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This document discusses two postulates for proving triangles congruent: the Angle-Side-Angle (ASA) postulate which states that if two angles and the included side of one triangle are congruent to those of another triangle, then the triangles are congruent. The Angle-Angle-Side (AAS) postulate similarly states that if two angles and the non-included side of one triangle match those of another, then the triangles are congruent.
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This document discusses various theorems and properties related to triangles. It explains the Basic Proportionality Theorem, also known as Thales' Theorem, which states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. It also covers similarity criteria like AAA, SSA, and SSS. The Area Theorem demonstrates that the ratio of areas of similar triangles equals the square of the ratio of corresponding sides. Additionally, it proves Pythagoras' Theorem, which relates the sides of a right triangle, and its converse. In summary, the document outlines key triangle theorems regarding proportional division, similarity, areas, and the Pythagorean relationship between sides.
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This document defines and describes different types of triangles:
- Equilateral triangles have three equal sides and three equal angles.
- Isosceles triangles have at least two equal sides.
- Scalene triangles have no equal sides.
- Right triangles have one 90 degree angle.
- Acute triangles have all angles less than 90 degrees.
- Obtuse triangles have one angle greater than 90 degrees.
It also describes three theorems used to prove triangle congruence: SSS (three equal sides), SAS (two equal sides and the included angle), and ASA (two equal angles and one included side).
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Triangles can be classified based on side lengths as equilateral, isosceles, or scalene triangles and based on angles as acute, obtuse, or right triangles. The three main properties of triangles are the angle sum property, exterior angle property, and Pythagorean theorem. Secondary parts of a triangle include the median, altitude, perpendicular bisector, and angle bisector. Triangles can be proven congruent using the SSS, SAS, ASA, AAS, or RHS criteria. Inequalities in triangles relate longer sides to larger angles and shorter sides to smaller angles. Important centers of a triangle include the incenter, circumcenter, centroid, and orthocenter.
Triangles
1) A triangle is a three-sided polygon with three vertices and three edges.
2) Triangles can be classified based on side lengths (equilateral, isosceles, scalene) or interior angles (right, acute, obtuse).
3) The interior angles of any triangle always sum to 180 degrees. Congruent triangles have the same shape and size, while similar triangles have the same angle measures but sides proportional in length.
Rectangle, rhombuses and square
This document outlines several geometry theorems about rectangles, rhombi, trapezoids, and polygons. It states that a parallelogram is a rectangle if its diagonals are congruent, and a rhombus if its diagonals are perpendicular and bisect opposite angles. For trapezoids, it provides theorems about the segment joining the midpoints of the non-parallel sides being parallel to the bases, and the diagonals being congruent in isosceles trapezoids. Regarding polygons, it notes that the sum of interior angles is (n-2)180 for a convex n-sided polygon, and the sum of exterior angles at each vertex is 360 degrees.
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This document provides information about triangles, including their basic properties, types, dimensions, and formulas for calculating perimeter and area. It defines triangles as three-sided, two-dimensional shapes where the sum of the interior angles is 180 degrees. Triangles are classified based on side lengths as equilateral, isosceles, or scalene, and based on angle measures as right, obtuse, or acute. The dimensions needed to calculate a triangle's perimeter and area are its three side lengths, base, and height. Formulas provided are that perimeter equals the sum of the three sides and area equals one-half the base times the height. Examples are given to demonstrate calculating perimeter and area of triangles.
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2) The Hypotenuse-Acute Angle (HyA) Congruence Theorem states that if the hypotenuse and an acute angle of one right triangle are congruent to the corresponding hypotenuse and acute angle of another right triangle, then the triangles are congruent.
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It also describes three theorems used to prove triangle congruence: SSS (three equal sides), SAS (two equal sides and the included angle), and ASA (two equal angles and one included side).
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1) A triangle is a three-sided polygon with three vertices and three edges.
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2) The Hypotenuse-Acute Angle (HyA) Congruence Theorem states that if the hypotenuse and an acute angle of one right triangle are congruent to the corresponding hypotenuse and acute angle of another right triangle, then the triangles are congruent.
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This document defines and describes various types of polygons including: convex polygons which have no interior angles, concave polygons which have interior angles, equilateral polygons with all sides congruent, equiangular polygons with all interior angles congruent, and regular polygons which are both equilateral and equiangular. It also defines vertices, diagonals, and explains the interior angle sum theorem for quadrilaterals.
8.6 terms
This document discusses proportional relationships in triangles. It states that if a line parallel to one side of a triangle intersects the other two sides, it divides them proportionally. It also states that if three parallel lines intersect two transversals, they divide the transversals proportionally. Finally, it notes that if a ray bisects an angle of a triangle, it divides the opposite side proportionally to the lengths of the other two sides.
11.2 terms
This document discusses how to calculate the areas of regular polygons such as equilateral triangles and other shapes. It explains that the area of an equilateral triangle is (side^2) * √3/4 and the area of a regular polygon is (1/2) * apothem * perimeter. The apothem is the distance from any side to the center and the central angle is found by dividing 360 by the number of sides.
