http://www.ck12.org Chapter 5. Relationships with Triangles
- Ifxis even andyis odd, thenx+yis odd.
- In 4 ABE, if^6 Ais a right angle, then^6 Bcannot be obtuse.
- IfA,B, andCare collinear, thenAB+BC=AC(Segment Addition Postulate).
- If a collection of nickels and dimes is worth 85 cents, then there must be an odd number of nickels.
- Hugo is taking a true/false test in his Geometry class. There are five questions on the quiz. The teacher gives
her students the following clues: The last answer on the quiz is not the same as the fourth answer. The third
answer is true. If the fourth answer is true, then the one before it is false. Use an indirect proof to prove that
the last answer on the quiz is true. - On a test of 15 questions, Charlie claims that his friend Suzie must have gotten at least 10 questions right.
Another friend, Larry, does not agree and suggests that Suzie could not have gotten that many correct. Rebecca
claims that Suzie certainly got at least one question correct. Ifonly oneof these statements is true, how many
questions did Suzie get right? - If one angle in a triangle is obtuse, then each other angle is acute.
- If 3x+ 7 ≥13, thenx≥2.
- If segment AD is perpendicular to segment BC, then^6 ABCis not a straight angle.
- If two alternate interior angles are not congruent, then the lines are not parallel.
- In an isosceles triangle, the median that connects the vertex angle to the midpoint of the base bisects the vertex
angle.
Summary
This chapter begins with an introduction to the Midsegment Theorem. The definition of a perpendicular bisector is
presented and the Perpendicular Bisector Theorem and its converse are explored. Now that the bisectors of segments
have been discussed, the definition of an angle bisector is next and the Angle Bisector Theorem and its converse are
presented. The properties of medians and altitudes of triangles are discussed in detail. The entire chapter builds to
a discovery of the relationships between the angles and sides in triangles as a foundation for the Triangle Inequality
Theorem. The chapter ends with a presentation of indirect proofs.
Chapter Keywords
- Midsegment
- Midsegment Theorem
- Perpendicular Bisector Theorem
- Perpendicular Bisector Theorem Converse
- Point of Concurrency
- Circumcenter
- Concurrency of Perpendicular Bisectors Theorem
- Angle Bisector Theorem
- Angle Bisector Theorem Converse
- Incenter
- Concurrency of Angle Bisectors Theorem
- Median
- Centroid
- Concurrency of Medians Theorem
- Altitude
- Orthocenter
- Triangle Inequality Theorem
- SAS Inequality Theorem
- SSS Inequality Theorem