1 Geometry Basics Concepts you must know!

2 How to find the distance between two points (this is the same as finding the length of a segment).Watch this video to review how to find the distance between two points.

3 How to find the midpoint of a segmentIn order to find the midpoint of segment AC, first find the slope from A to C. Slope = 𝑟𝑖𝑠𝑒 𝑟𝑢𝑛 = 𝑢𝑝 2 𝑟𝑖𝑔ℎ𝑡 6 Since we want to find the point in the middle, divide the rise and the run by two, to find the new slope that will take you to the midpoint. Slope to find midpoint= 𝑢𝑝 1 𝑟𝑖𝑔ℎ𝑡 3 The midpoint of 𝑨𝑪 is at (0,2).

4 Translations This transformation could be described in the following three ways: Using the rule (formula): 𝑥+9,𝑦−4 In words: translate the shape 9 units to the right and 4 units down. Using translation vector 9,−4 Watch video explaining this translation.

5 Reflection over the y axisTo reflect a point over the y axis, measure the distance from the point to the y axis and find the point on the other side of the y axis that is located that same distance from the y axis. So point C is 5 units to the right of the y axis, notice that its image, C’ is 5 units to the left of the y axis. Video explaining reflection over the y axis or over the x axis

6 Reflection over the x axisTo reflect over the x axis, find the distance from the point to the x axis and count that same distance on the other side of the x axis to find the location of the image. As you can see point B is 4 units above the x axis, and its image, B’ is also 4 units from the x axis but under it. Video explaining reflecting over the y axis or over the x axis.

7 Perpendicular bisectorWhen a point is reflected over a line, the line of reflection is the perpendicular bisector of the segment connecting the preimage (point P) to the image (point P’). So, 𝑃𝑀 ≅ 𝑀𝑃′ And right angles are formed at the intersection.

8 90º CCW Rotation Apply the formula to each point 𝑥 ,𝑦 −𝑦,𝑥The formula shows that the x value becomes the new y, and the opposite of the y becomes the new x value. So for example, the image of point C 4,5 is C’ −5,4 Video explaining how to rotate 90 CCW

9 180º Rotation Apply the 90º rotation formula to each point twice. The x value becomes the new y, and the opposite of the y becomes the new x value. 𝑥 ,𝑦 2 ,5 −𝑦,𝑥 −5,2 −𝑥 ,−𝑦 −2 ,−5 So the image of point C 2,5 is C’ −2,−5

10 Angles

11 Video explaining linear pair and vertical anglesLinear pair angles Linear pair angles are adjacent angles (next to each other) that together form a straight angle. These two angles are supplementary (their sum is 180º). So: 𝑚∢2+𝑚∢1=180° Video explaining linear pair and vertical angles

12 Vertical angles When two lines intersect, four angles are formed. The two angles that are opposite to each other are called vertical angles and they measure the same. So, ∢1=∢3 And ∢2=∢4 Video explaining linear pair and vertical angles

13 Angles formed by parallel linesWhen two parallel lines are intersected by a transversal, the angles formed are equal. Although these angles have specific names, the most important fact to know is that all the acute angles will be equal and all the obtuse angles will be equal. In the diagram you can see that all the acute angles measure x and all the obtuse angles measure 180-x. Video explaining angles formed by parallel lines and a transversal

14 Graphing lines To graph an equation of the form 𝑦=𝑚𝑥+𝑏:graph the y intercept, in this example the y intercept is 2, so put a point at 2 on the y axis Find other points by using the slope, in this case you find them by going up 2, to the right 3 or by going down 2 and to the left 3. Video explaining how to graph 𝑦= 𝑚𝑥+𝑏

15 Writing the equation of a line Video showing how to find equation of a lineIf you know a point on a line and the slope of the line, you can find the equation of the line by using the point-slope formula.

16 Definitions Plane: a flat surface Collinear: points on the same line Coplanar: points or shapes on the same plane Parallel lines: two coplanar lines that never intersect. The symbol for parallel is ∥ Perpendicular lines: two lines that intersect forming right angles. The symbol for perpendicular is ⊥

17 Slopes of parallel, perpendicular linesIf two lines are perpendicular, their slopes are opposite reciprocals. Here you can see the slope of one line is 𝟐 𝟑 while the other is − 𝟑 𝟐 If two lines are parallel, then their slopes are equal. Here you can see the slope of each line is 2.

18 How to find the equation of a line parallel Video showing how to find equation of parallel or perpendicular line

19 How to find the equation of a line perpendicular Video showing how to find equation of parallel or perpendicular line

20 Shortcuts to prove two triangles are congruent.Remember that AAA or SSA (the stinky one) cannot be used to prove that two triangles are congruent.

21 Triangle Angle Sum Example:

22 Isosceles Triangles An isosceles triangles has two equal sides called the legs. The side that is not equal is called the base. Example: The base angles of an isosceles triangle are equal.

23 Relationship between sides and anglesExample: Which is the smallest side in the triangle below? Since angle C is the smallest, the side opposite to it would be the shortest. So the answer is side AB.

24 Triangle Inequality The sum of any two sides must be greater than third side or else the three sides cannot form a triangle.

25 Parallelogram Properties