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In geometry, we start with basic ideas like points, lines, and planes. These are undefined terms because we don’t need to define them; they’re understood and used to create other shapes like rays, segments, and angles. A point is a location with no size, a line is a straight path that extends infinitely, and a plane is a flat surface that extends infinitely in all directions. For example, the problem asks which points are coplanar with D and E. Coplanar means points are on the same plane. So, we eliminate points C, B, F, and G, leaving D, A, H, and E. Since the question specifically asks for points that are coplanar with D and E, the answer is points A and H.

DISTANCE FORMULA: The distance formula is used to find the distance between two points in a coordinate plane. It’s based on the Pythagorean Theorem and can be written as the following:

d = √ (x₂ - x₁)² + (y₂ - y₁)²

Let's say that in the example given, we have two points: (1, 3) and (7, 6). Using the distance formula, plug the numbers in: d = √ (7 - 1)² + (6 - 3)². Next, simplify inside the parentheses: d = √ 6² + 3². Then, simplify the squares: d = √ 36 + 9. Finally, the distance is √ 45, which is your final answer.

MIDPOINT FORMULA: The midpoint formula is used to find the point exactly halfway between two given points in a coordinate plane. It is written as the following:

(x₁ + x₂)/2, (y₁ + y₂)/2.

Let's say that there is an example that is giving us two points: (4, 1) and (10, 5). Using the midpoint formula, plug the numbers in: (4 + 10)/2, (1 + 5)/2. Then, do solve one point at a time: (14/2, 6/2), simplify all the numbers to get the final answer:

( 7, 3).

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Conditional Statements and Venn Diagrams

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A conditional statement is an "if-then" statement, where the "if" part is the hypothesis and the "then" part is the conclusion. The inverse negates both parts, the converse switches them, and the contrapositive both negates and switches them. A biconditional is true when both the conditional and its converse are true, often written as "if and only if." A Venn diagram uses overlapping circles to show relationships between sets. If all elements of set p are in set q, p’s circle is inside q. If some elements are shared, the circles overlap. If there’s no relationship, the circles don’t overlap, meaning there are no common elements. In the image, the problem asks for a conditional statement that matches the Venn diagram. Based on the diagram, I would write, "If a shape is a square, then it is a rhombus." This is because the circle for "Squares" is inside the circle for "Rhombuses", meaning all squares are rhombuses.

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Parallel lines are lines in a plane that never intersect and are always the same distance apart, having the same slope. For instance, opposite sides of a rectangle are parallel. Perpendicular lines, on the other hand, intersect at a right angle. The slopes of two perpendicular lines are negative reciprocals of each other, meaning if one line has a slope of 2, the perpendicular line will have a slope of -1/2. In this example, I would start by converting the equation

8x+2y=10 from standard form to slope-intercept form. This gives y=-4x+5. Next, I would solve for b using the point (2, −1) and the slope -4. Substituting into the equation, −1=−4(2)+b, and simplifying gives −1=−8+b. Solving for b, I get b=7. Finally, the equation in slope-intercept form is y=−4x+7.

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Point-slope form is used to write the equation of a line when you know the slope m and a point (x1, y1) on the line. The formula is y−y1=m(x−x1), which makes it easy to find the equation or graph the line from a point and slope. Standard form is written as Ax+By=C, which is a way of writing a math concept, like an equation, number, or expression, following specific rules. This form is useful for solving systems of equations and finding the x- and y-intercepts. It’s often used when working with integer values or comparing different lines. In the example given, it asks which line is parallel to the line shown below. The first thing I did was convert all answer choices from standard form to point-slope form. This gave me:

A) y= -3/4x - 1, B) y= 3/4x - 3,

C) y= =4/3x - 2, and D) y= 4/3x -5.

Then, using the slope formula, I would find the slope of both points: (-1, -1) and (-2, -3). With this, the slope I get is 4/3. Therefore, my final answer would be D) y=3/4x - 3 because it has the same slope as the two points given.

