12 Chapter Congruence, and Similarity with Constructions Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

12-2 Additional Congruence Properties Angle, Side, Angle Property Congruence of Quadrilaterals and Other Figures 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 12-6 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 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. 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.

Properties of Quadrilaterals Trapezoid: A quadrilateral with at least one pair of parallel sides. 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. Isosceles trapezoid: A trapezoid with a pair of congruent base angles Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Properties of Quadrilaterals 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. Parallelogram: A quadrilateral in which each pair of opposite sides is parallel Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Properties of Quadrilaterals 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. Rectangle: A parallelogram with a right angle Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Properties of Quadrilaterals 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. Kite: A quadrilateral with two adjacent sides congruent and the other two sides also congruent. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Properties of Quadrilaterals 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. Rhombus: A parallelogram with two adjacent sides congruent Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Properties of Quadrilaterals A square has all the properties of a parallelogram, a rectangle, and a rhombus. Square: A rectangle with all sides congruent Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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

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

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

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

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

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 12-9 (continued) The diagonals of a rectangle are congruent, so CE = AB. Therefore 2CD = AB, or Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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