Section 7.1 – Triangle Application Theorems
description
Transcript of 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>.