1. L21_Ratios and Proportions Eleanor Roosevelt High School Chin-Sung Lin
Special Topics
Eleanor Roosevelt High School
Chin-Sung Lin

2. Similar Triangles ERHS Math Geometry Mr. Chin-Sung Lin
L22_Similar Triangles
ERHS Math Geometry
Similar Triangles
Mr. Chin-Sung Lin

3. Definition of Similar Triangles
L22_Similar Triangles
ERHS Math Geometry
Definition of Similar Triangles
Triangles are similar if their corresponding angles are equal and their corresponding sides are proportional
(The number represented by the ratio of similitude is called the constant of proportionality)
Mr. Chin-Sung Lin

4. Example of Similar Triangles
L22_Similar Triangles
ERHS Math Geometry
Example of Similar Triangles
A  X, B  Y, C  Z
AB = 6, BC = 8, and CA = 10
XY = 3, YZ = 4 and ZX = 5
Show that ABC~XYZ A B C 6 8 10 X Y Z 3 4 5
Mr. Chin-Sung Lin

5. Example of Similar Triangles
L22_Similar Triangles
ERHS Math Geometry
Example of Similar Triangles
A  X, B  Y, C  Z
AB BC CA
XY YZ ZX
Therefore ABC~XYZ = = = A B C 6 8 10 X Y Z 3 4 5
Mr. Chin-Sung Lin

6. Example of Similar Triangles
L22_Similar Triangles
ERHS Math Geometry
Example of Similar Triangles
The sides of a triangle have lengths 4, 6, and 8.
Find the sides of a larger similar triangle if the constant of proportionality is 5/2 ? 4 6 8
Mr. Chin-Sung Lin

7. Example of Similar Triangles
L22_Similar Triangles
ERHS Math Geometry
Example of Similar Triangles
Assume x, y, and z are the sides of the larger triangle, then
x y z = = = 4 6 8
y = 20
x = 10
z = 15
Mr. Chin-Sung Lin

8. Example of Similar Triangles
L22_Similar Triangles
ERHS Math Geometry
Example of Similar Triangles
In ABC, AB = 9, BC = 15, AC = 18.
If ABC~XYZ, and XZ = 12, find XY and YZ 9 15 18 A B C X Y Z ? 12
Mr. Chin-Sung Lin

9. Example of Similar Triangles
L22_Similar Triangles
ERHS Math Geometry
Example of Similar Triangles
Since ABC~XYZ, and XZ = 12, then
XY YZ = = 9 15 18 A B C X Y Z 6 10 12
Mr. Chin-Sung Lin

10. Example of Similar Triangles
L22_Similar Triangles
ERHS Math Geometry
Example of Similar Triangles
In ABC, AB = 4y – 1, BC = 8x + 2, AC = 8.
If ABC~XYZ, and XZ = 6, find XY and YZ
4y – 1
8x + 2 8 A B C X Y Z ? 6
Mr. Chin-Sung Lin

11. Example of Similar Triangles
L22_Similar Triangles
ERHS Math Geometry
Example of Similar Triangles
Since ABC~XYZ, and XZ = 6, then
XY YZ
4y–1 8x+2 8 = =
4y – 1
8x + 2 8 A B C X Y Z
3y–¾
6x+ 3/2 6
Mr. Chin-Sung Lin

12. Prove Similarity ERHS Math Geometry Mr. Chin-Sung Lin
L22_Similar Triangles
ERHS Math Geometry
Prove Similarity
Mr. Chin-Sung Lin

13. Angle-Angle Similarity Theorem (AA~)
L22_Similar Triangles
ERHS Math Geometry
Angle-Angle Similarity Theorem (AA~)
If two angles of one triangle are congruent to two corresponding angles of another triangle, then the triangles are similar
Given: ABC and XYZ with A  X, and C  Z
Prove: ABC~XYZ X Y Z A B C
Mr. Chin-Sung Lin

