
1. Plot the y-intercept on the y-axis
2. Use the slope in fraction form to get to the next point using
rise
run
3. Continue to make other points by repeating step 2
Section 3-8: Slopes of Parallel and Perpendicular Lines


Parallel and Perpendicular Lines
 Slopes of parallel lines are the same
 Slopes of perpendicular lines are opposite reciprocals of one another
Unit 3
Sections 4-1, 4-2, 4-3, & 4-6 : SSS, SAS, ASA, AAS & HL
SIDE-SIDE-SIDE (SSS)
 If three sides of one triangle are congruent to three sides of a second triangle, then the
two triangles are congruent.
SIDE-ANGLE-SIDE (SAS)
 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.
ANGLE-SIDE-ANGLE CONGRUENCE (ASA)
 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.
ANGLE-ANGLE-SIDE CONGRUENCE (AAS)
 If two angles and a NON-included side of one triangle are congruent to the
corresponding two angles and NON-included side of a second triangle then the two
triangles are congruent.
HYPOTENUSE-LEG CONGRUENCE (HL)
 If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg
of another right triangle, then the triangles are congruent.
o For example, in the diagram below ∆ABC @ ∆DEF, by HL.
Section 4-4: Using Corresponding Parts of Congruent Triangles
Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
 If you know two triangles are congruent, then you know that every pair of their
corresponding parts is also congruent.
Section 4-5: Isosceles & Equilateral Triangles
Isosceles Triangle: two congruent legs
Triangle Theorems
Triangle Sum Theorem
Description
The sum of the measures
of the angles in a triangle is 180°.
Exterior Angles Theorem
The measure of an exterior angle
of a triangle is equal to the sum of
the measure of the two
nonadjacent interior angles.
Isosceles Triangle Theorem 1
If two sides of a triangle are
congruent, then the angles
opposite them are congruent.
Image
If
Isosceles Triangle Theorem 2
AB @ AC , then ÐC @ ÐB
If two angles of a triangle are
congruent, then the sides opposite
them are congruent.
If ÐB @ ÐC, then
Equilateral Theorem 1
AC @ A B
If a triangle is equilateral, then it is
equiangular.
AB @ AC @ B C , then
ÐA @ ÐB @ ÐC
If
Equilateral Theorem 2
If a triangle is equiangular then its
equilateral.
If ÐB @ ÐC @ ÐA, then
AB @ AC @ B C
Section 5-1: Midsegments of Triangles

A midsegment of a triangle is a segment connecting the midpoints of two sides of the
triangle.
Section 5-2: Perpendicular & Angle Bisector
Angle Bisectors &
Perpendicular Bisectors
Angle Bisector Theorem
Converse of the Angle
Bisector Theorem
Description
If a point is on
the bisector of
an angle, then
the point is
equidistant from
the sides of the
angle.
If a point in the
interior of an
angle is
equidistant from
the sides of the
angle, then the
point is on the
angle bisector.
Image
Perpendicular Bisector
Theorem
Converse of the
Perpendicular Bisector
Theorem
If a point is on
the
perpendicular
bisector of a
segment, then it
is equidistant
from the end
points of the
segment.
If a point is
equidistant from
the endpoints of
a segments,
then it is on the
perpendicular
bisector of the
segment.
Section 5-3 & 5-4: Bisectors in Triangles, Medians & Altitudes
Triangle
Relationships
Median
Intersections of
Medians of a
Triangle
Description
Image
A segment from a
vertex to the midpoint
of the opposite side.
The medians of a
triangle intersect at the
centroid, a point that is
two thirds of the
distance from each
vertex to the midpoint
of the opposite side.
If P is the centroid of
Centroid
The point at which the
three medians of a
triangle intersect.
, then AP =
2
CP = CE
3
point P is the centroid
Circumcenter
The point at which the
three perpendicular
bisectors intersect.
2
2
AD, BP = BF , and
3
3