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Triangle properties are applied in a wide range of real-life situations. This lesson emphasizes similarity and proportionality principles in triangles. Applying these concepts are very useful when direct measurements are impossible to record.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Geometric Transformations and Similarity:

Triangles and Measurement:

Explore

## The Slope Triangle of a Line

Explore the slope triangle on the displayed line by dragging the point above the line to adjust the dimensions of the slope triangle.
Now, consider these questions.
• What is the ratio of the rise to the run?
• Observe how this ratio changes (if at all) when modifying the dimensions of the slope triangle.
• Based on these observations, what conclusions can be drawn about the relationship between the dimensions of the slope triangle and the ratio of the rise to the run?
Discussion

## Rules:Angle-Angle Similarity Theorem

If two angles of a triangle are congruent to two angles of another triangle, then the triangles are similar.

If and then

Pop Quiz

## Similar Triangles?

Determine if the given triangles are similar using the Angle-Angle Similarity Theorem. If necessary, apply the Interior Angles Theorem to find missing angles before identifying corresponding angles.

Discussion

## Indirect Measurement

The method of Indirect measurement uses the properties of similar triangles and proportionality to calculate distances or the dimensions of objects that are difficult to measure directly. For instance, visualize two flag poles under the sunlight; one feet tall pole casts a foot shadow and the other casts an foot shadow. What is the height of the second flagpole?

In what is called shadow reckoning, the angles made by the sun's rays off of different objects to a certain spot are equal. Similar triangles are formed by the flagpoles and their shadows as there are two pairs of congruent angles between them.
In similar triangles, the ratios between corresponding sides are equal. This means that the ratio of the lengths of the shadows of the flagpoles equals the ratio of the lengths of the flag poles. The following proportion can then be set.
This proportion can be solved to determine the height of the small flagpole.
The height of the small flagpole is feet. This example showcases one of the main uses of indirect measurement.
Example

## The Shadow of a Tree

Dylan is a skater who enjoys exploring the math behind the ramps, obstacles, and shadows at a local skate park. One sunny day, he took a break under a tall tree and noticed his shadow next to the tree's shadow. This sparked his curiosity about the tree's height, and he decided to use its shadow to calculate it.

Dylan positioned himself along the tree shadow so that the transversal from the top of the tree to the end of its shadow formed two similar triangles. Using the information provided in the diagram, help Dylan calculate the tree's height.

### Hint

The ratio of the lengths of corresponding sides of similar triangles is always the same.

### Solution

Begin by labeling the vertices for each of the triangles in the diagram.
In this diagram, corresponds to and corresponds to As the triangles and are similar, the ratio between their corresponding sides will be the same.
Note that equals the sum of and
In this case, and are known. Substitute these values into the proportion and solve for to find the height of the tree.
Dylan determined that the height of the tree is feet.
Example

Dylan is enjoying his time skating at the park. He drops-in on a triangular-shaped ramp. He begins his descent from a height of feet, which is also the height of the ramp. After he glides down the slope for feet, he realizes that he is now only as high as when he started.

He wants to know the total length of the slope, which means the total distance he will glide down. Help him find out this information.

### Hint

The ratio of the lengths of corresponding sides of similar triangles is always the same.

### Solution

The problem states that the height of a ramp is feet. Dylan rides down feet from the top of the ramp and lands at a height that is of his initial height. Dylan's current height can be found by multiplying his initial height by
This means that his current height is feet. Use the base of the ramp, its height, the distance that Dylan covered, Dylan's current height, and the slope of the ramp to draw a diagram of the situation.
Based on the diagram, it can be seen that both triangles and have a right angle each and share a common angle at vertex Therefore, these triangles are similar because of the Angle-Angle Similarity Theorem.
Consequently, corresponds to and corresponds to
Additionally, equals the sum of and Since the ratios between the lengths of corresponding sides of the triangles are equal, the following equation holds true.
In this equation, only is unknown. The known values can be substituting in this proportion to find the value of
After solving for the length of can be added to to determine the length of the slope.
Dyland found that the length of the slope is approximately feet.
Discussion

## Similarity of Slope Triangles

Slope triangles are right triangles that indicate the rise and run between two points on a line. The horizontal arrow denotes the run, while the vertical arrow represents the rise. Consider a pair of slope triangles on a line.
Because slope triangles are right triangles, and are congruent since they are both right angles. In addition, is parallel to and both are cut by the line, which is also a transversal. As a result, the corresponding angles and are also congruent. Therefore, the slope triangles are similar, as per the Angle-Angle Similarity Theorem.
This fact holds for any pair of slope triangles on the same line.
Example

## Comparing Slope Triangles

Later that day, Dylan returned home and realized that he still forgot about his math homework. He needs to finish it so that he can have free time on Saturday to go skateboarding again. His homework requires him to compare a pair of slope triangles that lie on the same line.

