![Page 1: A B C 12 We know ∠B = ∠C S TU 1214 We could write a proof to show ∠T ≠∠U *We could also prove that m ∠T > m ∠U, BUT theorem 1 tells us that!](https://reader035.vdocuments.mx/reader035/viewer/2022080915/56649db05503460f94a9e0b4/html5/thumbnails/1.jpg)
Inequalities for Triangles
![Page 2: A B C 12 We know ∠B = ∠C S TU 1214 We could write a proof to show ∠T ≠∠U *We could also prove that m ∠T > m ∠U, BUT theorem 1 tells us that!](https://reader035.vdocuments.mx/reader035/viewer/2022080915/56649db05503460f94a9e0b4/html5/thumbnails/2.jpg)
What we already know…
A
BC
1212
We know ∠B = ∠C
S
TU
1214
We could write a proof to show ∠T ≠∠U
*We could also prove that m∠T > m ∠U, BUT theorem 1 tells us that!
![Page 3: A B C 12 We know ∠B = ∠C S TU 1214 We could write a proof to show ∠T ≠∠U *We could also prove that m ∠T > m ∠U, BUT theorem 1 tells us that!](https://reader035.vdocuments.mx/reader035/viewer/2022080915/56649db05503460f94a9e0b4/html5/thumbnails/3.jpg)
A B C
Complete each statement by writing <, =, or >.
a. AC _____ AB + BCb. AC _____ ABc. BC _____ AC
Example 1 :
![Page 4: A B C 12 We know ∠B = ∠C S TU 1214 We could write a proof to show ∠T ≠∠U *We could also prove that m ∠T > m ∠U, BUT theorem 1 tells us that!](https://reader035.vdocuments.mx/reader035/viewer/2022080915/56649db05503460f94a9e0b4/html5/thumbnails/4.jpg)
Theorem 1
If one side of a triangle is longer than a second side, then the angle opposite the longer side is larger than the angle opposite the shorter side.
![Page 5: A B C 12 We know ∠B = ∠C S TU 1214 We could write a proof to show ∠T ≠∠U *We could also prove that m ∠T > m ∠U, BUT theorem 1 tells us that!](https://reader035.vdocuments.mx/reader035/viewer/2022080915/56649db05503460f94a9e0b4/html5/thumbnails/5.jpg)
Example 2:
Name the largest angle and the smallest angle of the triangle.
V
U
W
9
10
11
∠U is the LARGEST angle because it is opposite the LONGEST side (WV)
∠W is the SMALLEST angle because it is opposite the SHORTEST side (VU)
![Page 6: A B C 12 We know ∠B = ∠C S TU 1214 We could write a proof to show ∠T ≠∠U *We could also prove that m ∠T > m ∠U, BUT theorem 1 tells us that!](https://reader035.vdocuments.mx/reader035/viewer/2022080915/56649db05503460f94a9e0b4/html5/thumbnails/6.jpg)
Theorem 2
If one angle of a triangle is larger than a second angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.
![Page 7: A B C 12 We know ∠B = ∠C S TU 1214 We could write a proof to show ∠T ≠∠U *We could also prove that m ∠T > m ∠U, BUT theorem 1 tells us that!](https://reader035.vdocuments.mx/reader035/viewer/2022080915/56649db05503460f94a9e0b4/html5/thumbnails/7.jpg)
Example 3:
Name the longest and shortest side of the triangle.
A
B
C
90º
30º
**Always find the other angle BEFORE you answer the question!!
Side AB is the LONGEST side because it is opposite the LARGEST angle (∠C)
Side CB is the SHORTEST side because it is opposite the SMALLEST angle (∠A)
![Page 8: A B C 12 We know ∠B = ∠C S TU 1214 We could write a proof to show ∠T ≠∠U *We could also prove that m ∠T > m ∠U, BUT theorem 1 tells us that!](https://reader035.vdocuments.mx/reader035/viewer/2022080915/56649db05503460f94a9e0b4/html5/thumbnails/8.jpg)
Theorem 3
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
L
M
N
LM + MN > LN
MN + LN > LM
LN + LM > MN
![Page 9: A B C 12 We know ∠B = ∠C S TU 1214 We could write a proof to show ∠T ≠∠U *We could also prove that m ∠T > m ∠U, BUT theorem 1 tells us that!](https://reader035.vdocuments.mx/reader035/viewer/2022080915/56649db05503460f94a9e0b4/html5/thumbnails/9.jpg)
Example 4:
The lengths of two sides of a triangle are 3 and 5. The length of the third side must be greater than _____ but less than _____.
5
3
x
Let x be the length of the third side.
x + 3 > 5
x > 2
3 + 5 > x
8 > x
x + 5 > 3
x > -2
2 8