1. Congruent, Regular, and Similar Triangles Dr. Jason Gershman

2. Regular Triangles Regular Triangles, or triangles with all 3 sides equal in length and all 3 angles equal to 60 degrees is called equilateral An equilateral triangle has some nice properties – Give side length s, the perimeter of an equilateral triangle is 3s – Give side length s, the area of an equilateral triangle is

3. Regular Hexagons A regular hexagon is just 6 equilateral triangles tiled correctly – Take the construction paper in front of you – Measure equilateral triangles of side length 2 inches. – Cut out 6 such triangles – Tile them properly to create a regular hexagon – What is the area of this regular hexagon? – What is the perimeter of this regular hexagon?

4. Congruent Triangles Two triangles are congruent if they have the same side lengths and the same angle measures. Suppose triangles ABC and DEF are congruent. Then

5. Congruent Triangles: Problems Suppose triangles ABC and DEF are congruent. If angle A has measure 55° and angle E has measure 64°, then find the angle measurement of angle C, in degrees, if angles C and F are equal in measurement.

6. Proving Congruency Which of the following prove congruency: SSS SAS AAS AAA ASA SSA

7. Answer SSS SAS AAS ASA These all prove congruency

8. AAA: Similarity AAA does not prove congruency. Counterexample: 30-60-90 triangles…one triangle is 2, 2√3,4 and the other is 4, 4√3,8. AAA does prove similarity Two triangles are similar when all corresponding angles are congruent and all corresponding sides are proportional.

9. Similar Triangles Triangles ABC and DEF are similar. If A=40° B=90° Find the area of triangle DEF. Find the measure of angle F.

10. SSA Ambiguous Case? This is the “ambiguous case” Law of Sines shows why this is true. Suppose that a triangle has sides a and b and angle A

11. SSA Ambiguous Case? Suppose a=20, b=10 and A=120°. Solve the triangle By law of sines, B is 25.7 degrees. When A is right or obtuse, there is congruency.

12. SSA Ambiguous Case? Suppose b=5, c=10, C=42°. Solve the triangle. B=19.5° There is congruency when the side oppose the angle is greater than the adjacent side.

13. SSA Ambiguous Case? Suppose b=10, c=12, and B=51° C=68.8° or B=111.2 degrees This is the ambiguous case

14. Determine if Similar, Congruent or Neither, or No Such Triangle In triangles ABC and DEF – Angle B=40°, angle C=110° and angle D=30° – Side AB=14, side BC=10, side AC=28 – Angle B=120°, angle C=10° and angle A=60° – Sides AB=3, BC=4, angle B=90°, sides DE=12, EF=16, DF=20