1. 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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2. 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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3. 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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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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8. 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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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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11. 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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12. 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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14. 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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15. 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 (continued)
AD  CB because opposite sides of a parallelogram are congruent.
Since AD || BC, and
So by ASA.
because corresponding parts of congruent triangles are congruent.
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