Proofs Using Reflections
equidistant: Point P is equidistant from A and B if P is the same distance from A and B.
Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment,
then it is equidistant from the endpoints of the se
One-Step Congruence Proofs
Every conclusion in a proof needs to be justified.
Every justification is either a postulate, definition, or theorem.
If we're only using one direction of a biconditional, we can name the biconditional, but write
only the if-the
Isosceles Triangles
Vertex
Leg
Leg
Base angle
Base angle
Base
Note: Vertex is the meeting of the two equal legs.
Base is opposite vertex angle.
Base angles are opposite the equal legs.
The bisector of the vertex angle is the symmetry line of an isosceles
Corresponding Parts of Congruent Figures
A figure under an isometry is congruent to the preimage.
Angles and sides that are images of each other are called corresponding parts.
Notation: Figure F is congruent to figure G
F G
CPCFC: Corresponding Parts of
Congruence & Equality
Some properties of congruence ( ) are like properties of equality (=).
(1) Any figure is congruent to itself. F F
(Think of rl o rl - reflect a figure over a line, and back again. You'd get the
same exact figure.)
(2) For any figures
Proofs Using Transitivity
EX 1: Use a compass and straightedge to construct an equilateral triangle:
C
B
A
D
(1)
(2)
(3)
(4)
Put your compass center on A and your pencil in the hole at B. Draw circle A.
Now put your compass center on B and your pencil on
Types of Quadrilaterals
Hierarchy of Quadrilaterals
In a hierarchy, a figure on the bottom also fits all those connected above it and shares its properties.
For example, every parallelogram is a trapezoid (but not every trapezoid is a parallelogram).
Defi
Sums of Angle Measures in Polygons
Triangle Sum Theorem: The sum of the measures of the angles of a triangle is 180.
EX 1: In VABC , the angles are in the extended ratio 2:3:5. Find their measures.
Picture:
Equation:
3x
2x
B
A + B + C = 180
2x + 3x + 5x =
Reflection-Symmetric Figures
Figure F is a reflection-symmetric figure iff there is a line (m) such that rm(F) = F.
(Reflecting a figure over its line of symmetry yields exactly the same figure. The
image and preimage coincide.)
The reflection line is kno
Properties of Trapezoids
EX 1:
B
C
A
In this trapezoid, AD / BC .
Extend AB upward. Label those angles 1&2.
Note: AD and BC are bases.
D
Conclusion
Justification
(1) m1 + m2 = 180
Linear Pair Theorem
(2) m2 = mA
Parallel Lines => Corresponding Angles Cong