Teaching Textbooks Geometry
T5: Perpendicular lines intersect to form...
...4 right angles.
P2: Three noncollinear points determine...
...a unique plane.
P1: Two points determine...
...a unique straight line.
C17.1: The acute angles of a right triangle...
...are complementary.
T4: Pairs of vertical angles...
...are congruent (equal).
T6: All right angles...
...are congruent (equal).
P4: Protractor Postulate - 1. The rays in a half rotation (180 degrees)...
...can be numbered so that to every ray there corresponds exactly one real number called its coordinate.
P8: Through a given point not on a line...
...exactly one line may be drawn parallel to the line.
P5: Through a given point not on a line, there exists...
...exactly one perpendicular to the given line.
T17: The sum of the measures of the angles of a triangle...
...is 180°.
C17.2: The measure of each angle of an equiangular triangle...
...is 60.
T18: The measure of an exterior angle of a triangle...
...is equal to the sum of the measures of the two remote interior angles.
P3: Ruler Postulate - 4. The distance between two points...
...is the absolute value of the distance between their coordinates.
P4: Protractor Postulate - 4. The measure of the angle is...
...the absolute value of the difference between the coordinates of its rays.
T8: If the exterior sides of a pair of adjacent angles are perpendicular...
...the angles are complementary.
T10: If the two angles in a linear pair have equal measures (are congruent)...
...then each is a right angle.
T12: If two parallel lines are cut (crossed) by a transversal...
...then interior angles on the same side of the transversal are supplementary.
C23.1: If a triangle is equilateral...
...then it is also equiangular.
T23: If two sides of a triangle are congruent (equal)...
...then the angles opposite those sides are congruent (equal).
T16: If two lines are parallel to a third line...
...then the lines are parallel to each other.
P7: If two lines form congruent alternate interior angles with a transversal...
...then the lines are parallel.
T13: If two lines form congruent corresponding angles with a transversal...
...then the lines are parallel.
T14: If two lines form congruent alternate exterior angles with a transversal...
...then the lines are parallel.
T15: If two lines form supplementary interior angles on the same side of a transversal...
...then the lines are parallel.
T21: If a point lies on the perpendicular bisector of a segment...
...then the point is equidistant from the endpoints of the segment.
T22: It a point is equidistant from the endpoints of a segment...
...then the point lies on the perpendicular bisector of the segment.
C17.3: If two angles of a triangle are congruent to two angles of another triangle...
...then the remaining pair of angles are congruent.
T11: If two parallel lines are cut (crosses) by a transversal...
...then their alternate exterior angles are congruent (equal).
P6: If two parallel lines are cut by a transversal...
...then their alternate interior angles are congruent (equal).
T9: If two parallel lines are cut (crossed) by a transversal...
...then their corresponding angles are congruent (equal).
T20: If two triangles are congruent to the same triangle...
...then they are congruent to each other.
T1: If two angles are complementary to the same angle or equal (congruent) angles...
...then they are equal (congruent).
T2: If two angles are supplementary to the same angle or equal (congruent) angles...
...then they are equal (congruent).
T3: If two angles are a linear pair...
...then they are supplementary.
P3: Ruler Postulate - 2. To every real number...
...there corresponds exactly one point of the line.
P4: Protractor Postulate - 2. To every real number from 0 to 180..
...there corresponds exactly one ray.
P3: Ruler Postulate - 1. To every point on a line...
...there corresponds exactly one real number called its coordinate.
P3: Ruler Postulate - 3. To every pair of points...
...there corresponds exactly one real number called the distance between the points.
P4: Protractor Postulate - 3. To every pair of rays...
...there corresponds exactly one real number called the measure of the angle that they determine.
T7: Through a given point on a line...
...there exists exactly one perpendicular to the given line.
D14: Segment Bisector
A bisector of −AB is any line, ray, or line segment that passes through the midpoint of −AB.
D28: A Polygon
A geometric figure whose sides are line segments.
D27: Transversal
A line that intersects two or more lines in different points.
D22: Perpendicular Bisector
A line that is perpendicular to a line segment and intersects the line segment at its midpoint.
D20: Vertical Angles
A pair of nonadjacent angles formed by two intersecting lines.
D29: A Triangle
A polygon that has three sides.
D41: A Median of a Triangle
A segment drawn from any vertex of the triangle to the midpoint of the opposite side.
D40: An Altitude of a Triangle
A segment drawn from any vertex of the triangle, perpendicular to the opposite side, extending outside the triangle if necessary.
Undefined Term: Line
A set of points that extend indefinitely in either direction. A line has length, but no width or depth.
Undefined Term: Plane
A set of points that form a flat surface which extends forever in all directions. So a plane has length and width, but no depth.
D36: An Equiangular Triangle
All three angles with equal measures.
D33: An Acute Triangle
All three angles with measures of less than 90°.
