Post on 04-Jan-2016
Special Topics
Eleanor Roosevelt High School
Chin-Sung Lin
Similar Triangles
Mr. Chin-Sung Lin
ERHS Math Geometry
Definition of Similar Triangles
Mr. Chin-Sung Lin
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)
ERHS Math Geometry
Example of Similar Triangles
Mr. Chin-Sung Lin
A X, B Y, C Z
AB = 6, BC = 8, and CA = 10
XY = 3, YZ = 4 and ZX = 5
Show that ABC~XYZX
Y Z
3
4
5
A
B C
6
8
10
ERHS Math Geometry
Example of Similar Triangles
Mr. Chin-Sung Lin
A X, B Y, C Z
AB BC CA 2
XY YZ ZX 1
Therefore ABC~XYZ
= = =
ERHS Math Geometry
X
Y Z
3
4
5
A
B C
6
8
10
Example of Similar Triangles
Mr. Chin-Sung Lin
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 ?
?
?
ERHS Math Geometry
Example of Similar Triangles
Mr. Chin-Sung Lin
Assume x, y, and z are the sides of the larger triangle, then
x 5 y 5 z 5
4 2 8 2 6 2
4
6
8 x = 10
z = 15
y = 20
ERHS Math Geometry
= = =
Example of Similar Triangles
Mr. Chin-Sung Lin
In ABC, AB = 9, BC = 15, AC = 18.
If ABC~XYZ, and XZ = 12, find XY and YZ
ERHS Math Geometry
X
Y Z
?
?
129
15
18
A
B C
Example of Similar Triangles
Mr. Chin-Sung Lin
Since ABC~XYZ, and XZ = 12, then
XY YZ 12
9 15 18
X
Y Z
6
10
129
15
18
A
B C
ERHS Math Geometry
= =
Example of Similar Triangles
Mr. Chin-Sung Lin
In ABC, AB = 4y – 1, BC = 8x + 2, AC = 8.
If ABC~XYZ, and XZ = 6, find XY and YZ
ERHS Math Geometry
X
Y Z
?
?
64y – 1
8x + 2
8
A
B C
Example of Similar Triangles
Mr. Chin-Sung Lin
Since ABC~XYZ, and XZ = 6, then
XY YZ 6
4y–1 8x+2 8
X
Y Z
3y–¾
6x+ 3/2
64y – 1
8x + 2
8
A
B C
ERHS Math Geometry
= =
Prove Similarity
Mr. Chin-Sung Lin
ERHS Math Geometry
Angle-Angle Similarity Theorem (AA~)
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Example of AA Similarity Theorem
Mr. Chin-Sung Lin
Given: mA = 45 and mD = 45
Prove: ABC~DEC
45o
A
B
C 45o
D
E
ERHS Math Geometry
Example of AA Similarity Theorem
Mr. Chin-Sung Lin
Statements Reasons
1. mA = 45 and mD = 45 1. Given
2. A D 2. Substitution property
3. ACB DCE 3. Vertical angles
4. ABC~DEC 4. AA similarity theorem
45oA
B
C45o
D
E
ERHS Math Geometry
Side-Side-Side Similarity Theorem (SSS~)
Mr. Chin-Sung Lin
Two triangles are similar if the three ratios of corresponding sides are equal
Given: AB/XY = AC/XZ = BC/YZ
Prove: ABC~XYZ
ERHS Math Geometry
X
Y Z
A
B C
Side-Angle-Side Similarity Theorem (SAS~)
Mr. Chin-Sung Lin
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
X
Y Z
A
B C
ERHS Math Geometry
Example of SAS Similarity Theorem
Mr. Chin-Sung Lin
Prove: ABC~DEC
Calculate: DE
16A
B
CD
E
1012
8
6 ?
