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Theorem 24
The measure of an
exterior angle in a triangle is equal to
1.The measure of an exterior angle in a triangle is equal to the sum of the measure of its two remote angles.

Theorem 25
If a transversal is perpendicular to one of two parallel lines, it is also
1. If a transversal is perpendicular to one of two parallel lines, it is also perpendicular to the other line.

Theorem 26
In a parallelogram:
a. Each diagonal divides the parallelogram into two congruent triangles
b. Each pair of opposite sides is congruent.
c. The diagonals bisect each other.

Theorem 27
A quadrilateral in which
1. A quadrilateral in which each pair of opposite sides is congruent is a parallelogram.
2. A quadrilateral in which the diagonals bisect each other is a parallelogram.
3. A quadrilateral in which each pair of opposite angles is congruent is a parallelogram.
4. A quadrilateral in which a pair of opposite sides is parallel and congruent is a parallelogram.

Theorem 28
A parallel projection
preserves
1. A parallel projection preserves betweenness.

Theorem 29
A parallel projection preserves congruence of
1. A parallel projection preserves congruence of segments belonging to the same line.

Theorem 30
A line through the midpoint of one side of a triangle and parallel to the second side
1. A line through the midpoint of one side of a triangle and parallel to the second side bisects the third side.

Theorem 31: The Midsegment Theorem
1. The segment connecting the midpoints of two sides of a triangle is parallel to the third and half as long as that side.

Theorem 32
The medians of a triangle are concurrent in a point whose distance to a vertex is
1.The medians of a triangle are concurrent in a point whose distance to a vertex is two thirds of the length of the median from that vertex.

Theorem 33
Suppose the angle formed by a tangent and a chord is acute.
1. Prove the measure of this angle equals half the measure of the intercepted arc.

Theorem 34: The inscribed circle
1. The measure of an inscribed angle equals half the measure of the intercepted arc.

Theorem 35
The locus of appoint P that lies on one side of a line AB such that m<APB remains
1. The locus of appoint P that lies on one side of a line AB such that m<APB remains constant is the arc of a circle with end points A and B.

Theorem 36
The area of a rectangle with sides of length a and b is
1. The area of a rectangle with sides of length a and b is ab.

Theorem 37
The area of a parallelogram equals
1. The area of a parallelogram equals the product of the length of a base and the corresponding height.

Theorem 38
The area of a triangle is
1. The area of a triangle is half the product of the length of a base and the corresponding height.

Theorem 39
The area of a trapezoid whose bases have length a and b and whose height is h is
1. The area of a trapezoid whose bases have length a and b and whose height is h is given by 1/2(a + b)h.

Theorem 40
1. In a right triangle with legs of length a and b and hypotenuse of length c, we have
1. In a right triangle with legs of length a and b and hypotenuse of length c, we have
a^{2} + b^{2} = c^{2}

Theorem 41
If the sides of a triangle have lengths a, b and c such that a^{2} + b^{2} = c^{2}, then the triangle is
1. If the sides of a triangle have lengths a, b and c such that a^{2} + b^{2} = c^{2}, then the triangle is a right triangle with the right angle opposite the side of length c.

Theorem 42
Parallel projection preserves
 1.
 Parallel projection preserves ratios of lengths of segments.

Theorem 43: Side Splitting Theorem
1. A line parallel to a side of a triangle that intersects the other two sides in distinct points splits these sides into proportional segments.

Theorem 44: The Converse of the Side Splitting Theorem
1. If a line divides two sides of a triangle proportionally (the ratio of the segments on one side equals the ratio of the corresponding segments on the other side), then the line is parallel to the third side.

Theorem 45: The AA Similarity Condition for Triangles
1.If two angles of one triangle are congruent to two angles of another, then the triangles are similar.

Theorem 46: The SSS Similarity Condition
1. If corresponding sides of two triangles are proportional, then the corresponding angles are congruent and the triangles are similar.

Definition 10.3
A parallelogram is
1.A parallelogram is a quadrilateral with two pairs of opposite sides parallel.

Definition 11.1
If P is a point not on a line l, then
Suppose a line m is transversal to lines k and l and there is
1. If P is a point not on a line l, then the point P’, where the perpendicular through P to l intersects l, is called the vertical projection of P onto l.
2. Suppose a line m is transversal to lines k and l and there is a line through P parallel to m that intersects l at P’. We say that P’ is the projection of P on l parallel to m.

Definition 13.1
1 secant
2 chord
3 diameter
4 major arc minor arc
5 central angle
6 measure of a minor arc
7 If line PQ is a diameter then
8 Adjacent arcs
9 Congruent arcs
 1. A line that intersects a circle in two points is called a secant.
 2. The segment connecting those two points is called a chord.
 3. A chord containing the center of the circle is a diameter, which is twice the radius.
 4. The secant PQ divides the circle into two arcs. The larger of the two arcs is called the major arc and the smaller is called the minor arc.
 5. An angle whose vertex is the center of the circle is a central angle. If the arc PQ is in the
 interior of the angle, we say that arc PQ subtends <POQ.
 6. The measure of a minor arc is defined as the measure of its central angle.
 7. If line PQ is a diameter then arc PQ is a semicircle.
 8. Adjacent arcs in a circle are arcs that have exactly one point in common.
 9. Congruent arcs are arcs in the same circle or in congruent circles that have the same measure.

Definition 13.6
A tangent to a circle
1. A tangent to a circle O at a point P on the circle is the line through P, perpendicular to the radius line OP.

Definition 13.8
An inscribed angle is
 1.An inscribed angle is an angle whose vertex is on the circle and whose sides are chords of the
 circle.

Definition 15.5
Two polygons are similar if
1.Two polygons are similar if and only if there is a onetoone correspondence between the vertices of one polygon and the vertices of the other polygon such that the corresponding angles are congruent and the ratio of the lengths of corresponding sides is a constant.

