1) The triangle inequality theorem states that in any triangle, the largest side is opposite the largest angle and the smallest side is opposite the smallest angle. 2) If two sides of a triangle are congruent, then the angles opposite those sides are also congruent. Similarly, if two angles are congruent, the sides opposite them are also congruent. 3) In a triangle, if one side is longer than the other, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side.

1. Triangle Inequality
Jezelyn C. Fabelinia

2. Relationship Inequality Theorem
• In any triangle , the largest side and largest angle are
opposite one another. < 𝐵 𝑖𝑠 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐴𝐶.
• In any triangle , the mid-sized side and mid-sized angle
are opposite one another. < 𝐶 𝑖𝑠 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐵𝐴
• In any triangle, the smallest side and smallest angle are
opposite one another. < 𝐴 𝑖𝑠 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝐵𝐶

3. Relationship Inequality Theorem
• If two sides are congruent(equal in measure) then the
corresponding two angles will be congruent( Equal in
Measure).(Isosceles Theorem) < 𝐴 ≅< 𝐶, 𝐴𝐵 = 𝐵𝐶
• Alternately, if two angles are congruent (Equal in measure) then
the corresponding two sides will be congruent(equal in measure)
A𝐵 = 𝐵𝐶, < 𝐴 ≅< 𝐶

4. SIDE-ANGLE INEQUALITY THEOREM
• In a triangle, one side is longer than the other side ,
then the angle opposite the longer side has a
greater measure than the angle opposite the
shorter side.

5. Example
• ∆𝐴𝐵𝐶
SIDE MEASUREMENT
EF 15 cm
DE 10 cm
DF 8cm
ANGLE MEASUREMENT
< 𝐴 95°
< 𝐶 45°
< 𝐵 40°

6. Example
Give the following triangles, name the biggest angle and smallest angle if:
1.∆𝑋𝑌𝑍, 𝑋𝑌 = 10𝑐𝑚, 𝑌𝑍 = 30𝑐𝑚, 𝑍𝑋 = 20𝑐𝑚
SIDE MEASUREMENT ANGLE
YZ 30cm < 𝑋
ZX 20cm < 𝑌
XY 10cm < 𝑍

7. Draw and Give the following triangles, name the biggest angle and smallest angle if:
• ∆𝑀𝑁𝑂, 𝑀𝑁 = 10𝑐𝑚, 𝑁𝑂 = 15𝑐𝑚, 𝑂𝑀 = 24𝑐𝑚
• ∆𝐺𝐻𝐼, 𝐺𝐻 = 15𝑐𝑚, 𝐻𝐼 = 12𝑐𝑚, 𝐼𝐺 = 10𝑐𝑚
• ∆𝐷𝐸𝐹, 𝐷𝐸 = 25𝑐𝑚, 𝐸𝐹 = 34𝑐𝑚, 𝐹𝐷 = 20𝑐𝑚
𝐸𝑋𝐴𝑀𝑃𝐿𝐸
SIDE MEASUREMENT ANGLE

8. SEATWORK
• Draw and Give the following triangles, name the biggest angle and smallest angle if
1.∆𝐵𝐷𝑂, 𝐵𝐷 = 15𝑐𝑚, 𝐷𝑂 = 10𝑐𝑚, 𝑂𝐵 = 24𝑐𝑚
2. ∆𝐿𝑌𝑁, 𝑌𝑁 = 30𝑐𝑚, 𝐿𝑌 = 20𝑐𝑚, 𝑁𝐿 = 15𝑐𝑚
3.∆𝑂𝑃𝑄, 𝑂𝑃 = 20𝑐𝑚, 𝑃𝑄 = 10𝑐𝑚, 𝑄𝑂 = 25𝑐𝑚
SIDE MEASUREMENT ANGLE

9. ANGLE-SIDE INEQUALITY THEOREM
•In a triangle, one angle is bigger than the
other angle, then the side opposite the
bigger angle has a greater measure than the
side opposite the smallest angle.

10. Example
• ∆𝐴𝐵𝐶
ANGLE UNITS SIDE UNKTS
< 𝐴 95° EF 15 cm
< 𝐶 45° DE 10 cm
< 𝐵 40° DF 8cm