ERHS Math Geometry
Example of SAS Similarity Theorem
Mr. Chin-Sung Lin
Prove: ABC~DEC
Calculate: DE
16A
B
CD
E
1012
8
6 5
ERHS Math Geometry
Triangle Proportionality Theorem
Mr. Chin-Sung Lin
ERHS Math Geometry
Triangle Proportionality Theorem
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Triangle Proportionality Theorem
Mr. Chin-Sung Lin
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 BCD E
A
B C
= =
ERHS Math Geometry
=
Converse of Triangle Proportionality Theorem
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Example of Triangle Proportionality Theorem
Mr. Chin-Sung Lin
Given: DE || BC, AD = 4, BD = 3, AE = 6
Calculate: CE and BC
8
3
4 6
?D E
A
BC
?
ERHS Math Geometry
Example of Triangle Proportionality Theorem
Mr. Chin-Sung Lin
Given: DE || BC, AD = 4, BD = 3, AE = 6
Calculate: CE and BC
8
3
4 6
4.5D E
A
BC
14
ERHS Math Geometry
Example of Triangle Proportionality Theorem
Mr. Chin-Sung Lin
Given: DE || BC, AE = 6, BD = 4, DE = 8, and BC = 12
Calculate: EC and AD
8
4
? 6
?D E
A
BC
12
ERHS Math Geometry
Example of Triangle Proportionality Theorem
Mr. Chin-Sung Lin
Given: DE || BC, AE = 6, BD = 4, DE = 8, and BC = 12
Calculate: EC and AD
8
4
8 6
3D E
A
BC
12
ERHS Math Geometry
Example of Triangle Proportionality Theorem
Mr. Chin-Sung Lin
Given: DE || BC, BD = 5, AC = 10, DE = 8, and BC = 12
Calculate: AE and AB
8
5
?
10?
D E
A
BC
12
ERHS Math Geometry
Example of Triangle Proportionality Theorem
Mr. Chin-Sung Lin
Given: DE || BC, BD = 5, AC = 10, DE = 8, and BC = 12
Calculate: AE and AB
8
5
15
1020/3
D E
A
BC
12
ERHS Math Geometry
Pythagorean Theorem
Mr. Chin-Sung Lin
ERHS Math Geometry
Pythagorean Theorem
Mr. Chin-Sung Lin
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, mC = 90
if and only if
a2 + b2 = c2
ERHS Math Geometry
A
C Ba
bc
Pythagorean Example - Distance
Mr. Chin-Sung Lin
Find the distance between A and B.
ERHS Math Geometry
A (5, 3)
B(2, 1)
?
Pythagorean Example - Distance
Mr. Chin-Sung Lin
Find the distance between A and B.
ERHS Math Geometry
A (5, 3)
B(2, 1)C (5, 1)
| 3 – 1 | = 2
| 5 – 2 | = 3
?
Pythagorean Example - Distance
Mr. Chin-Sung Lin
Find the distance between A and B.
