1.
Foundations of geometry Undefined terms: The basic building blocks of geometry
are the undefined terms:Point: A location in space with no size or dimension.
Represented by a dot and labeled with a capital letter.Line: A straight path of
points that extends infinitely in two directions. Represented with a line and two
arrows, or with a lowercase letter.Plane: A flat surface that extends infinitely in
all directions. Represented by a shape like a parallelogram.Defined terms: Using
undefined terms, we define other concepts:Line segment: A part of a line with two
endpoints.Ray: A part of a line with one endpoint, extending infinitely in one
direction.Angle: Formed by two rays that share a common endpoint called the
vertex.Angle pairs:Complementary angles: Two angles whose measures add up to \
(90^{\circ }\).Supplementary angles: Two angles whose measures add up to \(180^{\
circ }\).Vertical angles: Two non-adjacent angles formed by intersecting lines.
They are always congruent (equal).Linear pair: Two adjacent angles that form a
straight line and are supplementary.2. TrianglesTriangles can be classified by
their sides as scalene (no equal sides), isosceles (at least two equal sides), or
equilateral (all sides equal). They can also be classified by their angles as acute
(all angles < \(90^{\circ }\)), obtuse (one angle > \(90^{\circ }\)), or right (one
angle = \(90^{\circ }\)). Key properties include the Triangle Angle-Sum Theorem
(interior angles sum to \(180^{\circ }\)) and, for right triangles, the Pythagorean
Theorem (\(a^{2}+b^{2}=c^{2}\)). The area of a triangle is \(A=\frac{1}{2}\times \
text{base}\times \text{height}\).3. QuadrilateralsQuadrilaterals are four-sided
polygons with interior angles summing to \(360^{\circ }\). Types include
parallelograms (opposite sides parallel and congruent), rectangles (parallelogram
with four right angles), rhombuses (parallelogram with all sides congruent),
squares (both a rectangle and rhombus), trapezoids (one pair of parallel sides),
and kites (two pairs of adjacent congruent sides).4. CirclesImportant circle terms
are center, radius (\(r\)), diameter (\(d=2r\)), circumference (\(C=2\pi r\) or \(\
pi d\)), area (\(A=\pi r^{2}\)), chord, and tangent. Key theorems involve central
angles, inscribed angles, and the relationship between a tangent line and the
radius at the point of tangency.5. TransformationsIsometries, or rigid
transformations, preserve size and shape and include translations (slides),
reflections (flips), and rotations (turns). Non-isometric transformations, like
dilations, enlarge or shrink a figure, resulting in similar but not congruent
shapes.6. Formulas to rememberCommon formulas in geometry include the distance,
slope, and midpoint formulas. Area formulas for a triangle (\(A=\frac{1}{2}bh\)),
circle (\(A=\pi r^{2}\)), and trapezoid (\(A=\frac{1}{2}h(b_{1}+b_{2})\)) are
useful. Volume formulas are used for figures like rectangular prisms (\(V=lwh\)),
cylinders (\(V=\pi r^{2}h\)), and spheres (\(V=\frac{4}{3}\pi r^{3}\)). The surface
area of a sphere is \(S=4\pi r^{2}\).AI responses may include mistakes. Learn
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proofs for triangle congruence theoremsExplain how to find the center and radius of
a circle from its equationExplain the difference between inscribed and central
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