8.5 terms
Two methods for proving triangles are similar are presented: SSS similarity, where all corresponding sides between two triangles are proportional, proving the triangles are similar; and SAS similarity, where two corresponding sides of two triangles are proportional and the included angle is congruent, also proving the triangles are similar.
8.2 terms
This document discusses proportions and their properties in geometry. A proportion is two equal ratios that can be set up and cross-multiplied to solve for unknown values. The terms of a proportion can be switched diagonally or by adding denominators to numerators while maintaining equality. The geometric mean is the value diagonal to itself in a proportion of two positive numbers.
6.2 terms
A parallelogram is a quadrilateral with both pairs of opposite sides parallel. The properties of a parallelogram are that if it is a parallelogram, its opposite sides are congruent, its opposite angles are congruent, consecutive angles are supplementary, and its diagonals bisect each other.
6.7 terms
This document discusses the formulas for calculating the areas of various shapes. It states that congruent polygons have the same area, and the area of a region is the sum of non-overlapping areas. It then provides the area formulas for a square, rectangle, parallelogram, triangle, trapezoid, kite, and rhombus. For a rhombus, it notes that the area can also be calculated as for a parallelogram using base times height.
6.4 terms
This document defines and compares properties of rhombuses, rectangles, and squares. A rhombus is a parallelogram with 4 congruent sides and perpendicular diagonals. A rectangle is a parallelogram with 4 right angles and congruent diagonals. A square has the combined properties of a rhombus, rectangle, and parallelogram - it is a rhombus and rectangle with 4 equal sides and right angles.
10.4 terms
If lines intersect within or on the edge of a circle, the measure of the angles formed can be determined based on the intercepted arcs. For intersections inside the circle, the angle measure is half the sum of the intercepted arcs. For intersections on the edge of the circle, the angle measure is half the intercepted arc. For intersections outside the circle, the angle measure is half the difference of the intercepted arcs.
10.2 terms
The document discusses key concepts relating to arcs and chords in circles. It defines a central angle as an angle whose vertex is the center of the circle, with the measure of its corresponding arc being equal to the angle's degrees. Arcs are classified as minor, major, or semicircles based on their angle measures. Adjacent arcs can be added together to find the measure of a larger arc. There are several theorems stated about relationships between arcs, chords, diameters, and angles, such as congruent arcs corresponding to congruent chords, diameters bisecting chords and angles if perpendicular to a chord, and chords being congruent if equidistant from the center.
9.6 terms
This document provides instructions for solving right triangles using trigonometric functions. It explains that solving a right triangle involves finding the measures of the three angles and lengths of the three sides. It notes that a calculator's inverse trigonometric functions allow working backwards from a trig ratio to the corresponding angle measure. Finally, it offers guidance for determining which inverse function to use based on the given information, such as using tangent when the hypotenuse is unknown or cosine when the opposite side is missing.
6.5 terms
This document defines and provides key details about trapezoids, isosceles trapezoids, the midsegment of a trapezoid, and kites. It states that a trapezoid has one pair of parallel sides called bases, and the other sides are called legs. An isosceles trapezoid has congruent legs and congruent base angles. The midsegment of a trapezoid connects the midpoints of the legs and is parallel to the bases. A kite has two pairs of consecutive congruent sides but opposite sides are not congruent, and its diagonals are perpendicular with one pair of opposite angles being congruent.
11.5 terms
The document discusses calculating the areas of circles and sectors. It states that the area of a circle is calculated as A = πr2, and the area of a sector is calculated as (Arc Measure/360)(area of the full circle), which finds the percentage of the total circular area that the sector represents by dividing the arc measure in degrees by 360.
11.1 terms
The document discusses angle measures in polygons. It states that the sum of interior angles in any polygon is (n-2)180, where n is the number of sides. It also explains that the sum of exterior angles is always 360, and each exterior angle is calculated as 360/n, where n is again the number of sides.
10.3 terms
This document discusses inscribed angles and their relationship to circles. It states that an inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. It also notes that an inscribed angle measures half the degrees of its intercepted arc and that two inscribed angles intercepting the same arc together measure the full arc. Additionally, it mentions that if a right triangle is inscribed in a circle, its hypotenuse is the diameter of the circle and that a quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
11.4 terms
This document discusses formulas for circumference and arc length. It states that circumference C equals 2πr or πd, where r is the radius and d is the diameter. It also explains that arc length is calculated as a percentage of the circumference by taking the arc measure in degrees and dividing by 360, then multiplying that fraction by the circumference.
9.5 terms
This document provides information about trigonometric ratios and using a calculator to evaluate trigonometric functions. It defines the sine, cosine, and tangent ratios using the SOHCAHTOA mnemonic. It also explains that calculators may require typing the trig function before or after the number, and some may not require hitting the equals sign, so it is important to know how to use the specific calculator.
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