Consider that the side of each square in the grid is unit. Help Dylan complete his homework.

a Write a proportion that compares the rise to the run for each of the similar slope triangles shown in the diagram by using the vertex labels.
b Write the numeric value for the proportion found in Part A without simplifying fractions.

### Hint

a Write the ratio between corresponding sides of the two slope triangles to create a proportion. Then, rewrite the proportion so the ratio of the rise to the run of one triangle is on one side, and the ratio of the rise to the run of the other triangle is on the other side.
b Count the number of units from its starting point to its endpoint to determine the length of a specific segment.

### Solution

a Consider that and are similar as they are slope triangles on the same line.
Due to this similarity, the ratio between corresponding sides is equal. In this case, corresponds to and corresponds to This information can be used to write the following proportion.
Since it is asked to compare the rise to the run for each of the similar slope triangles, this proportion will be rewritten to have the ratio of the rise to the run of on one side and the ratio of the rise to the run of on the other side. This indicates that the ratios of the rise to the run are equal in both slope triangles.
b In the previous part, the proportion that compares the rise to the run for each of the similar slope triangles was written.
The given diagram will be used to find the numerical value of this proportion. Count the number of units it takes to get from point to point to find the length of Similarly, count the number of units it takes to get from point to point to find the length of

This means that and Apply a similar reasoning to find the lengths of and

Once of the lengths in the proportion are known, substitute them into the proportion to find its numerical value.
Discussion

## Slope Triangles and the Slope of a Line

The slope of a line is given by the ratio of the vertical change or rise to the horizontal change or run. This ratio remains equal across any two slope triangles formed by that line.
This shows that the line's steepness is constant, no matter which points are used to form the slope triangles. Also recall that when two points — and — on the line are known, the Slope Formula can be used.
This formula directly applies the principle that the ratio of the rise to the run between any two points on a line remains constant.
Example

## Slope of a Skate Park Shelter

At Dylan's skate park, there is a cool shelter that covers the entire skateboard track, providing shade for everyone during their skate sessions. Seeing the roof line, Dylan thinks it is the perfect chance to apply his new school knowledge about slope triangles on the same line.

With the help of a friend, Dylan measured the rise and run at various points on one of the slopes of the roof and sketched a diagram.

Now, he is curious whether the slope of the roof line remains consistent over the different areas. Help Dylan figure out the roof's slope and see if it remains the same everywhere.

### Hint

Start by choosing two points on the roof line and find the slope using the Slope Formula. Next, verify that the slope is the same by selecting a different set of points.

### Solution

The slope of a line can be determined by substituting the coordinates of two points and into the Slope Formula.
In this case, and are two points on the roof line. Substitute these points into the formula to determine the roof's slope.
The slope of the line is A different set of points can be used to confirm that the slope of the line remains the same everywhere. In this case, the points and can be used to evaluate the Slope Formula.
This confirms that the slope of the line is constant no matter which pair of points are used to calculate it.
Closure

## Perimeter and Scale Factor

Dylan had a great time skateboarding today and now he is sitting down to draw two ramps he dreams of skating on someday. He names them Little Leap and Giant Glide. Dylan draws them as triangles that look similar on paper, but Giant Glide is just a bigger version of Little Leap.
He wonders if the way around the ramps, or their perimeters, are related. Help him find the following information to find out the answer to his question.
a What is the perimeter of the triangle formed by the Little Leap ramp?
b What is the perimeter of the triangle formed by the Giant Glide ramp?
c Is it true that the perimeter of the bigger triangle is equal to the perimeter of the smaller triangle multiplied by a scale factor?

### Hint

a Add up the lengths of the smaller ramp's triangle to determine its perimeter.
b Add up the lengths of the bigger ramp's triangle to determine its perimeter.
c Find the ratio between the corresponding side lengths of the bigger triangle and smaller triangle to find the scale factor.

### Solution

a The smaller ramp is known as the Little Leap ramp. It is formed by the legs of a triangle measuring feet and feet, with a hypotenuse of feet. Add up the length of all three sides to find the perimeter of the Little Leap ramp.
The perimeter of this triangle is equal to feet.
b The perimeter of the triangle formed by the Giant Glide ramp can be found in a similar way. The legs of the triangle are of lengths and feet, respectively, while the hypotenuse has a length of feet. Add these lengths to determine the perimeter of the triangle.
The perimeter of the triangle formed by the Giant Glide ramp is feet.
c Start by calculating the ratio between the corresponding side lengths of the triangles. Then, check if there exists the same scale factor between the perimeters of the triangles. In this case, use the base of the triangles, which measure feet and feet, respectively.
The scale factor is Recall that the perimeter of the smaller triangle is feet, and that of the bigger triangle is feet. Now, calculate the ratio between the perimeters of the triangles.
It can be concluded that the scale factors between the corresponding side lengths and perimeters of the triangles are the same. This refers that if the perimeter of the smaller triangle is multiplied by the scale factor, it equals to the perimeter of the bigger triangle.
Therefore, the given statement is true. If two figures are similar, then the ratio of their perimeters is equal to the scale factor.