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Triangle Properties
Phil Plante, Olney, MD
1. 3 sided, closed polygon
2. Angles:
a. Sum of interior angles:
mA+ mB+ mC = 180o
b. Sum of exterior angles = 2*sum of interior
angles = 360o
c. An exterior angle = the sum of the nonadjacent interior angles
3. Types of triangles:
a. Right = one right angle
b. Acute = all acute (<90o) angles
c. Obtuse = one obtuse angles (>90o)
d. Equiangular = Equilateral = 3 equal angles and sides
e. Isosceles = 2 equal sides; equal base angles
f. Scalene = no equal sides or angles
4. Perimeter and Area:
a. Perimeter = a + b + c
b. Area = (1/2)(base)(height) = (1/2)bh, where h is an altitude
5. Median:
a. Line segment from a vertex to the midpoint of opposite side
b. Point of concurrency = Centroid
 always inside the triangle
 balance point
 divides each median into two parts in 2:1 ratio, longer side near the vertex
c. Medians divide triangle into multiple triangles that all have the same area
6. Angle Bisector:
a. Line from each vertex that bisects the angle
b. Point of Concurrent = Incenter
 always inside
 equidistant from all sides
 center of inscribed circle
c. Theorem: An angle bisector of a triangle divides the opposite side into two segments that
are proportional to the other two sides of the triangle.
7. Perpendicular Bisector:
a. Line that is perpendicular to and bisects a side
b. Point of concurrency = Circumcenter
 Outside if obtuse, inside otherwise
 equidistant from all vertices
 center of circumcircle through all vertices
c. If it hits a vertex, it is also an angle bisector and the triangle is isosceles
Triangles
8. Altitudes:
a. Segment from a vertex perpendicular to opposite side
b. Point of concurrency = Orthocenter. Outside if obtuse; inside otherwise
9. Midline (Triangle Midpoint Theorem):
a. Line segment joining the midpoint of one side to the midpoint of another
b. The midline is parallel to the third side and one-half its length
c. The three midlines form 4 congruent triangles inside the larger one, each similar to the
larger triangle with scale factor (k) = ½.
10. Triangle Congruence: same shape, same
a. SSS = all three corresponding sides are congruent
b. SAS = two sides and the angle between them are congruent
c. ASA = two angles and the side between them are congruent
d. AAS = two angles and a non-included side
e. For right triangles:
 HL = The hypotenuse and one leg are congruent
 HA = The hypotenuse and one acute angle are congruent
 LL = both legs are congruent
 LA = One leg and one acute angle are congruent
f. CPCTC = corresponding parts of congruent triangles are congruent
11. Triangle Similarity: same shape, different size
a. Corresponding angles are equal; ratio of all corresponding sides = scale factor k
b. AA: if two angles are congruent, the triangles are similar
c. SSS: ratios of all corresponding sides equal the scale factor (k):
𝒔𝒊𝒅𝒆 𝒂′
𝒔𝒊𝒅𝒆 𝒃′
𝒔𝒊𝒅𝒆 𝒄′
=
=
=𝒌
𝒔𝒊𝒅𝒆 𝒂
𝒔𝒊𝒅𝒆 𝒃
𝒔𝒊𝒅𝒆 𝒄
d. SAS: Ratio of corresponding sides = k, included angles are equal
e. Ratio of perimeters = k
f. Ratio of areas = k2
g. Ratio of volumes = k3
12. Triangle Inequality Theorem:
a. The sum of the lengths of any two sides of triangle > the length of the third side
b. If two angles of a triangle are unequal, their opposite sides are unequal and the larger
side will be opposite the larger angle
c. If two sides of a triangle are unequal, their opposite angles are unequal and the larger
angle will be opposite the larger side
13. Right Triangles:
a. Pythagorean Theorem:
i. a2 + b2 = c2
ii. If c2 > a2+b2, triangle is obtuse; if c2 < a2+b2, triangle is acute
b. Pythagorean triple = right triangle with all three sides being whole numbers
i. Can scale up any triple by multiplying by another whole number (e.g., (3,4,5)*2 =
(6, 8,10)). Scaled up versions = same family.
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Triangles
ii. Common ones (not including same family): (3,4,5), (5,12,13), (7,24,25),
(8,15,17), (9.40,41), (11,60,61), … infinitely many more
14. Geometric Mean Theorem
a. If an altitude is drawn to the hypotenuse of a
right triangle, it creates two similar right
triangles, each similar to the original triangle
and to one another
b. ABD ~ BCD ~ ABC
c. In similar triangles, the ratios of two
corresponding sides are equal. For triangles
ABD and BCD then, we have:
𝐴𝐷
ℎ
=
ℎ
𝐶𝐷
Cross multiplying:
AD*CD = h2
√𝑨𝑫 ∗ 𝑪𝑫 = 𝒉
The altitude h is the geometric mean of lines AD and CD
d. Scale Factors (k) for triangles 2 and 3 wrt triangle 1:
k21 = 𝑨𝑩⁄𝑨𝑪
k31 = 𝑩𝑪⁄𝑨𝑪
15. Law of Sines and Cosines
a. Applies to non-right triangles
b. Law of Sines:
𝑺𝒊𝒏(𝑨)
𝒂
=
𝑺𝒊𝒏(𝑩)
𝒃
=
𝑺𝒊𝒏(𝑪)
𝒄
c. Law of Cosines:
𝒂𝟐 = 𝒃𝟐 + 𝒄𝟐 − 𝟐𝒃𝒄𝑪𝒐𝒔(𝑨)
𝒃𝟐 = 𝒂𝟐 + 𝒄𝟐 − 𝟐𝒂𝒄𝑪𝒐𝒔(𝑩)
𝒄𝟐 = 𝒂𝟐 + 𝒃𝟐 − 𝟐𝒂𝒃𝑪𝒐𝒔(𝑪)
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