Section 7.1 – Triangle Application Theorems

Stephanie Lalos

Theorem 50The sum of measures of the three angles of a

triangle is 180o

A

B

C

60

4080

180 CmBmAm o

ProofAccording to the parallel postulate, there exists exactly one line through point A parallel to BCBecause of the straight angle, we know that

Since and we may substitute to obtain Hence,

180321 mm

B

A

C

321

o

B1 ) .int (3 saltlinesbyC1802 CB o

180 CmBmAm o

Other Proofs

Right triangles are used to prove the sum of the angles of a triangle in a youtube video that can be seen here.

LemmaIf ABCD is a quadrilateral and <)CAB = <)DCA then AB and DC are parallel. ProofAssume to the contrary that AB and DC are not parallel.Draw a line trough A and B and draw a line trough D and C.These lines are not parallel so they cross at one point. Call this point E. Notice that <)AEC is greater than 0.Since <)CAB = <)DCA, <)CAE + <)ACE = 180 degrees.Hence <)AEC + <)CAE + <)ACE is greater than 180 degrees.Contradiction. This completes the proof. DefinitionTwo Triangles ABC and A'B'C' are congruent if and only if|AB| = |A'B'|, |AC| = |A'C'|, |BC| = |B'C'| and,<)ABC = <)A'B'C', <)BCA = <)B'C'A', <)CAB = <)C'A'B'.

DefinitionExterior angle – an angle of a polygon that is adjacent to

and supplementary to an interior angle of the polygonExamples - 1 is an exterior angle to the below triangles

1

1For alternative exterior angle help visit…

Regents Prep

Theorem 51The measure of an exterior angle of a triangle is equal to

the sum of the measures of the remote interior angles

1

CA

B

AmBmm 1

Theorem 52A segment joining the midpoints of two sides of a triangle

is parallel to the third side, and its length is one-half the length of the third side. (Midline Theorem)

A

B

C

D E

Given: D & E are midpointsTherefore, AD DB & BE EC

Prove: a. DE AC b. DE = (AC)

2

1

A

B

C

D E

CEFBED (vertical angles are congruent)

BED CEF (SAS)

(CPCTC)

Extend DE through to a point F so that EF DE. F is now established, so F and C determine FC.

F

FCEB FC DA (alt. int. Lines)

FC DA (transitive)

s

DFCA is a parallelogram, one pair of opposite sides is both congruent and parallel, therefore, DF ACOpposite sides of a parallelogram are congruent, so DF=AC, since EF=DE,DE= (EF) and by substitution DE= (AC).

2

1

2

1

Sample Problems

x + 100 + 60 = 180 55 + 80 + y = 180 x + y + z = 180

x + 160 = 180 135 + y = 180 20 + 45 + z = 180

x = 20 y = 45 z = 115

80

55

100

60y

z

x

substitution

The measures of the three angles of a triangle are in the ratio 2:4:6. Find the measure of the smallest angle.

2x

4x

6x

2x + 4x + 6x = 180 12x = 180 x = 15 2x = 30

80

BC

A

xx

yy

DBisectors BD and CD meet at DLet ABC = 2x and ABC = 2y

In EBC,x + y + = 18050 + = 180 (substitution) = 130

In ABC, 2x + 2y + 80 = 180 2x + 2y = 100 x + y = 50

EmEmEm

1A

B

C

, and the measure of is twice that of1501 o B A

Let = x and = (2x)A Boo

According to theorem 51, is equal to +1 B A

Find the measure of each angle of the triangle.

150 = x + 2x150 = 3x 50 = x

ABBCA

= 50o

= 100o

= 30o

Practice Problems

0

Find the measures of the numbered angles.

47o

86o

1

69

71o

2

3

4

o

5 65o

40o1 25o

2

95

3

1.

2.

o

D

E

F

16

Find: GH

G H

3.

A

B

C

D E

70o

Find: , , and

Bm BDEmBEDm

4.

Three triangles are in the ratio 3:4:5.Find the measure of the largest angle.

5.

A

B

C

D

50

Find: Dm

o

7.

4x+6

2x+4

x+24

6.

Find: RmQ

R

S

8. Always, Sometimes, Nevera. The acute angles of a right triangle are complementary.b. A triangle contains two obtuse angles.c. If one angle of an isosceles triangle is 60 , it is

equilateral.d. The supplement of one of the angles in a triangle is

equal in measure to the sum of the other two angles.

o

Answer Key1. 1 = 47 5. 75

2 = 40

3 = 93 6. 48

4 = 40

5 = 140 7. 115

2. 1 = 75 8. a. A

2 = 85 b. N

3 = 70 c. A

d. A

3. GH = 8

4. = 20

= 90

= 70

BmBDEmBEDm

Works Cited

"Exterior Angles of a Triangle." Regents Prep. 2008. 29 May 2008

<http://regentsprep.org/REgents/math/triang/LExtAng.htm>. Rhoad, Richard, George Milauskas, and Robert Whipple.

Geometry for Enjoyment and Challenge. Boston: McDougal Littell, 1991.

"Triangle." Apronus. 29 May 2008 <http://www.apronus.com/geometry/triangle.htm>.