D32: An Equilateral Triangle
All three congruent (equal) sides.
D37: An Exterior Angle of a Polygon
An angle that forms a linear pair with one of the interior angles of the polygon.
D10: A Right Angle
An angle with a measure of 90°.
D12: An Obtuse Angle
An angle with a measure of greater than 90° (and less than 180°).
D11: An Acute Angle
An angle with a measure of less than 90°.
D8: Congruent Angles
Angles that have equal measures.
D18: Adjacent Angles
Angles that have the same vertex, share a common side, and have no interior points in common.
D17: Supplementary Angles
Angles with measures that add to 180°.
D16: Complementary Angles
Angles with measures that add to 90°.
Reflexive Property
Any quantity is equal to itself: a=a
Equivalent Figures
Figures that have the same area.
Congruent Figures
Figures that have the same shape and the same size.
Similar Figures
Figures that have the same shape but are different sizes.
D5: Betweenness of Points
If F, G and H are collinear and FG + GH = FH, then G is between F and H.
Substitution Property
If a = b, then either a or b may be substituted for the other in any equation.
Addition Property
If equals are added to equals, the results are equal: If a = b, then a + c = b + c.
Division Property
If equals are divided by nonzero equals, their quotients are equal: If a = b, then a ÷ c = b ÷ c as long as c ≠ 0.
Multiplication Property
If equals are multiplied by equals, their products are equal: If a = b, then ac = bc.
Subtraction Property
If equals are subtracted from equals, the results are equal: If a = b, then a - c = b - c.
P11: Hypotenuse-Leg
If the vertices of two right triangles an be paired so that the hypotenuse and leg of one of them are congruent to the corresponding parts of the second right triangle , then the two right triangles are congruent.
D38: Congruent Triangles
If the vertices of two triangles can be paired in a correspondence so that all pairs of corresponding angles are congruent and all pairs of corresponding sides are congruent, then the triangles are congruent.
P12: Side-Side-Side (S.S.S)
If the vertices of two triangles can be paired so that three sides of one triangle are congruent to the corresponding sides of the second triangle, then the two triangles are congruent.
P10: Angle-Side-Angle (A.S.A)
If the vertices of two triangles can be paired so that two angles and the included side of one triangle are congruent to the corresponding parts of the second triangle, then the two triangles are congruent.
T19: Angle-Angle-Side (A.A.S)
If the vertices of two triangles can be paired so that two angles and the side opposite one of them in one triangle are congruent to the corresponding parts of the second triangle, then the two triangles are congruent.
P9: Side-Angle-Side (S.A.S)
If the vertices of two triangles can be paired so that two sides and the included angle of one triangle are congruent to the corresponding parts of the second triangle, then the two triangles are congruent.
Transitive Property
If two quantities are equal to the same quantity, then they are equal to each other: If a = b and b = c, then a = c.
D39: Corresponding Parts of Congruent Triangles are Congruent (C.P.C.T.C.)
If two triangles are congruent, then their vertices ca be paired so a correspondence so that all pairs of corresponding angles are congruent and all pairs of corresponding sides are congruent.
Undefined Term: Point
Indicates position, and has no length, width, or depth
D4: Congruent Line Segments
Line segments that have equal lengths.
D25: Parallel Lines
Lines that lie in the same plane (coplanar) and that never intersect.
D21: Perpendicular Lines
Lines which intersect to form right angles.
D26: Coplanar
Lines, segments, rays, or points which lie in the same plane.
D30: A Scalene Triangle
No congruent (equal) sides.
D35: An Obtuse Triangle
One angle with a measure of greater than 90°.
D34: A Right Triangle
One angle with a measure on 90°.
D6: A Ray
Part of a line consisting of a given point, called the end point, and the set of all points on ine side of the end point.
D3: A Line Segment
Part of a line consisting of two points, called end points, and the set of all points between them.
D1: Collinear Points
Points that lie on the same line.
D2: Noncollinear Points
Points the do not lie on the same line.
D23: Distance Between Two Points
The length of the line segment joining the points.
D24: Distance Between a Line and a Point Not on the Line
The length of the perpendicular segment drawn from the point o the line.
D13: Midpoint of a Line Segment
The point that divides the line segment into two congruent line segments.
Symmetric Property
The positions of the expressions on either side of an equals sign may be reversed: If a=b, then b=a
D7: An Angle
The union of two rays having the same end point. The end point is called the vertex of the angle; the rays are called the sides of the angle.
D19: Linear Pair
Two adjacent angles whose exterior sides forma straight line.
D31: An Isosceles Triangle
Two congruent (equal) sides.
D15: Angle Bisector
→OR is the bisector of ∠PON if R lies in the interior of ∠PON and m∠POR = m∠RON.
D9: Betweenness of Rays
→PS is between →PQ and →PR, if point S lies in the interior if ∠QPR and m∠SPR + m∠QPS = m∠QPR