Unit 5 Notes Triangle Properties. Definitions Classify Triangles by Sides.
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Transcript of Unit 5 Notes Triangle Properties. Definitions Classify Triangles by Sides.
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Unit 5 Notes
Triangle Properties
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Definitions
Classify Triangles by Sides
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Definitions
Classify Triangles by Angles
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Definitions
Interior and Exterior AnglesWhen the sides of a polygon are extended, other angles are formed. The original angles are the interior angles. The angles that form linear pairs with the interior angles are the exterior angles.
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Theorem
Triangle Sum TheoremThe sum of the measures of the interior angles of a triangle is 180°.
A
B C
180m A m B m C
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Theorem
Exterior Angle TheoremThe measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. A
B C D
m A m B m ACD
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• The two congruent sides of an isosceles triangle are called the legs.
• The angle formed by the legs is the vertex angle.
• The third side is the base of the isosceles triangle.
• The two angles adjacent to the base are called base angles.
Parts of an isosceles triangle
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Base angles theorem
• Two sides of a triangle are congruent if and only if the angles opposite them are congruent.
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Copy and complete each statement
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Corollary to the base angles theorem
• A triangle is equilateral if and only if it is equiangular.
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Find the values of x and y in the diagram
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Definition
Definition of MidsegmentA midsegment is a segment that connects the midpoints of two sides of a triangle. Every triangle has three midsegments.
Line segment BD is a midsegment of triangle AEC
ED
C
B
A
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Theorem
Midsegment TheoremThe segment connecting the midsegment is parallel to the third side and is half as long as that side.
BD = ½ AEBD is parallel to AEDB
E
C
A
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Definition: Median of a Triangle
•A median of a triangle is a segment from a vertex to the midpoint of the opposite side.
•The point of concurrency is called the centroid. •The centroid is the center of gravity for the triangle. • The medians must intersect inside the triangle.
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Definition: Altitudes of a triangle
•An altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side.•The point of concurrency is called the orthocenter. •It doesn’t have a special function. •The three altitudes of a triangle can intersect inside, on, or outside the triangle.
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Definition: Perpendicular bisector
• The segment that is perpendicular to a side of a triangle at it’s midpoint.
• The point of concurrency is called the circumcenter. • The circumcenter is the center of the circumscribed circle
making it equidistant from all three vertices. • The three perpendicular bisectors in a triangle can intersect
inside, on, or outside the triangle.
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Definition: Angle bisector:
• The segment that bisects an angle of a triangle. • The point of concurrency is called the incenter. • The incenter is the center of the inscribed circle making it
equidistant from the three sides of the triangle. • The 3 incenters can only intersect inside the triangle.
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Theorem
•Perpendicular Bisector Theorem•Any point on a perpendicular bisector is equally distant from the • endpoints of the segment it is bisecting.
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Theorem
•Angle Bisector Theorem•Any point on an angle bisector is equally distant from the two sides of the angle.
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Theorem
•Centroid Theorem• The distance from the centroid to the vertex is 2/3 the length of the entire
median. The distance from the centroid to the midpoint is 1/3 the length of the entire median.
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Side Angle Relationships in a Triangle
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Theorem
Triangle Inequality TheoremThe sum of the lengths of any two sides of a triangle is greater than the length of the third side.
A
AB + BC > AC
BC + AC > AB
B C AC + AB > BC