Proposed Problems |
Help: Proposed Problem 40. Level: High School, SAT
As is often true, there may be more
than one way to solve a problem.
Without giving away the solution, the following definitions and
propositions can help you get back on the track if you run
UNDEFINED TERMS OF GEOMETRY:. Point, Line, and Plane.
Geometry combines simple conceptual
building blocks to construct complex logical structures. These
building blocks include undefined terms, defined terms, and
postulates. Combining these components creates chains of
reasoning that support conclusions called theorems.
Point, Line, and Plane are
undefined terms of Geometry. These undefined terms underlie the
definitions of all geometric terms.
A point has position
only. It has no length, width, or thickness.
A line has length but
has no width or thickness. A line may be straight,
curved or a combination of these. In this website, the
word line will mean "straight line" unless you
are told otherwise.
A plane has length
and width but no thickness. A plane describes a flat
surface such as a floor, or desktop.
Plane geometry is the
geometry of plane figures. In this website, the
word figure will mean "plane figure" unless you
are told otherwise.
METHOD OF PROOF: Deductive Reasoning
Deductive Reasoning Method
enables us to derive true or acceptably true conclusions from
statements which are true or accepted as true such as undefined
terms, definitions, axioms, postulates and previously proven
theorems or propositions.
Using the deductive method, we start
with a few true statements (the axioms) and use them to prove
dozens, hundreds, or thousands, of other statements (the
Geometry was organized by the Greek
mathematician Euclid, and his deductive method is still used by
DEFINITION 1. Angle is the figure formed by two
rays with a common end point.
Congruent angles are angles
that have the exact same measure (the same number of degrees).
Angle Bisector is a ray that
divides the angle into two congruent adjacent angles.
DEFINITION 2. Triangle is a three side polygon.
Polygon is a closed plane figure with
sides. Altitude is the perpendicular line segment from
one vertex to the line that contains the opposite side.
PROPOSITION 1. The sum of the measures of the three
angles of a triangle is 180.
PROPOSITION 2. The measure of an exterior angle of a
triangle equals the sum of the measures of the two
non-adjacent interior angles.
PROPOSITION 3. The sum of the measures of the acute
angles of a right triangle is 90 (complementary).
PROPOSITION 4. Triangle Congruence A.S.A. If two
angles and the included side of one triangle are congruent to
the corresponding parts of another, then the triangles are
PROPOSITION 5. Any point on the bisector of
an angle is equidistant from the sides of the angle. Also
PROPOSITION 6. Triangle Congruence S.A.S. If
two sides and the included angle of one triangle are
congruent to the corresponding parts of another, then the
triangles are congruent.
PROPOSITION 7. Isosceles triangle: If two sides of
a triangle are congruent, the angles opposite these sides are
congruent. Also converse.
PROPOSITION 8. An equilateral triangle is equiangular: 60-60-60. Also converse.
DEFINITION 3. Circle is the set of all points in a
plane that are at the same distance from a fixed point
called the center.
Tangent of a circle is a
line that touches the circle at one and only one point no
matter how far produced.
PROPOSITION 9. If a line is
tangent to a circle, it is perpendicular to a radius at the
point of tangency.
PROPOSITION 10. The bisectors
AD, BF and CE of the angles of a triangle ABC meet in a
point I, which is equidistant from the sides of the
The incircle is the inscribed
circle of a triangle. The center of the incircle is called
the incenter, and the radius of the circle is called the
PROPOSITION 11. An excircle of the
triangle is a circle lying outside the triangle, tangent to
one of its sides and tangent to the extensions of the other
The center of an excircle is the
intersection of the internal bisector of one angle and the
external bisectors of the other two.
The center of an excircle is called
the excenter, and the radius of the circle is called the
PROPOSITION 12. A central angle is measured by its
PROPOSITION 13. An inscribed angle is measured by
one-half its intercepted arc.
DEFINITION 4. A cyclic quadrilateral is a
quadrilateral whose vertices all lie on a single circle.
Opposite angles of a cyclic
(inscribed) quadrilateral are supplementary. Also converse.
PROPOSITION 14.2 A quadrilateral is cyclic if one side subtends
congruent angles at the two opposite vertices. Also
DEFINITION 5. Similar Triangles
are triangles whose corresponding angles are congruent and whose
corresponding sides are in proportion.
PROPOSITION 15. Triangle Similarity AA. If two
angles of one triangle are congruent to two angles of another
triangle, the two triangles are similar.