© 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. 12-1Constructions, Congruence, and Similarity  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)

Slide Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 12-1Constructions, Congruence, and Similarity (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 © 2010 Pearson Education, Inc. All rights reserved. Slide Congruent fish and congruent birds have the same shape and same size. Similar triangles have the same shape but not the same size. Congruence and Similarity

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide Definition Congruent Segments and Angles

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Constructing a circle given its radius Geometric Constructions

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Circles An arc of a circle is any part of the circle that can be drawn without lifting a pencil. The center of an arc is the center of the circle containing the arc. If there is no ambiguity in a discussion, the endpoints of the arc are used to name the arc, otherwise three points are used.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Circles If the two arcs determined by a pair of points on the circle are the same size, each is a semicircle. If only two points are used in the arc name, then the minor arc is intended. A segment connecting two points on a circle is a chord of the circle. If a chord contains the center, it is a diameter.

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

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

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Congruent Triangles Corresponding parts of congruent triangles are congruent, abbreviated CPCTC. Definition

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Example 12-1 Write an appropriate symbolic congruence for each of the pairs.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide Example 12-2 Use SSS to explain why the given triangles are congruent.

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

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

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide A B C A′A′ C′C′ Constructing a Triangle Given Three Sides 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′. B″B″ B′B′ A′A′ C′ B′B′

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

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide 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 © 2010 Pearson Education, Inc. All rights reserved. Slide Example 12-3 Use SAS to show that the given pair of triangles are congruent.

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

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

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

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

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide By constructing the perpendicular bisectors of any two sides of triangle ABC, the point where the bisectors intersect is the circumcenter. Construction of a Circle Circumscribed About a Triangle