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) Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Definition Congruent Segments and Angles Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Triangle Congruence Two figures are congruent if it is possible to fit one figure onto the other so that matching parts coincide. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Definition Congruent Triangles Corresponding parts of congruent triangles are congruent, abbreviated CPCTC. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 12-1 Write an appropriate symbolic congruence for each of the pairs. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 12-2 Use SSS to explain why the given triangles are congruent. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Triangle Inequality The sum of the measures of any two sides of a triangle must be greater than the measure of the third side. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 12-3 Use SAS to show that the given pair of triangles are congruent. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. 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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 12-6 Show that the triangles are congruent. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 12-7 (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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.