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Triangle Congruence: SSS, SAS, ASA, AAS, HL

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Triangle congruence means two triangles are the same in size and shape. You can prove two triangles are congruent using different criteria: SSS (Side-Side-Side) means all three sides are equal, SAS (Side-Angle-Side) means two sides and the included angle are equal, ASA (Angle-Side-Angle) means two angles and the included side are equal, AAS (Angle-Angle-Side) means two angles and a non-included side are equal, and HL (Hypotenuse-Leg) applies to right triangles, where the hypotenuse and one leg are equal. These rules make it easier to prove triangles are congruent without checking every side and angle. In this example, the first step is to look at the given information. The first statement is that line AC is congruent to line AD, and the reason is Given. Next, write that line CB is congruent to line DB, also by the reason Given. The third statement is that line AB is congruent to line AB because it’s the same line. The reason for this is Reflexive Property. Finally, the last statement would be that triangle ABC is congruent to triangle ABD, and the reason is SSS (Side-Side-Side) because no angles were used.

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A parallelogram is a four-sided shape where opposite sides are parallel and congruent. Opposite angles are congruent, and consecutive angles add up to 180 degrees. The diagonals bisect each other, meaning they cut each other into equal parts. Special types of parallelograms include rectangles, rhombuses, and squares, each with unique properties related to angles or sides. In this example, the first thing I did was find JK, which is 21, since opposite sides are congruent. Then, I found KL, which is 29, also because opposite sides are congruent. After that, I determined that angle J is 127 degrees because opposite angles are congruent. Next, I found out that angle M was 53 degrees because consecutive angles are supplementary (180 - 127 = 53). Finally, I found that angle K is also 53 degrees because opposite angles are congruent.

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Rectangles are quadrilaterals with the same properties as parallelograms: opposite sides are congruent, opposite sides are parallel, opposite angles are congruent, consecutive angles are supplementary, and diagonals bisect each other. Also, rectangles have these unique properties: all four angles are right angles and both diagonals are congruent. In the problem shown above, the first thing I did was find the measurement of angle XWY. I did this by subtracting 64 degrees from 180 degrees, which gave me 116 degrees. Then, I divided by two because opposite angles in a triangle are congruent, and since angle XWY is part of an isosceles triangle, this gave me the measure of angle XWY as 58 degrees. Next, I found the measurement of angle YXZ. Since all angles in a rectangle are 90 degrees, I subtracted the 58 degrees I just calculated from 90 degrees. Therefore, 90 - 58 = 32, which is the measure of angle YXZ. Then, I found the measurement of angle WVZ. Since consecutive angles in a linear pair are supplementary, I subtracted 64 degrees from 180 degrees, which gave me 116 degrees as the measurement of angle WVZ. Next, I found the measurement of angle XWZ. Since it is a right angle, the measurement is 90 degrees. Finally, I found the measurement of angle XZY. Since opposite angles are congruent, the measurement of angle XZY is the same as angle XWY, which is 58 degrees.

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A rhombi is a quadrilateral that has the following properties: opposite sides are congruent, opposite sides are parallel, opposite angles are congruent, consecutive angles are supplementary, and diagonals bisect each other. Plus, all sides are congruent, diagonals are perpendicular, and diagonals bisect the opposite angles.In this example, the first thing I did was solve for side GF. I found that side GF was equal to 23 because side EF was also equal to 23, and opposite sides are congruent. Next, I solved for HF, which was 20. Since DF was equal to 40 and HF is half of DF, I divided 40 by 2 to get 20. Then, I used the Pythagorean theorem to solve for GH. The Pythagorean theorem states that a² + b² = c2 . I plugged in the values and set up the equation: x² + 20² = 23² . Simplifying the equation, I got: x² + 400 = 529. I subtracted 400 from both sides: x2 = 129. Taking the square root of both sides:

x = √129 = 11.4. So, GH = 11.4. Finally, to find GE, I used the fact that GH is half of GE. Therefore, I multiplied 11.4 by 2 to get GE: 11.4×2=22.8.

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A square is a quadrilateral with all four sides equal in length and all four angles equal to 90 degrees. It has all the properties of a parallelogram, rectangle, and rhombus: opposite sides are congruent, opposite sides are parallel, opposite angles are congruent, consecutive angles are supplementary, diagonals bisect each other, all four angles are right angles, diagonals are congruent, all four sides are congruent, diagonals are perpendicular, and diagonals bisect the opposite angles. In the problem abovem, they are asking you to find VT. The first thing I would do is try to solve x by setting up my equation: 2x + 13 = 8x - 41. They are equal to eachother because diagonals are congruent. Then, I would simplify the equation by moving the number with variables to one side and the numbers without variables to the other side leaving me with the following: -6x = -54. Next, I would simplify by dividing both sides by -6. This would give me x = 9. Finally, I would replace x in one of the equations given: 2(9) + 13. I would then simplify leaving me with 31. Lastly, I would multiply 31 x 2 because that is only half of what they are asking for. My final answer would be VT = 62.

