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1B Postulates, Theorems, and Constructions
Terms in this set (40)
Properties of Proportions
The proportion a/b=c/d is equivalent to the following: ad=bc, b/a=d/c, a/c=b/d, a+b/b+c+d/d
Angle-Angle (AA) Similarity Postulate
If two corresponding angles of two or more triangles are congruent, the triangles are similar.
The product of the extremes is equal to the product of the means; a:b =c:d where a and d are he extremes, and b and c are the means
Two polygons are similar if...
(1) corresponding angles are congruent and (2) corresponding sides are proportional
Side-Angle-Side (SAS) Theoren
If an angle of one triangle is congruent to an angle of a second triangle, and the sides including the two angles are proportional, then the triangles are similar.
Side-Side-Side (SSS) Similarity Theorem
If the corresponding side of two triangles are proportional, then the triangles are similar.
The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other.
Corollary 1 to Theorem 7-3
The length of the altitude to the hypotenuse if a right triangle is the geometric mean of the lengths of the segments of the hypotenuse.
In the proportion a/b=c/d...
B and C are the means.
Corollary 2 to Theorem 7-3
The altitude to the hypotenuse of a right triangle separates the hypotenuse so that the length of each leg of the triangle is the geometric mean of the length of the adjacent hypotenuse segment and the length of the hypotenuse.
Side Splitter Theorem
if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those two sides proportionally.
Corollary to Side Splitter Theorem
If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional.
Triangle-Angle Bisector Theorem
If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle.
3,4,5 and 5,12,13 and 8,15,17 and 7,24,25
If c² > a² + b², then the triangle is obtuse.
If c² < a² + b², then the triangle is acute.
45-45-90 Triangle Theorem
Hypotenuse = √2 x a leg
30-60-90 Triangle Theorem
Hypotenuse = 2 x shorter leg; longer leg = √3 times short leg
A translation or rotation is a composition of two reflections.
A composition of reflections across two parallel lines is a translation.
A composition of reflections across two intersecting lines is a rotation.
Fundamental Theorem of Isometries
In a plane, one of two congruent figures can be mapped onto the other by a composition of at most three reflections.
Area of Rhombus/Kite
Area of a Regular Polygon
Law of Sines
Law of Cosines
length of arc AB = mAB/360 x circumference
Area of a Sector of a Circle
Area of Sector AOB (m of AB/360) x πr²
The two segments tangent to a circle from a point outside the circle are congruent.
If two space figures have the same height and the same cross sectional area at every level, then they have the same volume.
Within a circle or in congruent circles,
(1) congruent central angles have congruent arcs and chords
(2) congruent chords have congruent arcs
Within a circle, or in congruent circles,
(1) chords equidistant from the center are congruent
(2) congruent chords are equidistant from the center
In a circle, a diameter that is perpendicular to a chord bisects the chord and its arcs.
In a circle, a diameter that bisects a chord (that is not a diameter) is perpendicular to the chord.
In a circle, the perpendicular bisector of a chord contains the center of the circle.
Inscribed Angle Theorem
The measure of an inscribed angle is half the measure of its intercepted arc.
m of angle B = 1/2 m of arc AC
Corollaries to the Inscribed Angle Theorem
(1) two inscribed angles that intercept the same arc are congruent
(2) the angle inscribed in a semicircle is a right angle
(3) the opposite angles of a quadrilateral inscribed in a circle are supplementary
The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.
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