Geometry Reviewer PDF

*Geometry Reviewer* ------------------- *by: Adrian Cedrick T. Calubaquib* **I. Collinearity, Betweenness and Congruent Segments** Collinearity -In geometry, collinearity refers to a property of points that lie on a single straight line. When three or more points are collinear, it means there is...

*Geometry Reviewer* ------------------- *by: Adrian Cedrick T. Calubaquib* **I. Collinearity, Betweenness and Congruent Segments** Collinearity -In geometry, collinearity refers to a property of points that lie on a single straight line. When three or more points are collinear, it means there is a line that passes through all of them. For example, if you have points A, B, and C, they are considered collinear if you can draw a straight line that intersects all three points. Ex: Points A, D and K are collinear because they lie on the same line. If you were to ask what is the betweenness, would you know it? Betweenness -Betweenness refers to the relative positions of points on a line segment. For instance, if you have three points, A, D, and K, point D is said to be \"between\" points A and K if all three points lie on a straight line (collinear) and AD + DK = AK. This means that D lies on the line segment connecting A and K. Ex: If line segment AD is 4x, DK is x+2 and AK is 9x-10, what is the value of x? And what is the measurement of the three segments? Solution: Given: -AK = 9x - 10 (the whole segment) -AD = 4x (a part of the segment) -DK = x + 2 (another part of the segment) We can set up the equation: AD+DK=AK Substituting the given values: 4x+(x+2)=9x−10 Simplifying the equation: 5x+2=9x−10 Transpose 5x in the right side, and −10 in the left side: 2+10=9x-5x *Take note that the sign of the number changes when it crosses the equivalent sign!* Simplify the equation: 12=4x Isolate the x by dividing 4 from both sides: 12/4=4x/4 Simplify the equation: 3=x Now let\'s find the measurement of the three segments: - - - So, the value of x is 3, and the measurements of the segments are: - - - Congruent Segments -Congruent segments are line segments that have the exact same length. When two segments are congruent, you can think of them as being identical in terms of their measurement, even if they are located in different positions or orientations. -Here\'s a formal definition: Two line segments AB and CD are said to be congruent if AB=CD. This means that the distance from point A to point B is equal to the distance from point C to point D. -In notation, this is often written as: AB≅CD -The symbol ≅ denotes congruence. Ex: Points L, M and O are collinear. M is the midpoint of L and O. Line segment LM is 3x and line segment MO is 2x+10. What is the measurement of the whole segment? Solution: Given: -LM=3x -MO=2x+10 Since M is the midpoint of line segment LO, LM is equal to MO Equation: LM=MO Substitute the given values: 3x=2x+10 Transpose 2x in the left side of the equation: 3x-2x=10 Simplify the equation: x=10 Now let's find the measurement of the segments. - - LM+MO=LO 30+30=60 Thus, segment LO is 60 units. *Take note: Angle bisectors do not necessarily split an angle into two congruent parts. It may vary based on the given value. Only perpendicular bisectors split an angle into two equal parts.* **II. Pythagorean Theorem** -One of the best known and most useful mathematical principles is the Pythagorean Theorem, credited to the Greek mathematician and philosopher Pythagoras. It is believed that he proved this theorem about 2500 years ago. -In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs (c^2^=a^2^+b^2^). Ex: A trapezoid is cut into three shapes. One square in the middle and 2 small right triangles from both sides. The base one measures 11cm and base two measures 21 cm. The hypotenuse of the trapezoid is 13cm. What is the height of the two triangles? Solution: We will use the Pythagorean Theorem to find the height of the triangles. C^2^=A^2^+B^2^ C would be our hypotenuse and A will be the length of the right triangle. But it is not given. How would we solve it? The solution is, we need to subtract base one from base 2 to get the length. But there are 2 triangles. So we need to get 2 lengths. B~2~-B~1~=2L 21 cm-11 cm=2L 10cm=2L 10cm/2=2L/2 5cm=L After finding the length of the triangles, we need to substitute it to our Pythagorean Theorem. (13)^2^=(5)^2^+B^2^ B is our height, which is the one with the missing value. 169=25+B^2^ Transpose 25 to the left side of the equation. 169-25=B^2^ 144=B^2^ To get our B, we need to extract the exponent by finding the square root of 144. [\$\\sqrt{}144 = \\sqrt{}B\$]{.math.inline}^2^ 12=B Thus, the height of the triangles is 12cm. **III. Relationship among the Angles of a Triangle** Angle Sum Theorem -The sum of the interior angles of any triangle is 180° (m\