Name:______________________________
Triangle Theorems
Date:_____ Period:____
Ms. Anderle
Triangle Theorems:
The exterior angle of a triangle is an angle that is adjacent and supplementary to
one of the angles of a triangle.
Exterior Angle Theorem:
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Example 1:
In the diagram below, if m<B=43˚ and m<C=115˚, what is the measure of exterior
<BAD?
Example 2:
Use the diagram below. If mC  x  10 , mD  3x  7 , and mABD  2x  25 ,
find the measure of ABD .
Triangle Inequality Theorems:
Triangle Inequality Theorem:
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In addition, in a triangle the smallest side is opposite the smallest angle.
Therefore, in a triangle, the largest side is opposite the __________________.
If two angles of a triangle are equal, then the sides opposite them are also equal.
Example 1:
Tell whether the given lengths may be the measures of the sides of a triangle:
a) 6 in., 4 in., 10 in.
b) 6 in., 4 in., 12 in.
c) 6 in., 4 in., 8 in.
Example 2:
In ΔABC, mC  125 , mB  35 . Which is the shortest side of the triangle.
*hint* draw a picture.
Example 3:
In triangle ABC angle C is 60˚ and AB is greater than AC. Angle B must be:
a) 60˚
b) greater than 60˚
c) less than 60˚
Examples:
Complete all work on a separate sheet of paper. Show all work.
1. The lengths of two sides of a triangle are 3 in. and 6 in. The length of the
third side may be:
a. 3 in.
b. 6 in .
c. 9 in.
d. 12 in.
2. Tell which of the following number triples may be used as the lengths of the
sides of a triangle:
a. (7,8,9)
b. (3,5,8)
c. (8,5,2)
d. (3,10,6)
e. (6,9,10)
3. In ΔABC, mC  90 and mB  35 . Name the shortest side of this
triangle.
4. In ΔABC, mA  74 and mB  58 . Which is the longest side of the
triangle?
5. If in ΔRST, mR  71 and mS  37 , then
a. ST  RS
b. RS  RT
c. RS  ST
d. RT  ST
6. In ΔDEF, mD  2x  4 , mE  6x  58 . The measure of an exterior angle
at F is represented by 5x.
a. Find the value of x.
b. Show that ΔDEF is a right triangle.
7. Side AC of ΔABC is extended through C to D. BCD measure 108 and the
mA is twice the mB . Triangle ABC is:
a. right
b. obtuse
c. isosceles
d. scalene
8. Find the measure of the vertex angle of an isosceles triangle if either of the
exterior angles formed by extending the base measures 112˚.
9. In ΔRST, if mS  90 and mT  30 , find the measure of an exterior
angle at R.
10. In ΔABC, AC  BC . The measure of an exterior angle at vertex C is
represented by 5x + 10. If mA  30 , find the value of x.
11. In ΔABC, the mA  2x  15 and mB  x  5 , if the measure of an
exterior angle at C is 148, what is the value of x?
12. In ΔJKL, the mJ  2x  27 and mK  2x  11 , if the measure of an
exterior angle at L is 100, what is the measure of angles J and K?
13. Which of the following sets of numbers cannot represent the lengths of the
sides of a triangle?
a. {9, 40, 41}
b. {7, 7, 3}
c. {4, 5, 1}
d. {6, 6, 6}
14. In ΔPQR, mP  51 and m Q  57 . Which expression is true?
a. QR  PQ
b. PQ  QR
c. PR  PQ
d. PQ  QR
15. The shortest distance between city A and city B is 150 miles. The shortest
distance between city B and city C is 350 miles. Which could be the shortest
distance, in miles, between city A and city C?
a. 175
b. 200
c. 250
d. 300
16. John wants to build a triangular pen for his pet rabbit. He has three lengths
of boards already cut that measure 7 feet, 8 feet, and 16 feet. Explain why
John cannot construct a pen in the shape of a triangle with sides of 7 feet,
8 feet, and 16 feet.
17. In ΔSAT, mA  53 and mT  76 . Which side of ΔSAT is the shortest?
Give a reason for your answer.
18. A box contains one 2-inch rod, one 3-inch rod, one 4-inch rod, and one 5-inch
rod. What is the maximum number of different triangles that can be made
using the full lengths of the rods as sides?