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A trapezoid is a quadrilateral with one pair of opposite sides parallel. A non-isosceles trapezoid is a trapezoid where the non-parallel sides are not equal in length. Its properties include: only one pair of opposite sides is parallel, and consecutive angles between the parallel sides are supplementary. On the other hand, an isosceles trapezoid is a trapezoid where the non-parallel sides (legs) are congruent. Its properties include the same as non-isosceles trapezoids, and the following: congruent non-parallel sides (legs), congruent diagonals, congruent base angles, and supplementary opposite angles. In the problem given, they are asking for the measurement of angle R. I would begin by identifying what type of trapezoid this is; it is an isosceles trapezoid. Then, I would add the two angle expressions provided because consecutive angles in an isosceles trapezoid are supplementary, leaving me with the following equation: (12x + 3) + (7x - 13) = 180. Simplifying the side on the left: 19x - 10 = 180. Next, I would add 10 to both sides: 19x = 190. Then, I would divide both sides by 19: x = 10. Finally, I would substitute x for 10 in the equation for angle R: 7(10) - 13 = 70 - 13 = 57 degrees. My final answer is 57 degrees.

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Slide 1-

https://www.ck12.org/assessment/ui/?test/view/practice/geometry/geometric-definitions-practice&branch=geometry&type=practice&referrer=google_search_practice&collectionHandle=geometry&collectionCreatorID=3&conceptCollectionHandle=geometry-::-geometry-terms&ep=https%253A%252F%252Fwww.ck12.org%252Fassessment%252Fui%252F%253Ftest%252Fdetail%252Fpractice%252Fgeometry%252Fgeometric-definitions-practice%2526collectionHandle%253Dgeometry%2526collectionCreatorID%253D3%2526conceptCollectionHandle%253Dgeometry-%253A%253A-geometry-terms%2526testType%253Dpractice%2526referrer%253Dgoogle_search_practice

Slide 2- https://quizlet.com/explanations/questions/write-a-conditional-statement-that-each-venn-diagram-illustrates-center-img-srchttpsslader-solution-uploadss3amazonawscom6f2e91aa-41ec-4883--0c47f8b9-b77b2124-f4ac-447f-b7a1-87c1ed4fd85a

Slide 3- https://youtu.be/omOXhj3HmNk?si=fykeigRBziA22F1t

Slide 4- Gina Wilson All Things Algebra Point-Slope and Standard Form Packet

Slide 5- Gina Wilson All Things Algebra Triangle Congruence: SSS, SAS, ASA, AAS, HL Packet

Slide 6- Gina Wilson All Things Algebra Parallelograms Packet

Slide 7- Gina Wilson All Things Algebra Rectangles Packet

Slide 8- Gina Wilson All Things Algebra Rhombi Packet

Slide 9- Gina Wilson All Things Algebra Squares Packet

Slide 10- Gina Wilson All Things Algebra Trapezoids Packet

Slide 11- Gina Wilson All Things Algebra Midsegment of a Trapezoid Packet

Slide 12-Gina Wilson All Things Algebra Kites Packet

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A kite is a type of quadrilateral with two pairs of adjacent sides that are the same length. It has one pair of opposite angles that are equal, which are the angles between the equal sides. The diagonals of a kite cross each other at a right angle, and one diagonal cuts the other into two equal parts. These special properties make a kite different from other shapes. In the example given, I started by finding angle GDE by adding the two congruent angles of 59 degrees each, resulting in 118 degrees. Then, to find angle DEH, I added 59 degrees and 90 degrees to get 149 degrees, and subtracted it from 180 degrees to get 31 degrees. Since opposite angles in a kite are congruent, angle DGH is also 31 degrees.

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The midsegment of a trapezoid is the line segment that connects the midpoints of the non-parallel sides. In other words, it's the line drawn between the midpoints of the legs of the trapezoid. In the example shown below, it states, "For trapezoid PQRS, Y and Z are midpoints of the legs. Find YZ." I would begin by setting up my equation: 5x - 19 = 1/2 (38 + x + 14). Then, I would simplify the equation even further: 10x - 38 = 52 + x. Next, I would solve for x: 9x = 90. Therefore, x = 10. Finally, I would substitue x into the equation of YZ: 5(10) - 19 = 31. Thus, my final answer is 31.