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Transcript
Geometry 2nd Semester Final Study Guide Chapter 6 Identify medians in triangles o Connects vertex to midpoint o Medians meet at the centroid Centroid is 2/3 of the way along a median Identify altitudes and perpendicular bisectors in triangles o Altitude: from vertex to opposite side and perpendicular o Perpendicular Bisector: Goes through the midpoint of a side, and is perpendicular to that side Identify and use angle bisectors in triangles o Goes through the vertex of a triangle, and splits the angle into 2 congruent parts Identify and use properties of isosceles triangles o Parts of isosceles triangles: legs, base, base angles, vertex angle o 2 congruent legs ↔ 2 congruent base angles Use tests for congruence of right triangles o LL, HA, LA, HL What do the H’s, L’s, and A’s stand for? Use the Pythagorean Theorem and its converse o If a triangle is a right triangle, then 𝑎2 + 𝑏2 = 𝑐 2 , c is the hypotenuse o Converse: If 𝑎2 + 𝑏2 = 𝑐 2 , then a triangle is a right triangle Find the distance between two points on the coordinate plane o 𝑑 = √(𝑥1 − 𝑥 2 )2 + (𝑦1 − 𝑦2 )2 Chapter 7 Apply inequalities to segment and angle measures o Inequality symbols: >, <, ≥, ≤, ≠, ≱, ≰ o Be able to compare segments on a number line o Be able to compare angles in a figure Identify exterior angles and remote interior angles of a triangle and use the Exterior Angle Theorem o What is an exterior angle? Remote interior angles? o Exterior angle = sum of the remote interior angles Identify the relationships between the sides and angles of a triangle o Biggest angle across from longest side o Shortest side across from smallest angle o Be able to list sides/angles in order when given the angles/sides Identify and use the Triangle Inequality Theorem o 2 sides added together > 3rd side o Find a range of possible values for the 3 rd side of a triangle when given 2 sides Small end: subtract 2 given sides Large end: add 2 given sides Chapter 9 Use ratios and proportions to solve problems o Always reduce ratios o Proportions: Cross products are always = Identify similar polygons o Same shape, different size o Angles are congruent o Sides are proportional Use AA~, SSS~, and SAS~ similarity tests for triangles o Check to see if angles are congruent or sides are proportional Identify and use the relationships between proportional parts of triangles o Small triangle and big triangle Use proportions to determine whether lines are parallel to sides of triangles o If segment in triangle is parallel to the side it doesn’t touch, then set up proportion to find missing part o If sides are split proportionally, then line is parallel to side it doesn’t intersect o Triangle Midsegments – connects midpoints of sides Parallel to side it doesn’t touch Half the length of side it doesn’t touch Identify and use the relationships between parallel lines and proportional parts o 3 parallel lines crossed by 2 transversals: transversals are split proportionally Identify and use proportional relationships of similar triangles o Perimeter can be used just like sides in similar triangles o Scale Factor: match sides of similar triangles, make a ratio, and reduce of = top, to = bottom Chapter 13 Multiply, divide, and simplify radical expressions o No perfect squares left under √ o No fractions under √ o No √ on the bottom of a fraction Multiply numerator and denominator by that √ Use the properties of 45-45-90 triangles o “leg times √ 2 = the hypotenuse” Use the properties of 30-60-90 triangles o “shorter leg times 2 = the hypotenuse” o “shorter leg times √ 3 = the longer leg” Use the sine, cosine, and tangent ratio to solve problems o SOH CAH TOA o Use inverses to find angles o Angles of elevation/depression Chapter 8 Identify and use the properties of parallelograms o Opposite sides are parallel o Opposite sides are congruent o Opposite angles are congruent o Consecutive angles are supplementary o Diagonals bisect each other Identify and use tests to show that a quadrilateral is a parallelogram o Are opposite sides parallel? o Are opposite sides congruent? o Are opposite angles congruent? o Do the diagonals bisect each other? o Is one pair of opposite sides both parallel and congruent? Identify and use the properties of rectangles, rhombi, and squares o Rectangles: All properties of a parallelogram + 4 right angles Congruent diagonals (forms 4 congruent parts on the diagonals) o Rhombi: All properties of a parallelogram + 4 congruent sides Perpendicular diagonals Diagonals bisect the opposite angles (forms 4 congruent angles across from each other) o Square: All properties of a parallelogram, rectangle and rhombus Chapter 10 Name polygons according to the number of sides and angles o Concave/convex o Regular polygon – both equilateral and equiangular o Parts of polygons Name consecutive/nonconsecutive sides, vertices, angles Name diagonals Find measures of interior and exterior angles of polygons o Interior Angles Sum of Interior Angles: 𝑆 = (𝑛 − 2)180 Each interior angle of a regular polygon: 𝐼 = ( 𝑛−2) 180 𝑛 o Exterior Angles Sum of Exterior Angles: always 360° (no matter how many sides) Each exterior angle of a regular polygon: 𝐸 = 360 𝑛 o Interior Angle + Exterior Angle = 180° (form a linear pair) Find the areas of triangles and trapezoids 1 o Triangle: 𝐴 = 2 𝑏ℎ 1 o Trapezoid: 𝐴 = 2 (𝑏1 + 𝑏2 )ℎ Estimate the areas of polygons o Split irregular polygons into shape you can find the area of 1 o Estimate: Full squares = 1 unit2 , partial squares = unit2 2 Find the areas of regular polygons o Apothem, a, connects center of polygon to a side and is perpendicular o Perimeter, P, is the number of sides, n, times the length of each side, s: 𝑃 = 𝑠 ∙ 𝑛 1 o Area: 𝐴 = 2 𝑎𝑃 o Area of shaded region: 𝐴𝑠ℎ𝑎𝑑𝑒𝑑 = 𝐴𝑤ℎ𝑜𝑙𝑒 − 𝐴𝑢𝑛𝑠ℎ𝑎𝑑𝑒𝑑