Explain how the triangle similarity postulates and theorems are alike and how they differ from triangle congruence postulates
Name something with a height that would be difficult to measure directly. Describe how you could measure it indirectly
Similarity theorems and postulates involve matching either angles only, or the ratios of corresponding side lengths. Congruence theorems and postultates invove actual lengths of corresponding sides. See http://www.mathwarehouse.com/geometry/similar/triangles/similar-triangle-theorems.php
An example of using trig to measure a height would be that of a tree that could not be climbed.
Which statement below is true about similar triangles?
The triangle similarity postulates and theorems are similar to triangle congruence postulates because they involve comparing the angles and sides of triangles. However, they differ in the criteria they use to establish similarity or congruence between triangles.
The triangle similarity postulates, which are AA (Angle-Angle), SAS (Side-Angle-Side), and SSS (Side-Side-Side), are used to determine if two triangles are similar. In these cases, only the angles and/or the ratio of the lengths of corresponding sides need to be equal. If two triangles satisfy any of these postulates, they are considered similar, which means they have the same shape but not necessarily the same size.
On the other hand, the triangle congruence postulates, which are SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and HL (Hypotenuse-Leg), are used to determine if two triangles are congruent. In these cases, corresponding sides and/or angles need to be equal. If two triangles satisfy any of these postulates, they are considered congruent, which means they have the same shape and size.
To measure something with a height that would be difficult to measure directly, such as the height of a tall building or a mountain, you can use indirect methods. One common indirect method is using a theodolite, which is a surveying instrument. The theodolite measures the angles between the observer, the top, and the base of the object. By taking these angle measurements from two different locations, you can use trigonometry to calculate the height indirectly using the principles of similar triangles.
First, you need to measure the distance between the two observation points accurately. Then, using the theodolite, measure the angles between the horizontal line (the ground) and the lines of sight to the top and the base of the object from each observation point. By forming triangles between the two observation points and the top and base of the object, you can create similar triangles.
With trigonometry, you can use the ratios between the measured angles and the known distance between the observation points to find the height of the object. This can be done by applying the tangent function to the angle measurements, as the tangent of an angle is equal to the ratio of the opposite side to the adjacent side in a right triangle.
By using these indirect measurement techniques and the principles of trigonometry, you can determine the height of an object that would be difficult to measure directly.