chapter 10

© 2010 Pearson Prentice Hall. All rights reserved.

CHAPTER 10

Geometry

© 2010 Pearson Prentice Hall. All rights reserved. 2

10.2

Triangles


Objectives

1. Solve problems involving angle relationships in triangles.

2. Solve problems involving similar triangles.

3. Solve problems using the Pythagorean Theorem.

3


Triangle

• A closed geometric figure that has three sides, all of which lie on a flat surface or plane.

• Closed geometric figures – If you start at any point and trace along the sides,

you end up at the starting point.

4


Euclid’s Theorem

Theorem: A conclusion that is proved to be true through deductive reasoning.

Euclid’s assumption: Given a line and a point not on the line, one and only one line can be drawn through the given point parallel to the given line.

5


Euclid’s Theorem (cont.)

Euclid’s Theorem: The sum of the measures of the three angles of any triangle is 180º.

6


Euclid’s Theorem (cont.)

Proof:m1 = m2 and m3 = m4 (alternate interior angles)Angles 2, 5, and 4 form a straight angle (180º)

Therefore, m1 + m5 + m3 = 180º .

7


Two-Column Proof A

Given: ABC△Prove: A + B + ACB = 180°∠ ∠ ∠

Statements

1. Through C, draw line m || to line AB.

2. ∠1 + 2 + 3 = 180°∠ ∠3. ∠1 = A; 3 = B∠ ∠ ∠4. ∠A + ACB + B = 180°∠ ∠

Reasons

1. Parallel postulate

2. Definition of straight angle

3. If 2 || lines, alt. int. ’s are equal∠4. Substitution of equals axiom

1

A B

Cm1

23

Theorem: The sum of ’s in a equals 180∠ △ °.


Exterior ∠ Theorem

Theorem: An exterior angle of a triangle equals the sum of the non-adjacent interior angles.

∠CBD = A + C ∠ ∠Since CBD + ABC = 180 -- def. of supplementary ∠ ∠ s∠

A + C + ABC = 180 -- sum of ∠ ∠ ∠ ’s interior △ s∠ A + C + ABC = CBD + ABC -- axiom∠ ∠ ∠ ∠ ∠ A + C = CBD -- axiom∠ ∠ ∠

1

A

C

B D


Base s of Isosceles ∠ △

• Theorem: Base s of an isosceles are equal∠ △

• Given: Isosceles ABC, which AC = BC△Prove: A = B∠ ∠

1

A

C

B


Two-Column Proof B

Given: Circle C with A, B, D on the circleProve: ACB = 2 • ADB∠ ∠

Statements1. Circle C, with central C, ∠

inscribed D∠2. Draw line CD, intersecting circle

at F

3. ∠y = a + b = 2 a∠ ∠ ∠x = c + d = 2 c∠ ∠ ∠ ∠

4. ∠x + y = 2( a + c)∠ ∠ ∠5. ∠ACB = 2 ADB∠

Reasons1. Given

2. 2 points determine a line (postulate)

3. Exterior ; Isosceles angles∠ △

4. Equal to same quantity (axiom)

5. Substitution (axiom)

1

Theorem: An formed by 2 radii subtending a chord is 2 x ∠an inscribed subtending the same chord. ∠ A

B

D

C

F

c

x

b

y

a

d


Example 1: Using Angle Relationships in Triangles

Find the measure of angle A for the triangle ABC.

Solution:

mA + mB + mC =180º

mA + 120º + 17º = 180º

mA + 137º = 180º

mA = 180º − 137º = 43º

12


Example 2: Using Angle Relationships in Triangles

Find the measures of angles 1 through 5.

Solution:m1+ m2 + 43º =180º

90º + m2 + 43º = 180º

m2 + 133º = 180º

m2 = 47º

13


Example 2 continued

m3 + m4+ 60º =180º 47º + m4 + 60º = 180ºm4 = 180º − 107º = 73º

m4 + m5 = 180º 73º + m5 = 180°

m5 = 107º

14


Triangles and Their Characteristics

15


Similar Triangles

1

A

B

C

Y

ZX

△ABC ~ XYZ iff△• Corresponding angles

are equal• Corresponding sides are

proportional

• ∠A = X; B = Y; C = Z∠ ∠ ∠ ∠ ∠• AB/XY = BC/YZ = AC/XZ

• If 2 corresponding s of 2 s ∠ △are equal, then s are similar. △


Example 1

• Given:– AB || DE– AB = 25; CE = 8

• Find AC • Solution• x 25

--- = ----- 8 12

• 8(25) 25 50x = ------- = 2 • ---- = ----- ≈ 16.66 12 3 3

1

A

E

DC

B

25

8

12

x


Example 2

• How can you estimate the height of a building, if you know your own height (on a sunny day)?

1


Example 2

1

x

400

610

6 10--- = ------ x 400

6 • 400 = 10 • x

x = 240


Similar Triangles

• Similar figures have the same shape, but not necessarily the same size.

• In similar triangles, the angles are equal but the sides may or may not be the same length.

• Corresponding angles are angles that have the same measure in the two triangles.

• Corresponding sides are the sides opposite the corresponding angles.

20


Similar Triangles (continued)

CB and FE

Triangles ABC and DEF are similar:

Corresponding Angles Corresponding Sides

Angles A and D Sides

Angles C and F Sides

Angles B and E Sides AB and DE

AC and DF

21


Example 3: Using Similar Triangles

Find the missing length x.

Solution: Because the triangles are similar, their corresponding sides are proportional:

22


Example 3: Using Similar Triangles

5 7

85 8 7

5 56

5 511.2

xx

x

x

Find the missing length x.

The missing length of x is 11.2 inches.23


Pythagorean Theorem

The sum of the squares of the lengths of the legs of a right triangle equals the square of the length of the hypotenuse.

If the legs have lengths a and b and the hypotenuse has length c, then

a² + b² = c²

24


Example 5: Using the Pythagorean Theorem

Find the length of the hypotenuse c in this right triangle:

Solution:

Let a = 9 and b = 122 2 2

2 2 2

2

2

9 12

81 144

225

225 15

c a b

c

c

c

c

C

25

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