Math Reviewer PDF

Classify triangles according to their angles and sides. Illustrate, name, and identify different kinds of triangle and define the terms associated with a triangle. Types of triangles according to sides: Equilateral = all sides are equal Isosceles = two sides are equal Scalene = none of the sides ar...

Classify triangles according to their angles and sides. Illustrate, name, and identify different kinds of triangle and define the terms associated with a triangle. Types of triangles according to sides: Equilateral = all sides are equal Isosceles = two sides are equal Scalene = none of the sides are equal Types of triangle according to angles: Right triangle ( one right angle and two acute) Obtuse triangle ( one obtuse angles and two acute) Acute triangle (three acute angles) Basic parts of a triangle Vertex (connects two sides of a triangle) Height (self explanatory) Sides (self explanatory) Interior Angle (inside the triangle) Exterior Angle (outside the triangle) Secondary parts of a triangle Angle bisector (divides the angle in two equal parts) Perpendicular bisector (goes through the midpoint and forms a right angle) Median (connects vertex and midpoint) Altitude (self explanatory) Lesson 2: Illustrate theorems on triangle inequalities (Exterior Angle Inequality Theorem, Triangle Inequality Theorem, Hinge Theorem) Exterior Angle (angle that is adjacent and supplementary to the triangle) Remote interior angle (not adjacent to the exterior angle) Adjacent interior angle ( adjacent to exterior angle and forms a linear pair 180 degrees) Exterior angle theorem ( is equal to the sum of the remote interior angles) Lesson 2.5 IMPORTANT to solve for angles that are in a equation style just combine them like 1+2=3-4 1.​ then you tranpose(switch) the second values of each side (dont forget to change the signs to the opposite) 2.​ you simplify 3.​ then divide both sides with the left side 4.​ example: 5x/5 = 10/5 so it becomes x = 2 5.​ its that simple if its like a corresponding thing just combine the second value (1-2) with the value of the one thats given (1-2) (72) then you divide with the left side so basically same steps 1/1=74/1 = 74 boom Lesson 3: Illustrate triangle congruence. Corresponding( same position) Congruence ( same size and shape) CPCTC ( Corresponding parts of a congruent triangle are congruent) Included angle ( the angle between two sides) Included side ( the side between two angles) Reflexive property (1=1) Symmetric property (1=2 2=1) Transitive property (1=2 2=3 1=3) Illustrate the SAS, ASA and SSS congruence postulates, and SAA congruence theorem. SSS (all three sides are congruent to the other triangle) SAS (two sides and included angle are congruent to the other triangle) ASA (two angles and included side are congruent to the other triangle) SAA (two angles and the opposite side are congruent to the other triangle) Lesson 4: Two column proof Lesson 5: Isosceles triangle (two equal sides) Isosceles triangle theorem (if two sides are congruent then the angles opposite of them are congruent) Converse of the isosceles triangle theorem (if two angles are congruent then the sides opposite of them are congruent) Corollary 28.1 (every equilateral triangle is equiangular) Corollary 29.1 (every equiangular triangle is equilateral) Lesson 6: Hypotenuse (Longest part of a right triangle) Leg (the ones that aren’t the longest) Hypotenuse leg theorem (if they are congruent then the other triangle also is congruent with that) Leg leg theorem ( if they are congruent then the other triangle also is congruent with that) Hypotenuse acute angle ( if they are congruent then the other triangle also is congruent with that) Leg acute angle ( if they are congruent then the other triangle also is congruent with that) (just read the name to understand basically it means whatever is in the name thats whats congruent) Lesson 7: Linear pair postulate (if two angles form a linear pair then they equal 180 degree) Line postulate (through any two points there is a line) Perpendicular bisector (a line that bisects a line segment at a right angle) Theorems on triangle inequalities - this theorem states that, for any given triangle, the sum of the lengths of any two sides of