Chapter 5 Review Flashcards
perpendicular bisector
a bisector that is also perpendicular

Theorem 5.1: Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
EX. If CD is a ⊥ bisector of AB, then AC = BC.

Theorem 5.2: Converse of the Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
EX. If AC = BC, then C lies on CD, the ⊥ bisector of AB.

concurrent lines
when three or more lines intersect at a common point

point of concurrency
the point where concurrent lines intersect

circumcenter
the point of concurrency of the perpendicular bisectors

Theorem 5.3: Circumcenter Theorem
The perpendicular bisectors of a triangle intersect at a point called the circumcenter that is equidistant from the vertices of the triangle.
EX. If P is the circumcenter of ΔABC, then PB = PA = PC.

Theorem 5.4: Angle Bisector Theorem
If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.
EX. If CP bisects ∠ACB, PA ⊥ CA, and PB ⊥ CB, then AP = PB.

Theorem 5.5: Converse of the Angle Bisector Theorem
If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle.
EX. If PA ⊥ CA, PB ⊥ CBCP, and AP = PB, then BF bisects ∠ACB.

incenter
the point of concurrency for the angle bisectors of a triangle

Theorem 5.6: Incener Theorem
The angle bisectors of a triangle intersect at a point called the incenter that is equidistant from the sides of the triangle.
EX. If K is the incenter of ΔMNO, then KT = KU = KV.

median
a segment with endpoints being a vertext of a triangle and the midpoint of the opposite side

centroid
the point of concurrency on the medians of a triangle

Theorem 5.7: Centroid Theorem
The medians of a triangle intersect at a point called the centroid that is two thirds of the distance from each vertex to the midpoint of the opposite side.
EX. If P is the centroid of ΔABC, then AP = 2/3AL, BP = 2/3 BM, and CP = 2/3 CN.

altitude
a segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side

orthocenter
the point of intersection for the lines containing the altitudes of a triangle

Inequality
For any real numbers a and b, a > b if and only if there is a positive number c such that a = b + c.
EX. If 5 = 2 + 3, then 5 > 2 and 5 > 3.
Comparison Property of Inequality
a < b, a = b, or a > b.
Transitive Property of Inequality
- If a < b and b < c, then a < c.
- If a > b and b > c, then a > c.
Addition Property of Inequality
- If a > b, then a + c > b + c.
- If a < b, then a + c < b + c.
Subtraction Property of Inequality
- If a > b, then a - c > b - c.
- If a < b, then a - c < b - c.
Theorem 5.8: Exterior Angle Inequality
The measure of an exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles.
EX. m∠1 > m∠2
m∠1 > m∠3

Theorem 5.9: Angle-Side Relationships in Triangles
If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side.
EX. CB > AC, so m∠A > m∠B

Theorem 5.10: Angle-Side Relationships in Triangles
If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.

indirect reasoning
Reasoning that assumes that the conclusion is false and then shows tha tthis assumption leads to a contradiction of the hypothesis like a postulate, theorem, or corollary. Then, since the assumption has been proved false, the conclusion must be true.
indirect proof; proof by contradiction; proof by negation
In an indirect proof, one assumes that the statement to be proved is false. One then uses logical reasoning to deduce that the statement contradicts a postulate, theorem, or on of the assumptions. Once a contradiction is obtained, one concludes that the statement assumed false must in fact be true.
Theorem 5.11: Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Theorem 5.13: Hinge Theorem
If two sides of a triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.
EX. If AB ≅ FG, AC ≅ FH, and m∠A > m∠F, then BC > GH.

Theorem 5.14: Converse of the Hinge Theorem
If two sides of a triangle are congruent to two sides of another triangle, and the third side in the first is longer than the third side in the second triangle, then the included angle measure of the first triangle is greater than the included angle measure in the second triangle. Converse of the Hinge Theorem
EX. If JL ≅ PR, KL ≅ QR, and PQ > JK, then m∠R > m∠L.
