PROPERTIES AND ATTRIBUTES OF
TRIANGLES
By: Mariana Beltranena 9-5
Perpendicular Bisector
 The perpendicular bisector of a segment is aligned
perpendicular to other segment and to its midpoint.
Perpendicular Bisector Theorem…
 If a point is in the perpendicular bisector of a segment, then it is
equidistant from both endpoints.
… and its Converse
 If a point is equidistant from both endpoints of a segment then it is
the perpendicular bisector of a segment.
Angle Bisector
 Is a line that divides an angle into two congruent angles.
Angle Bisector Theorem…
 If a point is on the angle bisector of an angle then the
perpendicular distance to each side of the angle is the same.
… and its Converse
 If the perpendicular distance from a point to both sides of an angle is
the same, then the point is on the angle bisector of that angle.
Concurrency of Perpendicular Bisectors
 Concurrent means the coincidence on a point. Where three
or more lines intersect at one point.
 The concurrency of bisectors is when the three perpendicular bisectors of a triangle are on a
point called circumcenter. This point can be inside or outside of the triangle. Also around the
triangle it could be drawn a circle touching all of the corners.
Circumcenter Therorem
 The circumcenter is equidistant from the 3 vertices of the
triangle.
Concurrency of Angle Bisectors
 The three angle bisectors of a triangle are concurrent on a
point called incenter, which is always in the triangle.
Incenter Theorem
 Is always in and is where the angle bisector meets. The perpendicular distance
from the incenter to the three sides of the triangle is the same.
Medians
 The median of a side of a triangle is the line from the midpoint of that side to the vertex
opposite to it.
 Centroid:
 the point of concurrency of the medians of a triangle. The centroid is always inside of
the triangle.
Centroid Theorem
 The distance from the vertex to the centroid is 2/3 of the distance
from the vertex to the opposite side midpoint.
Concurrency of altitudes of a triangle
theorem
 The three altitudes of a triangle are concurrent in a point called orthocenter
which has no special properties.
 The altitude of a triangle is the line from the vertex to its opposite side or the
prolongation of that side if the triangle is obtuse.
Midsegments
 The midsegment of a triangle is the line that joins the
midpoints of two of its sides.
 Midsegment Theorem
 The midsegment of a triangle is parallel to the other side and
measures half the measure of that side.
Midsegment examples
the relationship between the longer and shorter
sides of a triangle and their opposite angles
 In the same triangle or in congruent triangles with no congruent angles the side
opposite to the biggest angle is the biggest and the side opposite to the smallest
angle is the smallest side.
Triangle Inequality
 In a triangle inequality the sum of two sides is always greater
than the third side length.
Triangle inequality examples
1.
Can 8,6,10 be the measures of a triangle? If so tell if it is
acute obtuse or right.
 8+6>10
 14>10
 Yes
 8²+6²=10²
 64+36=100
 100=100
 It is a right triangle
2.
Can 5,6,11 be a triangle?
2.
3.
5+6 11….11= 11. No, because the two short sides have to add
up a greater number than the longer side.
The measures of the sides of a triangle are 5 and 9. Find all the
possible measures of the third side.
2.
3.
4.
5+9= 14
9-5=3
x>3, x<14… possible measures are 4,5,6,7,8,9,10,11,12,13
Indirect Proofs
 To write an indirect proof we first assume that the opposite
of the statement that we want to prove is true. Then we work
until we get a false conclusion. Later we conclude that if the
opposite is false, then our statement must be true.
Examples of indirect proofs
1) Prove that the three angles of a triangle add up to 180 degrees by
indirect proof.
• Given triangle ABC assume that m<A + m<B+ m<C doesn’t equal 180
degrees.
• 1) trace a parallel line to BC through A
• Euclides postulate
• 2) m< DAB + m<A+ m< CAF = 180
• Angle addition postulate
3) < DAB congruent to <B
<CAF congruent to <C
- alt. interior angles
4) m< DAB= m< B, m<CAF = m<F
-deff. of congruent angles
5) m<B +m<A+m<C= 180
- substitution prop.
-This is a statement that contradicts our original assumption. Then our
original assumption must be false.
2) Write an indirect proof that a right triangle cannot have an obtuse
angle.
Given: Triangle JKL is a right triangle
Prove: Triangle JKL does not have an obtuse angle
Assume triangle JKL does have an obtuse angle.
 m<K+m<L = 90 the acute angles of a rt. Triangle are
complementary
 m<K= 90 – m<L subtr. Prop. of =
 m<K > 90  def of obtuse.
 90-m<L > 90 subs. 90 –m<L for m<K
 m<L < 0 degrees subtract 90 from both sides and solve for m<L
 By the protractor postulate a triangle cannot have an angle less than 0.
So the original conjecture is true and an right triangle cannot have an
obtuse angle.
3) Write an indirect proof that the supplement of an acute angle
cannot have and acute angle.
Statement
Reason
Assuming that the supplement of
an acute angle is also an acute
angle.
given
Two angles add up to form 180
degrees.
Deff of supplementary angles.
If the angle is less than 90
Deff. Of acute angles
89 + < 89 doesn’t equal 180
addition postulate
The last< step
is the contradiction Angle
of our
assumption
Hinge Theorem…
 If two triangles have two congruent sides, then the one with the biggest angle
between those sides will have the biggest third side.
…and its converse.
 If two triangles have two congruent sides and one different side
the biggest third side will have the biggest angle between the 2
congruent sides.
Right Triangles
 45-45-90:
 Two legs are congruent and the hypotenuse equals the
measurement of one leg times the squared root of 2.
 30-60-90:
 In this type of right triangle there is a minor leg in front of the
30 degrees angle. A major leg in front of the 60 degrees angle
and the hypotenuse.
 The hypotenuse is twice the minor leg, and the major leg equals
the minor leg times the squared root of 3.
45-45-90
30-60-90
Exterior Angle Inequality
 The measure of an exterior angle of a triangle is greater than
the measure of either opposite interior angle.
The end!!!