PROPERTIES OF PLANE FIGURES

1. PROPERTIES OF PLANE FIGURES ANGLES & TRIANGLES

2. Angles Angles are formed by the intersection of 2 lines, 2 rays, or 2 line segments. The point at which the lines, rays, or line segments intersect is called the vertex. Vertex Vertex Vertex

3. Measuring Angles The measure of an angle is the amount of rotation in degrees about the vertex from one side to the other. The measure of any angle is between 0º and 180º. The wider the “mouth,” the greater the measure of the angle.

4. Classifying Angles There are 4 types of angles: acute angle – angle that is greater than 0° and less than 90° right angle – angle that is exactly 90° obtuse angle – angle that is greater than 90° and less than 180° straight angle – angle that is exactly 180°

5. acute right obtuse straight

6. Naming Angles Angles are named according to their vertex and the points through which each side passes. Angles can be labeled by only the letter or number that represents their vertex. If more than 1 angle share a vertex, then label the angle with the points on each side of the angle and the vertex. Make sure that the vertex is always the middle letter.

7. Homework Complete all of the problems on “PPF Practice Problems 1.” Also, answer the following questions: What do acute people and acute angles have in common? What do obtuse people and obtuse angles have in common?

8. Angle Addition Postulate When more than 1 angle share a vertex, the sum of the measure of the smaller angles equal the measure of the largest angle. Example 1

9. Another Example of Applying the Angle Addition Postulate Example 2

10. You Try

11. Angle Relationships Adjacent Angles – two angles that are next to each other Complementary Angles – two adjacent angles that form a right angle Supplementary Angles – two adjacent angles that form a straight angle Vertical Angles – two angles that are opposite of each other when two lines cross

12. More Angle Relationships The following relationships are formed when a transversal line intersects two parallel lines: Corresponding Angles – two angles that are on the same side of the transversal line, but one is inside and the other is outside of the parallel lines Alternate Exterior Angles – two angles that are on opposite sides of the transversal line, but both are outside of the parallel lines Alternate Interior Angles – two angles that are on opposite sides of the transversal line, but both are inside of the parallel lines Consecutive (Co-interior) Angles – two angles that are on the same side of the transversal line and are both inside of the parallel lines

13. Adjacent Angles CAB and BAD are adjacent angles

14. Complementary Angles

15. Supplementary Angles

16. Vertical Angles

17. Corresponding Angles

18. Alternate Exterior Angles

19. Alternate Interior Angles

20. Consecutive (Co-Interior) Angles

21. You Try Identify the relationship Identify the relationship

22. Homework Complete all of the problems on PPF Practice Problems 2.

23. More Info On Angle Relationships Complementary Angles – add up to 90° Supplementary Angles – add up to 180° Vertical Angles – have the same measure Alternate Exterior Angles – have the same measure Alternate Interior Angles – have the same measure Corresponding Angles – have the same measure Consecutive Angles – add up to 180°

24. You Try Find the measure of angle b Find the measure of angle b

25. You Try More Find the measure of angle b Solve for x

26. Homework Complete all of the problems on PPF Practice Problems 3.

27. Geometric Notations

28. More Geometric Notations

29. Congruent Angles and Segments

30. Parallel Lines

31. Perpendicular Lines

32. Homework Complete all of the problems on PPF Practice Problems 4

33. Basic Facts About Triangles • Three-sided polygon • Each “corner” is a vertex (vertices for plural) • The area of a triangle is ½ · base · height • The angles in a triangle add up to 180° • Triangles are named by their vertices • Triangles are classified by their sides and angles

34. Classifying Triangles by Angles Acute Triangle – all 3 angles are acute Right Triangle – has 1 right angle Obtuse Triangle – has 1 obtuse angle

35. Classifying Triangles by Sides Equilateral Triangle – all sides and all angles are congruent Isosceles Triangle – 2 sides are congruent and base angles are congruent Scalene Triangle – no sides are congruent

36. Classifying Triangles by Angles and Sides Right Scalene Acute Scalene Right Isosceles Obtuse Scalene Acute Isosceles Obtuse Isosceles

37. You Try: Classify Triangles by Their Angles and Sides 4 1 2 5 6 3

38. Triangle Angle Sum The measure of the unknown angle is 180° – (81° + 58°) = 41° The measure of the unknown angle is 180° – (90° + 40°) = 50°

39. Triangle Angle Sum Extended This angle is also 102° due to the property of vertical angles This angle is 180° – (102° + 52°) = 26° The measure of the unknown angle is 180° – 26° = 154° due to the property of supplementary angles.

40. You Try: Find the Measure of the Unknown Angles 1 2 3

41. Homework Complete all of the problems on PPF Practice Problems 5 and 6. For extra credit you may complete the “Additional Triangle Angle Sum Practice Problems” sheet.

42. Triangle Congruence • Congruent triangles are exactly the same size (same side lengths and angle measures) • Corresponding sides are the congruent sides of congruent triangles • Corresponding angles are the congruent angles of congruent triangles

43. Examples of Congruent Triangles Note: We must name the congruent triangles correctly according to the corresponding angles!

44. You Try: Write a Statement Indicating the Pairs of Triangle Are Congruent 1 2

45. Proving Triangle Congruence SSS (side-side-side) – if all 3 pairs of corresponding sides are congruent, then the triangles are congruent SAS (side-angle-side) – if 2 pairs of corresponding sides and the pair of corresponding angles between them are congruent, then the triangles are congruent ASA (angle-side-angle) – if 2 pairs of corresponding angles and the pair of corresponding sides between them are congruent, then the triangles are congruent AAS (angle-angle-side) – if 2 pairs of corresponding angles and the pair of corresponding sides not between them are congruent, then the triangles are congruent

46. Examples of Proving Triangle Congruence SSS ASA SSS ASA SAS AAS SAS

47. You Try: State Why the Triangles Are Congruent 1 2

48. Proving Right Triangle Congruence LA (leg-angle) – if a pair of corresponding legs and a pair of corresponding angles other than the right angles are congruent, then the right triangles are congruent LL (leg-leg) – if 2 pairs of corresponding legs are congruent, then the right triangles are congruent HA (hypotenuse-angle) – if the pair of hypotenuses and a pair of corresponding angles other than the right angles are congruent, then the right triangles are congruent HL (hypotenuse-leg) – if the pair of hypotenuses and a pair of corresponding legs are congruent, then the right triangles are congruent

49. Examples of Proving Right Triangle Congruence LA HA HL HL LL

50. You Try: State Why the Right Triangles Are Congruent 1 2 3