a triangle is always greater than the length of the third side. -a+b>c b+c>a a+c>b SIDE-ANGLE INEQUALITY THEOREM -​ If one side of a triangle is longer than the second side, then the measure of the angle opposite the longer side is greater than the measure of the angle opposite the shorter side. ANGLE-SIDE INEQUALITY THEOREM -If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer the side opposite the smaller angle. EXTERIOR ANGLE INEQUALITY THEOREM -states that "the measure of any exterior angle of a triangle is greater than both of the non-adjacent interior angles (two remote interior angles) HINGE THEOREM -(which may also be referred to as the SAS Inequality Theorem) If two sides of one triangle are congruent to two sides of a second triangle and the included angle of the first triangle is GREATER than the included angle of the second triangle, then the third side of the first triangle is LONGER than the third side of the second triangle. (if the included angle (angle between two sides) of the first triangle is bigger than the second triangle then it means the third side of the first triangle is bigger, bigger angle bigger length) CONVERSE HINGE THEOREM -(which may also be referred to as the SSS Inequality Theorem) If two sides of one triangle are congruent to two sides of a second triangle, and the third side of the first triangle is LONGER than the third side of the second triangle, then the included angle of the first triangle is LARGER than the included angle of the second triangle.( the same as hinge theorem but instead of checking the angle we check the third side, bigger side = bigger included angle) RATIOS: As said, ratios are two quantities just put beside each other To simplify ratios, we divide both sides by their greatest common factor (something that can divide both sides properly into a whole number, 10/5 can be divided by 5 (Their GCF) so it becomes 2/1)  Describe a proportion. They are ratios that are equal This means that when you cross multiply, the answers you get should be equal to each other Simplify Fractions (If Possible): ​ Start by simplifying any fractions to make the numbers easier to work with. Cross Multiply: ​ Multiply diagonally across the equal sign to eliminate the fractions. ​ Example: For ab=cd\frac{a}{b} = \frac{c}{d}, cross-multiply to get a×d=b×ca \times d = b \times c. Distribute (If Necessary): ​ If you have expressions in parentheses, distribute the multiplication. ​ Example: k(m+n)=pk(m + n) = p becomes km+kn=pkm + kn = p. Transpose to other side: ​ Use addition or subtraction to move terms around, aiming to isolate the variable on one side. ​ Example: 6x+8=86x + 8 = 8 becomes 6x=06x = 0 by subtracting 8 from both sides. Solve for the Variable: ​ Divide or multiply to get the variable by itself. (divide using the coefficient aka the number with a variable) ​ Example: 6x=06x = 0 becomes x=0x = 0 by dividing both sides by 6. TRIANGLE PROPORTIONALITY THEOREM Theorem 7-4-1 This means that if a line is parallel to a side of a triangle and intersects two sides, then the sides are divided proportionally( the sides are divided equally to each other, ae= af, eb=fc) Theorem 7-4-2 The line that divides the triangle becomes parallel to the third side (ef = bc) Theorem 7-4-4 This means that when the angle bisector divides the opposite side, it is proportional to the other two sides in length, (bd = ab, dc = ac) TRIANGLE SIMILARITY Always cross multiply, just write corresponding side divided by corresponding side, AB/DE AC/DF just like that then substitute then multiply then divide by the left side, (u can see 8DE = 64 then it becomes DE= 8 u just divide by the left side then boom) If unsure what to do then transpose ok now with what u learned try to answer this keep practicing or else ur cooked Almost done bro next lesson 🔥 Quick refresh RIGHT TRIANGLE SIMILARITY Memorize these GEOMETRIC MEAN Just multiply both numbers then squareroot Analyze and find a pattern PYTHAGOREAN + SPECIAL RIGHT TRIANGLES BASICALLY special right triangles Just remember and understand what it means 2x is 2(X) and x times square root of 2 or 3 is what it seems just substitute for x always And we are done bro its over now review the whole thing again until u master it