14. Example of AA Similarity Theorem
L22_Similar Triangles
ERHS Math Geometry
Example of AA Similarity Theorem
Given: mA = 45 and mD = 45
Prove: ABC~DEC
45o A B C D E
Mr. Chin-Sung Lin

15. Example of AA Similarity Theorem
L22_Similar Triangles
ERHS Math Geometry
Example of AA Similarity Theorem
45o A B C D E
Statements Reasons
1. mA = 45 and mD = Given
2. A  D Substitution property
3. ACB  DCE 3. Vertical angles
4. ABC~DEC AA similarity theorem
Mr. Chin-Sung Lin

16. Side-Side-Side Similarity Theorem (SSS~)
L22_Similar Triangles
ERHS Math Geometry
Side-Side-Side Similarity Theorem (SSS~)
Two triangles are similar if the three ratios of corresponding sides are equal
Given: AB/XY = AC/XZ = BC/YZ
Prove: ABC~XYZ A B C X Y Z
Mr. Chin-Sung Lin

17. Side-Angle-Side Similarity Theorem (SAS~)
L22_Similar Triangles
ERHS Math Geometry
Side-Angle-Side Similarity Theorem (SAS~)
Two triangles are similar if the ratios of two pairs of corresponding sides are equal and the corresponding angles included between these sides are congruent
Given: A  X, AB/XY = AC/XZ
Prove: ABC~XYZ A B C X Y Z
Mr. Chin-Sung Lin

18. Example of SAS Similarity Theorem
L22_Similar Triangles
ERHS Math Geometry
Example of SAS Similarity Theorem
Prove: ABC~DEC
Calculate: DE 16 A B C D E 10 12 8 6 ?
Mr. Chin-Sung Lin

19. Example of SAS Similarity Theorem
L22_Similar Triangles
ERHS Math Geometry
Example of SAS Similarity Theorem
Prove: ABC~DEC
Calculate: DE 16 A B C D E 10 12 8 6 5
Mr. Chin-Sung Lin

20. Triangle Proportionality Theorem
L22_Similar Triangles
ERHS Math Geometry
Triangle Proportionality Theorem
Mr. Chin-Sung Lin

21. Triangle Proportionality Theorem
L22_Similar Triangles
ERHS Math Geometry
Triangle Proportionality Theorem
If a line parallel to one side of a triangle intersects the other two sides, then it divides them proportionally
Given: DE || BC
Prove: AD AE
DB EC D E A B C =
Mr. Chin-Sung Lin

22. Triangle Proportionality Theorem
L23_Triangle Angle Bisector Theorem
ERHS Math Geometry
Triangle Proportionality Theorem
If a line parallel to one side of a triangle intersects the other two sides, then it divides them proportionally
DE || BC
AD AE
DB EC
AD AE DE
AB AC BC D E A B C = = =
Mr. Chin-Sung Lin

23. Converse of Triangle Proportionality Theorem
L22_Similar Triangles
ERHS Math Geometry
Converse of Triangle Proportionality Theorem
If the points at which a line intersects two sides of a triangle divide those sides proportionally, then the line is parallel to the third side
Given: AD AE
DB EC
Prove: DE || BC D E A B C =
Mr. Chin-Sung Lin

24. Example of Triangle Proportionality Theorem
L22_Similar Triangles
ERHS Math Geometry
Example of Triangle Proportionality Theorem
Given: DE || BC, AD = 4, BD = 3, AE = 6
Calculate: CE and BC 8 3 4 6 ? D E A B C
Mr. Chin-Sung Lin

25. Example of Triangle Proportionality Theorem
L22_Similar Triangles
ERHS Math Geometry
Example of Triangle Proportionality Theorem
Given: DE || BC, AD = 4, BD = 3, AE = 6
Calculate: CE and BC 8 3 4 6
4.5 D E A B C 14
Mr. Chin-Sung Lin

