Download 5.1 Angle Relationships in a Triangle

5.1 Angle Relationships in a
Triangle
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Triangles can be classified by the measure of their
angles. These classifications include acute triangles,
obtuse triangles, right triangles, and equiangular
triangles.
The longest side of a triangle is opposite the largest
interior angle and the shortest side of a triangle is
opposite the smallest interior angle.
The measure of an exterior angle of a triangle is equal to
the sum of the measures of the two remote interior
angles of a triangle.
The Exterior Angle Inequality Theorem states that the
measure of an exterior angle of a triangle is greater than
the measure of either of its remote interior angles.
Lesson 5.2
Side Relationships of a Triangle
Triangles can be classified by the lengths
of their sides. These classifications
include scalene triangles, isosceles
triangles, and equilateral triangles.
 The Triangle Inequality Theorem states
that the sum of the lengths of any two
sides of a triangle is greater than the
length of the third side of the triangle.
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Lesson 5.3
Points of Concurrency
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An angle bisector is a line segment, or ray that divides
an angle into two smaller angles of equal measures.
Concurrent lines are three or more lines that intersect at
the same point.
The incenter of a triangle is the point at which the three
angle bisectors intersect.
A segment bisector is a line, segment, or ray that divides
a segment into two smaller segments of equal length.
The circumcenter of a triangle is the point at which the
three perpendicular bisectors intersect.
Lesson 5.3 Cont’d
A median of a triangle is line segment that
connects a vertex to the midpoint of the side
opposite the vertex.
 The centroid of a triangle is the point at which
the three medians intersect
 An altitude of a triangle is a perpendicular line
segment that is drawn from a vertex to the
opposite side.
 The orthocenter of a triangle is the point at
which the three altitudes intersect.
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Lesson 5.4
Direct and Indirect Proofs
A two-column formal proof is a way of writing a
proof such that each step is listed in one column
and the reason for each step is listed in the
other column.
 A proof of contradiction begins with a negation
of the conclusion, meaning that you assume the
opposite of the conclusion. When a
contradiction is developed, then the conclusion
must be true.
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Lesson 5.5
Proving Triangles Congruent: SSS and SAS
If two triangles are similar, then the ratios of the lengths
of the corresponding sides are proportional and the
measures of the corresponding angles are equal.
 If two triangles are congruent, then the triangles are
similar and the ratios of the lengths of the corresponding
sides are equal to 1.
 The Side-Side-Side Congruence Theorem states that if all
corresponding sides of two triangles are congruent, then
the triangles are congruent.
 The Side-Angle-Side Congruence Theorem states that if
two sides and the included angle of one triangle are
congruent to two sides and the included angle of a
second triangle, the then triangles are congruent
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Lesson 5.6
Proving Triangles Congruent: ASA and AAS
The Angle-Side-Angle Congruence Postulate
states if two angles of one triangle are
congruent to two angles of another triangle,
then the triangles are congruent.
 The Angle-Angle-Side Congruence Theorem
states if two angles of one triangle are
congruent to two angles of another triangle and
two corresponding non-included sides are
congruent, then the triangles are congruent.
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Lesson 5.7
Proving Triangles Congruent: HL
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The Hypotenuse-Leg Congruence
Theorem states if the hypotenuse and a
leg of a right triangle are congruent to the
hypotenuse and leg of another triangle,
then the triangles are congruent.