TheXSquaredFactor

In order to determine which pairs of figures are indeed similar, you must compare the ratios of one side to another.

A) \(\frac{18}{27}=\frac{10}{15}\)

Simplifying both the fractions into simplest terms, \(\frac{2}{3}=\frac{2}{3}\), which indeed proves both figures to be similar.

B) Do the same thing with the rectangles.

\(\frac{7}{10}=\frac{14}{18}\)

Doing the exact process, \(\frac{7}{10}\neq\frac{7}{9}\), so these figures here are not congruent.

C) All equilateral triangles are similar.

D) Yet again, check the similarity by comparing the ratio.

\(\frac{18}{24}=\frac{24}{32}\)

Simplifying in simplest terms, \(\frac{3}{4}=\frac{3}{4}\), so this pair of figures are similar. 

A, C, and D are all pairs of figures that are similar.

1) Triangle Sum Theorem

To find \(m\angle 3\), one must understand that the sum of the measures of all the interior angles of a triangle equals 180, which is what the triangle sum theorem states.

2) Corresponding

\(\angle 3\) and \(\angle 4\) are both corresponding angles because both angles maintain the relative positions at the intersection of two lines. 

3) Linear Pair Postulate

4) Subtraction Property of Equality

The final step requires elementary subtraction to figure out the measure of the remaining angle. 

1) Definition of a rectangle

A rectangle is a quadrilateral with 4 right angles. That is all. 

2) \(\overline{JM}\cong \overline{KL}\)

These are the only pair of segments listed that are opposite.

3) \(\overline{ML}\cong\overline{ML}\)

4) Side-Angle-Side Triangle Congruence Theorem

For this particular proof, you are utilizing one side, an included angle, and another side to prove the triangles congruent. 

To find the midpoint of a segment, find the average of the x-coordinates and y-coordinates.

The triangle sum theorem states that the sum of measure of all the interior angles in a triangle equals 180; in this case, the fact that both lines are parallel has little significance to this particular problem.

There is a theorem called the Exterior Angle Theorem, which states that the measure of the exterior angle of a triangle equals the sum of the measure of the two nonadjacent angles (also known as remote interior angles).

Knowing this information above, we can set up an equation and solve for x:

Yes, you are right that the isosceles triangle theorem would end this problem in one step.

Standard form of a quadratic is in the form \(ax^2+bx+c\), and vertex form is in the form of \(a(x-h)^2+k\).

a) The method of factoring requires a quadratic to be in standard form in order to use its procedure, so standard form is desired.

b) Vertex form is friendlier when it comes to graphing. One can quickly identify the vertex (hence its name vertex form), graph the points to the right, and then reflect about the axis of symmetry. Computationally speaking, it seems to be easier to indentify coordinates with vertex form. 

c) As aforementioned, vertex form allows one to identify the vertex simply by looking at the equation because \((h,k)\) is the vertex. In standard form, one would have to calculate \(\left(-\frac{b}{2a},f\left(-\frac{b}{2a}\right)\right)\), which requires more work.

d) In order to solve a quadratic using its corresponding formula, the formula requires the function be written in standard form. 

This might help you!
 

This may help you!