Chapter 3 Summary
Definitions
- Congruent triangles
- All pairs of corresponding parts (angles and sides) are congruent. (p. 111)
- Congruent polygons
- All pairs of corresponding parts (angles and sides) are congruent. (p. 112)
- Notation: △ABC ≅ △DEF
- This means that the two triangles are congruent and also sets up the correspondence; ∠A ≅ ∠D; ∠B ≅ ∠E; ∠C ≅ ∠F; segment AB ≅ segment DE; etc.
- Included angle or side
- The included angle between two sides of a triangle is the angle between those two sides of a triangle; the included side between two angles of a triangle is the side between those two angles. (p.115)
- CPCTC
- An abbreviation for "congruent parts of congruent triangles are congruent", which is true by the definition of congruent triangles. (p. 125)
- Circle
- The set of all points at distance r from some point O. r is called the radius and O is called the center. Since O is not of distance r from itself, it is not considered to be part of the circle. (p. 125)
- Median (of a triangle)
- A line segment drawn from any vertex to the midpoint of the opposite side (p. 131)
- Altitude (of a triangle)
- A line segment drawn from any vertex to the opposite side, extended if necessary, and perpendicular to that side (p. 132)
- Scalene triangle
- A triangle with no congruent sides (p.142)
- Isosceles triangle
- A triangle with at least two congruent sides. The congruent sides are called the legs; the other side is called the base; the angles opposite the congruent sides are the base angles and the angle opposite the base is the vertex angle (p.142).
- Equilateral triangle/polygon
- A triangle/polygon with all sides congruent (p.142)
- Equiangular triangle/polygon
- A triangle/polygon with all angles congruent (p.143)
- Acute triangle
- A triangle with all acute angles (p.143)
- Right triangle
- A triangle with one right angle. The side opposite the right angle is the hypotenuse and the other sides are the legs (p.143).
- Obtuse triangle
- A triangle with one obtuse angle (p.143)
- Angle-side theorems
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- Theorem 20, 21: Two sides of a triangle are congruent if and only if the angles opposite the sides are congruent. (p.148,149)
- If two sides of a triangle are not congruent, then the angles opposite them are also not congruent, and the larger angle is opposite the larger side.
- If two angles of a triangle are not congruent, then the sides opposite them are also not congruent, and the larger side is opposite the larger angle.
- A triangle is equilateral if and only if it is equiangular.
- HL Theorem (called a postulate in the book)
- If two right triangles have one set of congruent legs and also congruent hypotenuses, then the triangles are congruent (p. 156).
Postulates
- Reflexive property: Any segment or angle is congruent with itself. (p. 112)
- SSS: If there is a correspondence between two triangles such that all three sides of one triangle are congruent to the corresponding sides in the other triangle, then the triangles are congruent. (p. 116)
- SAS: If there is a correspondence between two triangles such that two sides of one triangle and their included angle are congruent to the corresponding parts the other triangle, then the triangles are congruent. (p. 116)
- ASA: If there is a correspondence between two triangles such that two angles of one triangle and their included side are congruent to the corresponding parts the other triangle, then the triangles are congruent. (p. 117)
- Two points determine a line (or ray or segment). That is, given two distinct points A and B, there is one and only one line that can be drawn through A and B; there is only one segment with A and B as endpoints, and there is only one ray with endpoint A going through B (of course there is another ray with endpoint B going through A) (p. 132).
Theorems
- Area and circumference
- The area of a circle is πr2; the circumference is 2πr. (p. 126)
- All radii of a circle are congruent.
- (p. 126)