26. Example of Triangle Proportionality Theorem
L22_Similar Triangles
ERHS Math Geometry
Example of Triangle Proportionality Theorem
Given: DE || BC, AE = 6, BD = 4, DE = 8, and BC = 12
Calculate: EC and AD 8 4 ? 6 D E A B C 12
Mr. Chin-Sung Lin

27. Example of Triangle Proportionality Theorem
L22_Similar Triangles
ERHS Math Geometry
Example of Triangle Proportionality Theorem
Given: DE || BC, AE = 6, BD = 4, DE = 8, and BC = 12
Calculate: EC and AD 8 4 6 3 D E A B C 12
Mr. Chin-Sung Lin

28. Example of Triangle Proportionality Theorem
L22_Similar Triangles
ERHS Math Geometry
Example of Triangle Proportionality Theorem
Given: DE || BC, BD = 5, AC = 10, DE = 8, and BC = 12
Calculate: AE and AB 8 5 ? 10 D E A B C 12
Mr. Chin-Sung Lin

29. Example of Triangle Proportionality Theorem
L22_Similar Triangles
ERHS Math Geometry
Example of Triangle Proportionality Theorem
Given: DE || BC, BD = 5, AC = 10, DE = 8, and BC = 12
Calculate: AE and AB 8 5 15 10
20/3 D E A B C 12
Mr. Chin-Sung Lin

30. L23_Triangle Angle Bisector Theorem
ERHS Math Geometry
Pythagorean Theorem
Mr. Chin-Sung Lin

31. L24_Right Triangle Altitude Theorem
ERHS Math Geometry
Pythagorean Theorem
A triangle is a right triangle if and only if the square of the length of the longest side is equal to the sum of the squares of the lengths of the other two sides
ABC, mC = 90
if and only if
a2 + b2 = c2 A C B a b c
Mr. Chin-Sung Lin

32. Pythagorean Example - Distance
L24_Right Triangle Altitude Theorem
ERHS Math Geometry
Pythagorean Example - Distance
Find the distance between A and B.
A (5, 3) ?
B(2, 1)
Mr. Chin-Sung Lin

33. Pythagorean Example - Distance
L24_Right Triangle Altitude Theorem
ERHS Math Geometry
Pythagorean Example - Distance
Find the distance between A and B.
A (5, 3) ?
| 3 – 1 | = 2
B(2, 1)
C (5, 1)
| 5 – 2 | = 3
Mr. Chin-Sung Lin

34. Pythagorean Example - Distance
L24_Right Triangle Altitude Theorem
ERHS Math Geometry
Pythagorean Example - Distance
Find the distance between A and B.
A (5, 3)
√13
| 3 – 1 | = 2
B(2, 1)
C (5, 1)
| 5 – 2 | = 3
Mr. Chin-Sung Lin

35. L21_Ratios and Proportions
ERHS Math Geometry
Parallelograms
Mr. Chin-Sung Lin

36. Theorems of Parallelogram
L14_Properties of a Parallelogram
ERHS Math Geometry
Theorems of Parallelogram
Theorem of Dividing Diagonals
Theorem of Opposite Sides
Theorem of Opposite Angles
Theorem of Bisecting Diagonals
Theorem of Consecutive Angles
Mr. Chin-Sung Lin

37. Criteria for Proving Parallelograms
L15_Proving Quadrilaterals Are Parallelograms
ERHS Math Geometry
Criteria for Proving Parallelograms
Parallel opposite sides
Congruent opposite sides
Congruent & parallel opposite sides
Congruent opposite angles
Supplementary consecutive angles
Bisecting diagonals
Mr. Chin-Sung Lin

38. L17_Rectangles Rhombuses and Squares
ERHS Math Geometry
Rectangles
Mr. Chin-Sung Lin

39. L17_Rectangles Rhombuses and Squares
ERHS Math Geometry
Rectangles
A rectangle is a parallelogram containing one right angle A B C D
Mr. Chin-Sung Lin