ERHS Math Geometry
A (5, 3)
B(2, 1)
√13
C (5, 1)
| 3 – 1 | = 2
| 5 – 2 | = 3
Parallelograms
Mr. Chin-Sung Lin
ERHS Math Geometry
Theorems of Parallelogram
Mr. Chin-Sung Lin
Theorem of Dividing Diagonals
Theorem of Opposite Sides
Theorem of Opposite Angles
Theorem of Bisecting Diagonals
Theorem of Consecutive Angles
ERHS Math Geometry
Criteria for Proving Parallelograms
Mr. Chin-Sung Lin
Parallel opposite sides
Congruent opposite sides
Congruent & parallel opposite sides
Congruent opposite angles
Supplementary consecutive angles
Bisecting diagonals
ERHS Math Geometry
Rectangles
Mr. Chin-Sung Lin
ERHS Math Geometry
Rectangles
Mr. Chin-Sung Lin
A rectangle is a parallelogram containing one right angle
A B
CD
ERHS Math Geometry
Properties of Rectangle
Mr. Chin-Sung Lin
The properties of a rectangle
All the properties of a parallelogram
Four right angles (equiangular)
Congruent diagonals A B
CD
ERHS Math Geometry
Proving Rectangles
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Rhombuses
Mr. Chin-Sung Lin
ERHS Math Geometry
Rhombus
Mr. Chin-Sung Lin
A rhombus is a parallelogram that has two congruent consecutive sides
A
B
C
D
ERHS Math Geometry
Properties of Rhombus
Mr. Chin-Sung Lin
The properties of a rhombus
All the properties of a parallelogram
Four congruent sides (equilateral)
Perpendicular diagonals
Diagonals that bisect opposite pairs of angles
A
B
C
D
ERHS Math Geometry
Proving Rhombus
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Application Example
Mr. Chin-Sung Lin
ABCD is a parallelogram. AB = 2x + 1, DC = 3x - 11, AD = x + 13
Prove: ABCD is a rhombusA B
D C
2x+1
3x-11
x+13
ERHS Math Geometry
Application Example
Mr. Chin-Sung Lin
ABCD is a parallelogram. AB = 2x + 1, DC = 3x - 11, AD = x + 13
Prove: ABCD is a rhombus
x = 12AB = AD = 25ABCD is a rhombus
A B
D C
2x+1
3x-11
x+13
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
ERHS Math Geometry
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Mr. Chin-Sung Lin
Squares
Mr. Chin-Sung Lin
ERHS Math Geometry
Squares
Mr. Chin-Sung Lin
A square is a rectangle that has two congruent consecutive sides
A B
CD
ERHS Math Geometry
Squares
Mr. Chin-Sung Lin
A square is a rectangle with four congruent sides (an equilateral rectangle)
ERHS Math Geometry
A B
CD
Squares
Mr. Chin-Sung Lin
A square is a rhombus with four right angles (an equiangular rhombus)
ERHS Math Geometry
A B
CD
Squares
Mr. Chin-Sung Lin
A square is an equilateral quadrilateral
A square is an equiangular quadrilateral
ERHS Math Geometry
A B
CD
Squares
Mr. Chin-Sung Lin
A square is a rhombus
A square is a rectangle
ERHS Math Geometry
A B
CD
Properties of Square
Mr. Chin-Sung Lin
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
CD
ERHS Math Geometry
Proving Squares
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Application Example
ABCD is a square, mA = 4x - 30, AB = 3x + 10 and BC = 4y. Solve x and y
A B
CD
ERHS Math Geometry
Mr. Chin-Sung Lin
Application Example
ABCD is a square, mA = 4x - 30, AB = 3x + 10 and BC = 4y. Solve x and y
4x – 30 = 90
x = 30
y = 25
A B
CD
ERHS Math Geometry
Mr. Chin-Sung Lin
Trapezoids
Mr. Chin-Sung Lin
ERHS Math Geometry
Trapezoids
Mr. Chin-Sung Lin
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
CD
Upper base
Lower base
LegLeg
ERHS Math Geometry
Isosceles Trapezoids
Mr. Chin-Sung Lin
A trapezoid whose nonparallel sides are congruent is called an isosceles trapezoid
ERHS Math Geometry
A B
CD
Upper base
Lower base
LegLeg
Properties of Isosceles Trapezoids
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Trapezoids
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Isosceles Trapezoids
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Numeric Example of Trapezoids
Mr. Chin-Sung Lin
Isosceles Trapezoid ABCD, AB || CD and AD BC
Solve for x and yA B
CD
2xo
xo 3yo
ERHS Math Geometry
Numeric Example of Trapezoids
Mr. Chin-Sung Lin
Isosceles Trapezoid ABCD, AB || CD and AD BC
Solve for x and y
x = 60
y = 20
A B
CD
2xo
xo 3yo
ERHS Math Geometry
Q & A
Mr. Chin-Sung Lin
ERHS Math Geometry
The End
Mr. Chin-Sung Lin
ERHS Math Geometry