© 2010 Pearson Education, Inc. All rights reserved Constructions, Congruence, and Similarity Chapter 12

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide NCTM Standard: Geometry In grades preK–2, all students should  relate ideas in geometry to ideas in number and measurement. (p. 396) In grades 3–5, all students should  explore congruence and similarity;  build and draw geometric objects;  use geometric models to solve problems in other areas of mathematics…;  recognize geometric ideas and relationships and apply them to other disciplines …. (p. 396)

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide NCTM Standard: Geometry In grades 6–8, all students should  use coordinate geometry to represent and examine the properties of geometric shapes;  use coordinate geometry to examine special geometric shapes, such as regular polygons or those with pairs of parallel or perpendicular sides;  draw geometric objects with specified properties, such as side lengths or angle measures. (p. 397)

Slide Copyright © 2010 Pearson Addison-Wesley. All rights reserved Other Congruence Properties  Angle, Side, Angle Property  Congruence of Quadrilaterals and Other Figures

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 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. Angle, Side, Angle (ASA) Property

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 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. Angle, Angle, Side (AAS)

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Example 12-5a Show that the triangles are congruent.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Example 12-5b Show that the triangles are congruent.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Example 12-6a 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 © 2010 Pearson Education, Inc. All rights reserved. Slide Example 12-6a (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 © 2010 Pearson Education, Inc. All rights reserved. Slide Properties of Quadrilaterals Trapezoid: A quadrilateral with at least one pair of parallel sides.  Consecutive angles between parallel sides are supplementary. Isosceles trapezoid: A trapezoid with a pair of congruent base angles  Each pair of base angles are congruent.  A pair of opposite sides are congruent.  If a trapezoid has congruent diagonals, it is isosceles.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Properties of Quadrilaterals Parallelogram: A quadrilateral in which each pair of opposite sides is parallel  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.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Kite: A quadrilateral with two disjoint pairs of congruent adjacent sides  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. Properties of Quadrilaterals

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 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. Properties of Quadrilaterals

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Square: A rectangle with all sides congruent  A square has all the properties of a parallelogram, a rectangle, and a rhombus. Properties of Quadrilaterals

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Example 12-7 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 © 2010 Pearson Education, Inc. All rights reserved. Slide Example 12-7 (continued) are supplements because ABCD is a parallelogram. Since these angles are supplementary and congruent, each must be a right angle. by SSS since (opposite sides of a parallelogram are congruent) and (given).

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Example 12-8 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 © 2010 Pearson Education, Inc. All rights reserved. Slide Example 12-8 (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 © 2010 Pearson Education, Inc. All rights reserved. Slide Example 12-8 (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 © 2010 Pearson Education, Inc. All rights reserved. Slide Example 12-8 (continued) The diagonals of a rectangle are congruent, so CE = AB. Therefore 2CD = AB, or

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 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.