40. Properties of Rectangle
L17_Rectangles Rhombuses and Squares
ERHS Math Geometry
Properties of Rectangle
The properties of a rectangle
All the properties of a parallelogram
Four right angles (equiangular)
Congruent diagonals A B C D
Mr. Chin-Sung Lin

41. L17_Rectangles Rhombuses and Squares
ERHS Math Geometry
Proving Rectangles
To show that a quadrilateral is a rectangle, by showing that the quadrilateral is equiangular or a parallelogram
that contains a right angle, or
with congruent diagonals
If a parallelogram does not contain a right angle, or doesn’t have congruent diagonals, then it is not a rectangle
Mr. Chin-Sung Lin

42. L17_Rectangles Rhombuses and Squares
ERHS Math Geometry
Rhombuses
Mr. Chin-Sung Lin

43. L17_Rectangles Rhombuses and Squares
ERHS Math Geometry
Rhombus
A rhombus is a parallelogram that has two congruent consecutive sides A B C D
Mr. Chin-Sung Lin

44. L17_Rectangles Rhombuses and Squares
ERHS Math Geometry
Properties of Rhombus A B C D
The properties of a rhombus
All the properties of a parallelogram
Four congruent sides (equilateral)
Perpendicular diagonals
Diagonals that bisect opposite pairs of angles
Mr. Chin-Sung Lin

45. L17_Rectangles Rhombuses and Squares
ERHS Math Geometry
Proving Rhombus
To show that a quadrilateral is a rhombus, by showing that the quadrilateral is equilateral or a parallelogram
that contains two congruent consecutive sides
with perpendicular diagonals, or
with diagonals bisecting opposite angles
If a parallelogram does not contain two congruent consecutive sides, or doesn’t have perpendicular diagonals, then it is not a rectangle
Mr. Chin-Sung Lin

46. L15_Proving Quadrilaterals Are Parallelograms
ERHS Math Geometry
Application Example
ABCD is a parallelogram. AB = 2x + 1, DC = 3x - 11, AD = x + 13
Prove: ABCD is a rhombus A B
2x+1
x+13 D
3x-11 C
Mr. Chin-Sung Lin

47. L15_Proving Quadrilaterals Are Parallelograms
ERHS Math Geometry
Application Example
ABCD is a parallelogram. AB = 2x + 1, DC = 3x - 11, AD = x + 13
Prove: ABCD is a rhombus
x = 12
AB = AD = 25
ABCD is a rhombus A B
2x+1
x+13 D
3x-11 C
Mr. Chin-Sung Lin

48. ERHS Math Geometry
Application Example
ABCD is a parallelogram, AB = 3x - 2, BC = 2x + 2, and CD = x + 6. Show that ABCD is a rhombus A B C D
Mr. Chin-Sung Lin

49. ERHS Math Geometry
Application Example
ABCD is a parallelogram, AB = 3x - 2, BC = 2x + 2, and CD = x + 6. Show that ABCD is a rhombus
x = 4
AB = BC = 10
ABCD is a rhombus A B C D
Mr. Chin-Sung Lin

50. L17_Rectangles Rhombuses and Squares
ERHS Math Geometry
Squares
Mr. Chin-Sung Lin

51. L17_Rectangles Rhombuses and Squares
ERHS Math Geometry
Squares
A square is a rectangle that has two congruent consecutive sides A B C D
Mr. Chin-Sung Lin

52. L17_Rectangles Rhombuses and Squares
ERHS Math Geometry
Squares
A square is a rectangle with four congruent sides (an equilateral rectangle) A B C D
Mr. Chin-Sung Lin

53. L17_Rectangles Rhombuses and Squares
ERHS Math Geometry
Squares
A square is a rhombus with four right angles (an equiangular rhombus) A B C D
Mr. Chin-Sung Lin

