Download 1. The Converse of the Pythagorean Theorem 2. Pythagorean

Pythagorus Converse
GEOMETRY
NAME_________________________
DATE __________ Per.___________
Trigonometry
1. The Converse of the Pythagorean Theorem
GIVEN: ∆ABC with CA = b, BC = a, and AB = c and a 2 + b2 = c2 . PROVE: ∆ABC is a right triangle.
Let ∆XYZ be a right triangle with right angle Z and sides YZ = a and ZX = b.
B
a
C
a) Then by the Pythagorean Theorem, z2 =
a 2 + b2 = c2
c
b) From the GIVEN, a 2 + b2 =
b
2
c) By the Transitive Property, c =
A
2
2
e) Since (− z) = ( −c ) , why can’t part d)’s answer be –c = –z?
Y
a
Z
d) Taking the square root of both sides of part c gives:
f) What justifies ∆XYZ ≅ ∆ABC?
z
b
X
g) Then ∠Z ≅ ∠___ because __ __ __ __ __.
h) So ∠C is right and from the Definition of right triangles,
2. Pythagorean Inequalities
b) Let BD = YZ, DA = ZX, and AB < XY.
B
Y
a
d
a
c
b
b
D
A
Z
Then by the Side-Side-Side
Triangle Inequality, m∠D < ?
2
If c > d, then c > ___ .
2
2
2
c) If c > a + b , then ∆XYZ is:
2
2
2
a) If d = a + b , then ∆ABD is:
d) Let BD = YZ, DA = ZX, but AB > XY.
B
Y
a
X
d
a
b
D
A
Then m∠Z < m∠D by ?
Z
c
b
2
If c < d, then ___ < d .
2
2
2
e) If c < a + b , then ∆XYZ is:
Use the SAS ∆ Inequality to complete each of the converse statements with =, <, or >.
a 2 + b2 ___ c 2 .
b
C
f) If ∆ABC is right,
.
f)
If
∆ABC
is
acute,
then
A
2
2
2
a
2
2
2
a
+
b
___
c
c
then
g) If ∆ABC is obtuse, then a + b ___ c
B
3. Classify each ∆ as right, acute, or obtuse.
a) a = 3, b = 4, c = 5
b) a = 4, b = 5, c = 6
c) a = 2, b = 3, c = 4
4. Find the missing integer to make a Pythagorean
b) 17, 15, and ?
c) 7, 25, and ?
triple.a) 5, 12, and ?
5. Find x to
make a right
triangle.
a)
b)
2 x
1
x
3
1
X