Congruence Constructions and Similarity 12 1 Congruent Triangles

Congruence, Constructions and Similarity 12. 1 Congruent Triangles 12. 2 Constructing Geometric Figures 12. 3 Similar Triangles

12. 1 Congruent Triangles Slide 12 -2

DEFINITION: CONGRUENT TRIANGLES Two triangles are congruent if, and only if, there is a correspondence of vertices of the triangles such that the corresponding sides and corresponding angles are congruent. Slide 12 -3

PROPERTY: SIDE-SIDE (SSS) If the three sides of one triangle are respectively congruent to the three sides of another triangle, then the two triangles are congruent. Slide 12 -4

PROPERTY: SIDE-ANGLE-SIDE (SAS) If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. Slide 12 -5

ISOSCELES TRIANGLE THEOREM The angles opposite the congruent sides of an isosceles triangle are congruent. Slide 12 -6

THALES’ THEOREM Any triangle ABC inscribed in a semicircle with diameter has a right angle at point C. Slide 12 -7

PROPERTY: ANGLE-SIDE-ANGLE (ASA) If two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the two triangles are congruent. Slide 12 -8

CONVERSE OF THE ISOSCELES TRIANGLE THEOREM If two angles of a triangle are congruent, then the sides opposite them are congruent; that is, the triangle is isosceles. Slide 12 -9

PROPERTY: ANGLE-SIDE (AAS) If two angles and a nonincluded side of one triangle are respectively congruent to two angles and the corresponding nonincluded side of a second triangle, then the two triangles are congruent. Slide 12 -10

12. 2 Constructing Geometric Figures Slide 12 -11

CONSTRUCTING A RHOMBUS Slide 12 -12

PROPERTIES OF A RHOMBUS • The diagonals are angle bisectors. • The diagonals are perpendicular. • The diagonals intersect at their common midpoint. • The sides are all congruent to each other. • The opposite sides are parallel. Slide 12 -13

EQUIDISTANCE PROPERTY OF THE PERPENDICULAR BISECTOR A point lies on the perpendicular bisector of a line segment if, and only if, the point is equidistant from the endpoints of the segment. Slide 12 -14

EQUIDISTANCE PROPERTY OF THE ANGLE BISECTOR A point lies on the bisector of an angle if, and only if, the point is equidistant from the sides of the angle. Slide 12 -15

12. 3 Similar Triangles Slide 12 -16

DEFINITION: SIMILAR TRIANGLES AND THE SCALE FACTOR Triangle ABC is similar to triangle DEF, written ABC ~ DEF if, and only if, corresponding angles are congruent and the ratios of lengths of corresponding sides are all equal. That is, ABC ~ DEF if, and only if, Slide 12 -17

THE AA SIMILARITY PROPERTY If two angles of one triangle are congruent respectively to two angles of a second triangle, then the triangles are similar. Slide 12 -18

Example 12. 9: Making an Indirect Measurement with Similarity A tree and point T is a line with a stake at point L when viewed across the river from point N. Use the information in the diagram to measure the width x of the river. Slide 12 -19

Example 12. 9: continued By the vertical angle theorem, Also, By the AA similarity property, Thus, since the ratios of the lengths of the corresponding sides are equal. We use and solve the proportion Slide 12 -20

THE SSS SIMILARITY PROPERTY If the three sides of one triangle are proportional to the three sides of a second triangle, then the triangles are similar. That is, if then ABC ~ DEF Slide 12 -21

THE SAS SIMILARITY PROPERTY If, in two triangles, the ratios of any two pairs of corresponding sides are equal and the included angles are congruent, then the two triangles are similar. That is, if then ABC ~ DEF. Slide 12 -22