Section 12-1 – Congruence Through Constructions
Congruence versus Similarity
In mathematics, similar objects have the same shape but not necessarily the same size.
Congruent objects have the same size and they are similar.
Two line segments are only congruent if they have the same length.
Two angles are only congruent if they have the same measure.
Circle: the set of all points in a plane that are the same distance (radius) from a given point
(center).
Arc: any part of the circle that can be drawn without lifting a pencil. (can be major or minor)
If the major and minor arcs are the same size, then each is a semicircle.
The center of the arc is the center of the circle containing the arc.
Triangle Congruence:
B′
B
A
C
A
A'
AB  A ' B '
B  B'
C  C'
AC  A ' C '
BC  B ' C '
A′
C′
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SSS Property: If the three sides of one triangle are congruent, respectively, to the three sides of a
second triangle, then the two triangles are congruent.
SAS Property: If two sides and the included angle of one triangle are congruent to two sides and
the included angles of another triangle, respectively, then the two triangles are congruent.
Examples: Determine if the following pairs of triangles are congruent.
A
B
Statement
Reason
Statement
Reason
C
E
D
F
120°
120°
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HL Property: If the hypotenuse and leg of one right triangle are congruent, respectively, to the
hypotenuse and leg of another right triangle, then the two triangles are congruent.
A line that is perpendicular to a segment and bisects it is the perpendicular bisector of the line
segment.
Theorems:
1. Any point equidistant from the endpoints of a segment is on the perpendicular bisector
of the segment.
2. Any point on the perpendicular bisector of a segment is equidistant from the endpoints
of the segment.
Theorems for Isosceles Triangles:
1. The angles opposite the congruent sides are congruent (base angles are congruent)
2. The angle bisector of an angle formed by two congruent sides is the perpendicular bisector
of the third side of the triangle.
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The altitude of a triangle is the perpendicular segment from a vertex of the triangle to the line
containing the opposite side of the triangle.
Section 12-2 – Other Congruence Properties
ASA Property: If two angles and the included side of one triangle are congruent to two angles and
the included side of another triangle, respectively, then the two triangles are congruent.
AAS Property: If two angles and a corresponding side of one triangle are congruent to two angles
and a corresponding side of another triangle, respectively, then the two triangles are congruent.
Examples: Determine if the following pairs of triangles are congruent.
B
75°
Statement
Reason
40°
A
C
D
E
65°
75°
F
4
Statement
Reason
A
B
C
D
E
There is no SSA property. Why not?
Go through the quadrilateral properties on pg. 772
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Constructions (from Sections 12-1 and 12-3)
Construct a Line Segment congruent to another line segment
Construct an Angle congruent to another angle
Construct a Triangle given the lengths of the three sides (4cm 5cm 8cm)
Can a triangle be constructed with sides 2cm, 3cm, and 5cm? Explain…..
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Construct a Perpendicular Bisector of a line segment
Example:
Construct the perpendicular bisectors of the sides of the triangle below
The point at which the perpendicular bisectors meet is called the _______________________.
From this point, you can draw a __________________________________________.
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Constructing Parallel Lines
Constructing a Perpendicular Line Through a Given Point
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Constructing an Angle Bisector
Property of Angle Bisectors:
Any point P on an angle bisector is equidistant from the sides of the angle (and vice versa)
Example:
Construct the angle bisectors of the triangle below
The point at which the angle bisectors meet is called the ___________________________.
From this point, you can draw an ______________________________________________.
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Example: For the following drawing:
Segment AX bisects angle A.
Triangle ABC is isosceles. Angle C is 72°.
Find the length of segment BX.
B
A
X
5′′
C
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Section 12-4 – Similar Triangles and Similar Figures
Two figures that have the same shape but not necessarily the same size are similar figures.
The ratio of the corresponding side lengths is called the scale factor.
In similar triangles, all three angles will be congruent.
Angle-Angle (AA) Property:
If two angles of one triangle are congruent, respectively, to two angles in a second triangle, then the
triangles are similar.
Example: Determine if the following pairs of triangles are similar.
A
Statement
B
D
Reason
C
E
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Example: Solve for x given that angles B and D are right angles:
D
14 cm
A
C
3 cm
7 cm
E
x
B
Example: On a sunny day, a tall tree casts a 40-m shadow. At the same time, a 10-m flagpole casts
a 50-m shadow. How tall is the tree?
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The segment connecting the midpoints of two sides of a triangle or two adjacent sides of a
quadrilateral is a midsegment.
Example: Construct a midsegment in the following triangle.
The Midsegment Theorem: The midsegment of a triangle is parallel to the third side of the
triangle and is half as long.
Theorem: If a line bisects one sides of a triangle and is parallel to a second side, then it bisects the
third side and is therefore a midsegment.
Section 12-5 – Lines and Linear Equations in a Cartesian Coordinate System
Label the following:
Origin
x-axis
y-axis
x-coordinate
y-coordinate
graph
(x, y)
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Vertical lines are of the form:
x = a (crosses the x-axis at a)
Horizontal lines are of the form:
y = b (crosses the y-axis at b)
Example: Find the equation of the line containing the point  2, 3 and parallel to the y-axis.
Example: Find the equation of the line containing the point 1, 4  and parallel to the x-axis.
Examples: Graph the following. Find at least 3 points.
a) y =
1
x3
2
b) y = 3x  2
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Using Triangles to Determine Slope:
Example: Find the slope of the following lines using triangles
#1
#3
#2
Slope of Line #1:
Slope of Line #2:
Slope of Line #3:
Slope Formula:
m
y2  y1
x2  x1
Example: Find the slope of the line going through the points
 5,3
and  4, 1 .
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Slope-Intercept form of a line:
y = mx + b
Example: Find the slope, y-intercept, and x-intercept of the line 6 x  4 y  18
Solving a System of Linear Equations Using The Substitution Method:
 2x  3y  9
Example: Solve 
 x  4 y  10
Solving a System of Linear Equations Using The Elimination Method:
 5x  3 y  7
Example: Solve 
2 x  6 y  10
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Solving a System of Linear Equations Using Geometry:
 25 x  y  125
Example: Solve 
 50 x  y  200
















Properties:
1. If the equations intersect in exactly one point, the system will have a unique solution.
2. If the equations represent parallel lines, the system will have no solution.
3. If the equations represent the same line, the system will have infinitely many solutions.
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