Chapter 14

1. Chapter 14 Geometry Using Triangle Congruence and Similarity

2. 14.1 Congruence of Triangles If two lines are of the same length, or if two angles have the same measure, we call them congruent lines or congruent angles, respectively. Q U B S P A D R T C

3. A D C B F E Suppose we have two triangles, ΔABC and ΔDEF, and that we pair up the vertices, A↔D, B↔E and C↔F. In this fashion, we have formed a correspondence between ΔABC and ΔDEF. Side AB corresponds to side DE, BC corresponds to EF, and CA corresponds to FD. Similarly, angle CAB corresponds to angle FDE, etc.

4. Definition: Suppose that ΔABC and ΔDEF are such that under the correspondence A↔D, B↔E and C↔F all corresponding sides are congruent and all corresponding vertex angles are congruent. Then ΔABC is congruent to ΔDEF and we write

5. Property: Side-Angle-Side (SAS) Congruence If two sides and the included angle of a triangle are congruent, respectively, to two sides and the included angle of another triangle, then the triangles are congruent. Property: Angle-Side-Angle (ASA) Congruence If two angles and the included side of a triangle are congruent, respectively, to two angles and the included side of another triangle, then the triangles are congruent.

6. Theorem: Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent.

7. Property: Side-Side-Side (SSS) Congruence If three sides of a triangle are congruent, respectively, to three sides of another triangle, then the triangles are congruent.

8. 14.2 Similarity of Triangles Two geometric figures that have the same shape but are not necessarily the same size are called similar. Definition: Suppose that ΔABC and ΔDEF are such that under the correspondence A↔D, B↔E and C↔F all corresponding sides are proportional and all corresponding vertex angles are congruent. Then ΔABC is similar to ΔDEF and we write

9. Similarity Properties of Triangles Two triangles, ΔABC and ΔDEF, are similar if and only if at least one of the following three statements is true. • Two pairs of corresponding sides are proportional and their included angles are congruent (SAS similarity). • Two pairs of corresponding angles are congruent (AA similarity). • All three pairs of corresponding sides are proportional (SSS similarity).

10. 14.3 Basic Euclidean Constructions Compass and Straightedge Properties • For every positive number r and for every point C, a circle of radius r and center C can be constructed. A connected portion of a circle is called an arc. • Ever pair of points can be connected by our straightedge to construct a line, a segment or a ray. • A line l can be constructed if and only if we have located two points that are on l.

11. Constructions • Copy a line segment. • Copy an angle. • Construct a perpendicular bisector of a line segment. • Bisect an angle. • Construct a line perpendicular to a given line through a specified point on the line. • Construct a line perpendicular to a given line through a point not on the line. • Construct a line parallel to a given line through a specified point not on the line.

12. 14.4 Additional Euclidean Constructions 8. Construct the circumscribed circle of a triangle. 9. Construct the inscribed circle of a triangle. 10. Construct an equilateral triangle.

13. Fermat Primes and Regular Polygon Constructions Definition: A Fermat prime is a prime number of the form Gauss’ Theorem for Constructible Regular n-gon A regular n-gon can be constructed with a straightedge and a compass if and only if the only odd prime factors of n are distinct Fermat Primes.