1. Congruence, Constructions and Similarity
12.1 Congruent Triangles
12.2 Constructing Geometric Figures
12.3 Similar Triangles

2. 12.1
Congruent Triangles

3. Construction 1: CONSTRUCT A LINE SEGMENT CONGRUENT TO A GIVEN SEGMENT

4. 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.

5. PROPERTY: SIDE-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.

6. Construction 2: CONSTRUCT AN ANGLE CONGRUENT TO A GIVEN ANGLE
Draw arcs of the same radius centered at D and at Q; let E and F be the points at which the arc intersects the sides of the given angle, and let R be the point at which the arc intersects

7. Construction 2: CONSTRUCT AN ANGLE CONGRUENT TO A GIVEN ANGLE
Place the point of the
compass at E and adjust it to draw an arc through F. Draw an arc of the same radius centered at R to locate S.

8. 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.

9. ISOSCELES TRIANGLE THEOREM
The angles opposite the congruent sides of an isosceles triangle are congruent.

10. THALES’ THEOREM
Any triangle ABC inscribed in a semicircle with diameter has a right angle at point C.

11. 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.

12. 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.

13. PROPERTY: ANGLE-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.

14. Constructing Geometric Figures
12.2
Constructing Geometric Figures
Slide 12-14

15. CONSTRUCTING A RHOMBUS

16. 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.

17. 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.

18. 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.

19. 12.3
Similar Triangles
Slide 12-19

20. 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,

21. 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.

22. 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 12-22 22

23. By the vertical angle theorem, Also, By the AA similarity property,
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 12-23 23

24. 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

25. 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.