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MATH 241
Midterm Review
Know these things. When appropriate be able to draw a diagram to illustrate the fact. Be able to use these facts. Be able to prove the items that
are starred.
Know and be able to state all theorems that have a name! You may be
asked to state and prove such a theorem (e.g. “State and prove the angle
side inequality theorem.”), if you do not know what the theorem by name
you will not be able to answer the question.
Chapter 1
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Playfair’s Parallel Axiom and the parallel postulate.
opposite angles in an X are equal.
Know what it means for two things to be congruent.
SAS, SSS, ASA, SAA, SSA* and HSR.
The triangle inequality.
The isosceles triangle theorem and it’s converse.
The angle side inequality and it’s converse.
Theorem 1.3.1/2 (parallel lines)
The angles in a triangle sum to 180 degrees.
The exterior angle theorem and the exterior angle inequality.
Thales’ Theorem
Thales’ corollary (1.3.7) and its converse (1.3.8).
Know what a convex, simple or non-simple quadrilateral is.
The interior angles of a simple quadrilateral sum to 360◦ .
What it means for a shape to be cyclic.
A quad is cyclic if and only if it’s opposite interior angles sum to
180◦ .
Characterization of the right bisector Theorem.
Characterization of the angle bisector Theorem.
Construction problems 1.5.1 to 1.5.8
Assignments 2 and 3.
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Chapter 2
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l1 k l2 , m1 ⊥ l1 and m2 ⊥ l2 then m1 k m2 .
m1 ⊥ l1 and m2 ⊥ l2 then l1 k l2 if and only if m1 k m2 .
A triangle’s three perpendicular bisectors are concurrent.
Know what is, and how to construct a triangle’s circumcircle.
All of theorem 2.2.1 (chords of circles).
A triangle’s three angle bisectors are concurrent.
Know what a triangle’s incircle is, how it’s center is found and how
the incircle would be constructed.
Each pair of exterior angle bisectors of a triangle intersect.
The external angle bisectors of two angles of a triangle and the internal angle bisector of the third are concurrent.
Know what a triangle’s excircle is, how it’s center is found and how
the excircle would be constructed.
In a parallelogram 1) opposite sides are congruent, 2) opposite angles
are congruent, 3) the diagonals bisect each other.
A simple quadrilateral is a parallelogram, if any of the following
are true: 1) Opposite sides are congruent, 2) opposite angles are
congruent, 3) one pair of opposite sides are congruent and parallel,
4) the diagonals bisect each other.
The three altitudes of a triangle are concurrent.
Know what the orthocenter of a triangle is.
The midline theorem. (any proof)
The three medians are concurrent and trisect each other.
In 4ABC if P is the midpoint of AB and Q is on AC such that P Q
is parallel to BC, then Q is the midpoint of AC.
Construct a parallelogram given three of the four vertices.
Construct a line Through a point P parallel to a given line l.
For a point P outside of a circle C(0, r), construct a line Through P
tangent to C(0, r).
What a Thales’ Locus is.
Construction problem 2.5.6.
Assignments 4 and 5.
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Chapter 3
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What it means for two objects to be similar.
Lemma 3.2.1.
Theorems 3.2.2 and 3.2.3.
In 4ABC, if D and E are on the segments AB and AC respectively,
and if DE k BC, then 4ABC ∼ 4ADE.
AAA, AA and sss for similar triangles.
In 4ABC, if D and E are interior points on the segments AB and
AC respectively, and if AD/DB = AE/EC then DE k BC.
sAs for similar triangles.
Pythagoras’ Theorem.
Converse of Pythagoras’ Theorem.
The Angle bisector theorem.
The medians of a triangle trisect each other and are concurrent.
Theorem 3.4.7.
Theorems 3.6.1 and 3.6.3.
Theorems 3.6.2, 3.6.4 and 3.6.5.
The lily pad problem and the earth’s diameter problem.
Assignment 6.
Chapter 4
• directed distance/ratios and properties, (including ideal points).
F Ceva’s and Menelaus’ Theorems. (I will not ask for the whole proof
of either theorem, but I may ask for one piece of one of the theorems.)
• Given a drawing know how to find the cevian product - know what
a cevian is, what a menelaus point is and what a transversal is.
• Two cevians that pass through the interior of a triangle are not
parallel.
F The medians of a triangle are concurrent.
F the internal angle bisectors of a triangle are concurrent.
• The altitudes are concurrent.
F Example 4.3.6.
F Examples 4.4.1 and 4.4.3
F Assignment 7.
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Chapter 5
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The 4 area Postulates.
The area of a parallelogram and of a triangle.
Area of a triangle - inradius formula
Pythagoras’ Theorem.
Theorems 5.2.1 to 5.2.4.
The three medians of a triangle are concurrent and trisect each other.
Example 5.2.6.
The angle bisector theorem (the if and only if version)
Theorem 5.2.10
Law of sines
Area formulas for SAS, ASA area
Heron’s formula
Assignment 8 and question 1 of Assignment 9.
Chapter 6
F Given a segment of length 1, for p and q some positive integers,
√
construct the lengths: p and q, p ± q, p/q, pq and p.
• Know that π, e, 21/3 , and cos 20◦ are not constructible.
F Construct regular 3, and 4 polygons.
• Know what a fermat prime is and what the first three Fermat primes
are.
• Theorem 6.3.2.
F Corollary 6.3.3.
F Example 6.3.4.
F Assignment 9.