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Chapter
12
Congruence, and
Similarity with
Constructions
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12-1 Congruence Through Constructions
Geometric Constructions
Constructing Segments
Triangle Congruence
Side, Side, Side Congruence Condition (SSS)
Constructing a Triangle Given Three Sides
Constructing Congruent Angles
Side, Angle, Side Property (SAS)
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12-1 Congruence Through Constructions
(continued)
Constructions Involving Two Sides and an
Included Angle of a Triangle
Selected Triangle Properties
Construction of the Perpendicular Bisector of a
Segment
Construction of a Circle Circumscribed About a
Triangle
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Congruence and Similarity
Congruent fish and congruent
birds have the same shape
and same size.
Similar triangles have the
same shape but not the
same size.
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Congruence and Similarity
Similar objects () have the same shape but not
necessarily the same size.
Congruent objects () have the same shape and
the same size.
Congruent objects are similar, but similar objects
are not necessarily congruent.
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Definition
Congruent Segments and Angles
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Euclidean Constructions
1. Given two points, a unique straight line can be
drawn containing the points as well as a unique
segment connecting the points. (This is
accomplished by aligning the straightedge
across the points.)
2. It is possible to extend any part of a line.
3. A circle can be drawn given its center and
radius.
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Euclidean Constructions
4. Any number of points can be chosen on a
given line, segment, or circle.
5. Points of intersection of two lines, two circles,
or a line and a circle can be used to construct
segments, lines, or circles.
6. No other instruments (such as a marked ruler,
triangle, or protractor) or procedures can be
used to perform Euclidean constructions.
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Geometric Constructions
Constructing a circle given its radius
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Constructing Segments
There are many ways to draw a segment
congruent to another segment – using a ruler,
tracing the segment, and the method shown below
using a straightedge and compass.
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Triangle Congruence
Two figures are congruent if it is possible to fit one
figure onto the other so that matching parts coincide.
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Definition
Congruent Triangles
Corresponding parts of congruent triangles are
congruent, abbreviated CPCTC.
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Example 12-1
Write an appropriate symbolic congruence for
each of the pairs.
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Side, Side, Side Congruence Condition
(SSS)
If the three sides of one triangle are congruent,
respectively, to the three sides of a second
triangle, then the triangles are congruent.
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Example 12-2
Use SSS to explain why the given triangles are
congruent.
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Constructing a Triangle Given
Three Sides
B′
B
A
C
B′
C′
A′
A′
C′
B″
First construct A′C′ congruent to AC.
Locate the vertex B′ by constructing two circles
whose radii are congruent to AB and CB.
Where the circles intersect (either point) is B′.
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Triangle Inequality
The sum of the measures of any two sides of a
triangle must be greater than the measure of the
third side.
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Constructing Congruent Angles
With the compass and straightedge alone, it is
impossible in general to construct an angle if given only
its measure. Instead, a protractor or a geometry drawing
utility or some other measuring tool must be used.
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Side, Angle, Side Property (SAS)
If two sides and the included angle of one triangle
are congruent to two sides and the included angle
of another triangle, respectively, then the two
triangles are congruent.
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Constructions Involving Two Sides and
an Included Angle of a Triangle
First draw a ray with endpoint A′, and then construct
A′C′ congruent to AC. Then construct
Mark B′ on the side of
not containing C′ so that
Then connect B′ and C′ to complete the
triangle.
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Example 12-3
Use SAS to show that the given pair of triangles
are congruent.
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Hypotenuse Leg Theorem
If the hypotenuse and a leg of one right triangle
are congruent to the hypotenuse and a leg of
another right triangle, then the triangles are
congruent.
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Selected Triangle Properties
Any point equidistant from the endpoints of a
segment is on the perpendicular bisector of the
segment.
Any point on the perpendicular bisector of a
segment is equidistant from the endpoints of the
segment.
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Selected Triangle Properties
For every isosceles triangle:
The angles opposite the congruent sides are
congruent. (Base angles of an isosceles
triangle are congruent).
The angle bisector of an angle formed by two
congruent sides contains an altitude of the
triangle and is the perpendicular bisector of the
third side of the triangle.
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Selected Triangle Properties
An altitude of a triangle is the perpendicular
segment from a vertex of the triangle to the line
containing the opposite side of the triangle.
The three altitudes
in a triangle intersect
in one point.
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Construction of the Perpendicular
Bisector of a Segment
Construct the perpendicular bisector of a segment
by constructing any two points equidistant from the
endpoints of the segment.
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Construction of a Circle Circumscribed
About a Triangle
By constructing the perpendicular bisectors of any
two sides of triangle ABC, the point where the
bisectors intersect is the circumcenter.
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