54. L17_Rectangles Rhombuses and Squares
ERHS Math Geometry
Squares
A square is an equilateral quadrilateral
A square is an equiangular quadrilateral A B C D
Mr. Chin-Sung Lin

55. L17_Rectangles Rhombuses and Squares
ERHS Math Geometry
Squares
A square is a rhombus
A square is a rectangle A B C D
Mr. Chin-Sung Lin

56. L17_Rectangles Rhombuses and Squares
ERHS Math Geometry
Properties of Square
The properties of a square
All the properties of a parallelogram
All the properties of a rectangle
All the properties of a rhombus A B C D
Mr. Chin-Sung Lin

57. L17_Rectangles Rhombuses and Squares
ERHS Math Geometry
Proving Squares
To show that a quadrilateral is a square, by showing that the quadrilateral is a
rectangle with a pair of congruent consecutive sides, or
a rhombus that contains a right angle
Mr. Chin-Sung Lin

58. ERHS Math Geometry
Application Example
ABCD is a square, mA = 4x - 30, AB = 3x + 10 and BC = 4y. Solve x and y A B C D
Mr. Chin-Sung Lin

59. ERHS Math Geometry
Application Example
ABCD is a square, mA = 4x - 30, AB = 3x + 10 and BC = 4y. Solve x and y
4x – 30 = 90
x = 30
y = 25 A B C D
Mr. Chin-Sung Lin

60. L17_Rectangles Rhombuses and Squares
ERHS Math Geometry
Trapezoids
Mr. Chin-Sung Lin

61. L18_Trapezoids
ERHS Math Geometry
Trapezoids
A trapezoid is a quadrilateral that has exactly one pair of parallel sides
The parallel sides of a trapezoid are called bases. The nonparallel sides of a trapezoid are the legs A B C D
Upper base
Lower base
Leg
Mr. Chin-Sung Lin

62. L18_Trapezoids
ERHS Math Geometry
Isosceles Trapezoids
A trapezoid whose nonparallel sides are congruent is called an isosceles trapezoid A B C D
Upper base
Lower base
Leg
Mr. Chin-Sung Lin

63. Properties of Isosceles Trapezoids
L18_Trapezoids
ERHS Math Geometry
Properties of Isosceles Trapezoids
The properties of a isosceles trapezoid
Base angles are congruent
Diagonals are congruent
The property of a trapezoid
Median is parallel to and average of the bases
Mr. Chin-Sung Lin

64. L18_Trapezoids
ERHS Math Geometry
Proving Trapezoids
To prove that a quadrilateral is a trapezoid, show that two sides are parallel and the other two sides are not parallel
To prove that a quadrilateral is not a trapezoid, show that both pairs of opposite sides are parallel or that both pairs of opposite sides are not parallel
Mr. Chin-Sung Lin

65. Proving Isosceles Trapezoids
L18_Trapezoids
ERHS Math Geometry
Proving Isosceles Trapezoids
To prove that a trapezoid is an isosceles trapezoid, show that one of the following statements is true:
The legs are congruent
The lower/upper base angles are congruent
The diagonals are congruent
Mr. Chin-Sung Lin

66. Numeric Example of Trapezoids
L18_Trapezoids
ERHS Math Geometry
Numeric Example of Trapezoids
Isosceles Trapezoid ABCD, AB || CD and AD  BC
Solve for x and y A B C D
2xo xo
3yo
Mr. Chin-Sung Lin

67. Numeric Example of Trapezoids
L18_Trapezoids
ERHS Math Geometry
Numeric Example of Trapezoids
Isosceles Trapezoid ABCD, AB || CD and AD  BC
Solve for x and y
x = 60
y = 20 A B C D
2xo xo
3yo
Mr. Chin-Sung Lin

68. L21_Ratios and Proportions
ERHS Math Geometry
Q & A
Mr. Chin-Sung Lin

69. L21_Ratios and Proportions
ERHS Math Geometry
The End
Mr. Chin-Sung Lin