Download Chapter 12 Section 2

Chapter
12
Congruence, and
Similarity with
Constructions
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12-2 Additional Congruence Properties
Angle, Side, Angle Property
Congruence of Quadrilaterals and Other Figures
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Angle, Side, Angle (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 triangles are congruent.
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Angle, Angle, Side (AAS)
If two angles and a side opposite one of these two
angles of a triangle are congruent to the two
corresponding angles and the corresponding side in
another triangle, then the two triangles are
congruent.
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Example 12-6
Show that the triangles are
congruent.
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Example 12-6
Show that the triangles are
congruent.
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Example 12-7
Using the definition of a parallelogram and the
property that opposite sides in a parallelogram are
congruent prove that the diagonals of a
parallelogram bisect each other.
We need to show that
BO  DO and AO  CO.
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Example 12-7
(continued)
AD  CB because
opposite sides of a
parallelogram are
congruent.
Since AD || BC,
So
and
by ASA.
because corresponding
parts of congruent triangles are congruent.
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Properties of Quadrilaterals
Trapezoid: A quadrilateral
with at least one pair of
parallel sides.
Isosceles trapezoid: A
trapezoid with a pair of
congruent base angles
 Consecutive angles
between parallel sides
are supplementary.
 Each pair of base angles are
congruent.
 A pair of opposite sides are
congruent.
 If a trapezoid has congruent
diagonals, it is isosceles.
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Properties of Quadrilaterals
Parallelogram: A
quadrilateral in which
each pair of opposite sides
is parallel
 A parallelogram has all the
properties of a trapezoid.
 Opposite sides are
congruent.
 Opposite angles are
congruent.
 Diagonal bisect each other.
 A quadrilateral in which the
diagonals bisect each other
is a parallelogram.
 Consecutive interior angles
are supplementary.
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Properties of Quadrilaterals
B
A
C
D
Rectangle: A
parallelogram with a right
angle
 A rectangle has all the
properties of a parallelogram.
 All the angles of a rectangle are
right angles.
 A quadrilateral in which all the
angles are right angles is a
rectangle.
 The diagonals of a rectangle
are congruent and bisect each
other.
 A quadrilateral in which the
diagonals are congruent and
bisect each other is a rectangle.
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Properties of Quadrilaterals
Kite: A quadrilateral with
two adjacent sides
congruent and the other
two sides also congruent.
 Lines containing the
diagonals are perpendicular
to each other.
 A line containing one
diagonal is a bisector of the
other diagonal.
 One diagonal bisects
nonconsecutive angles.
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Properties of Quadrilaterals
Rhombus: A parallelogram
with two adjacent sides
congruent
 A rhombus has all the
properties of a parallelogram
and a kite.
 A quadrilateral in which all
the sides are congruent is a
rhombus.
 The diagonals of a rhombus
are perpendicular to and
bisect each other.
 Each diagonal bisects
opposite angles.
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Properties of Quadrilaterals
 A square has all the
properties of a parallelogram,
a rectangle, and a rhombus.
Square: A rectangle with all
sides congruent
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Example 12-8
Prove that a quadrilateral in which the diagonals
are congruent and bisect each other is a rectangle.
If ABCD is a quadrilateral
whose diagonals bisect each
other, it is a parallelogram.
We must show that one of the angles in ABCD is a
right angle.
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Example 12-8
(continued)
are supplements because
ABCD is a parallelogram.
by SSS since
(opposite sides of a
parallelogram are congruent)
and
(given).
Since these angles are supplementary and
congruent, each must be a right angle.
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Example 12-9
In the figure,
is a right triangle, and CD is a
median (segment joining a vertex to the midpoint
of the opposite side) to the hypotenuse AB. Prove
that the median to the hypotenuse in a right
triangle is half as long as the hypotenuse.
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Example 12-9
(continued)
We are given that
is a
right angle and D is the
midpoint of AB. We need to
prove that 2CD = AB.
Extend CD to obtain
CE = 2CD.
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Example 12-9
(continued)
Because D is the midpoint of
AB and also the midpoint of
CE (by the construction), the
diagonals of ACBE bisect
each other, and ACBE is a
parallelogram.
Since
is a right angle, ACBE is a rectangle.
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Example 12-9
(continued)
The diagonals of a rectangle
are congruent, so CE = AB.
Therefore 2CD = AB, or
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Congruence of Quadrilaterals and
Other Figures
One way to determine a quadrilateral is to give
directions for drawing it. Start with a side and tell
by what angle to turn at the end of each side to
draw the next side (the turn is by an exterior
angle).
Thus, SASAS seems to be a valid congruence
condition for quadrilaterals. This condition can be
proved by dividing a quadrilateral into triangles and
using what we know about congruence of
triangles.
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