Download Theorems – Angle-Side Relationships in Triangles

5-5
-- the positions of the longest and shortest sides of a triangle are related to the positions of the
largest and smallest angles
Theorems – Angle-Side Relationships
in Triangles
Hypothesis
If two sides of a triangle are not
congruent, then the larger angle is
opposite the longer side. (In ∆, larger
is opp. longer side)
If two angles of a triangle are not
congruent, then the longer side is
opposite the larger angle. (In ∆,
longer side is opp. larger )
Conclusion
m C>m A
XY > XZ
Ex. 1:
A. Write the angles in order from smallest to largest.
1.)
2.)
________, ________, ________
________, ________, ________
B. Write the sides in order from shortest to longest.
1.)
2.)
________, ________, ________
________, ________, ________
-- a triangle is formed by three segments – but can every set of 3 segments form a triangle? ____
Triangle Inequality Theorem:
The sum of any two side lengths of a triangle is greater than the third side length.
AB + BC > AC
BC + AC > AB
AC + AB > BC
Ex. 2: Tell whether a triangle can have sides with the given lengths.
A. 7, 10, 19
B. 8, 13, 21
C. 2.3, 3.1, 4.6
D. 6.2, 7, 9
E. n + 6, n2 – 1, 3n, when n = 4
F. t – 2, 4t, t2 + 1, when t = 4
Ex. 3: The lengths of two sides of a triangle are 8 in. and 13 in. Find the range of possible
lengths for the third side (we’ll call it “s”).
Add the two known side lengths:
 the third side must be shorter than this
Subtract the two known side lengths:
 the third side must be longer than this
Combine these two numbers for your answer as a range of possible length: ____ in. < s < ____ in.
Ex. 4: The figure shows the approximate distances between cities of California. What is the
range of distances from San Francisco to Oakland?
Add the two known side lengths:
Subtract the two known side lengths:
 the third side must be shorter than this
 the third side must be longer than this
Combine these two numbers for your answer as a range of possible length: ____ mi < s < ____ mi