4.10: Triangle Similarity using AA and SSS
Learning Objectives
- Understand that SSA does not necessarily prove triangles are congruent.
- Understand that AA does not necessarily prove triangles are congruent.
- Determine whether triangles are similar.
- Identify corresponding angles and sides of similar polygons from a statement of similarity.
- Calculate and apply scale factors.
- Understand and apply the AA Similarity Postulate.
- Understand and apply the SSS Similarity Postulate.
- Solve problems about similar triangles.
Proving Triangles are Congruent – Why SSA does NOT always work.
SSA relationships do not necessarily prove congruence. In other words, if you have two sides and a non-included angle (an angle that is not between them), then you cannot prove congruence.
Observe the triangles below. Two sides and a non-included angle from triangle \begin{align*}ABC\end{align*} are congruent to the corresponding sides and non-included angle of triangle \begin{align*}PQR\end{align*}. In fact, the triangles appear to be congruent.
However, below is a diagram that serves as a counterexample to the notion that SSA might work. Again, we see two sides and a non-included angle of triangle \begin{align*}ABC\end{align*} which are congruent to their corresponding parts in triangle \begin{align*}PQR\end{align*}, yet clearly the two triangles are not congruent to one another.
SSA does not prove that two triangles are ________________________________.
A counterexample proves that a statement is not ________________________.
Example 1
Can you prove that the two triangles below are congruent?
Note: Figure is not to scale.
The two triangles above look congruent, but are labeled, so you cannot assume that how they look means that they are congruent!
There are two sides labeled congruent, as well as one angle. Since the angle is not between the two sides, however, this is a case of SSA. You cannot prove that these two triangles are congruent. Also, it is important to note that although two of the angles appear to be right angles, they are not marked that way, so you cannot assume that they are right angles.
Reading Check:
1. True/False: SSA is another way of proving that two triangles are congruent.
2. True/False: You CANNOT always trust the drawings, so even if triangles look congruent, you CANNOT assume that they are congruent!
Proving Triangles are Congruent – Why AA does NOT always work.
Based on the marks in the diagram below, you can see that the three angles of triangle \begin{align*}ABC\end{align*} are congruent to the three corresponding angles of \begin{align*}PQR\end{align*}, yet the triangles are not congruent to one another. Triangles like this that are the same shape but different sizes are called similar triangles.
Similar triangles have the ________________ shape but ___________________________ sizes.
Similar Polygons
Similar figures have the same shape, but they may have different sizes.
Look at the triangles below:
- The triangles on the left are not similar because they are not the same shape.
- The triangles in the middle are similar. They are all the same shape, no matter what their sizes.
- The triangles on the right are similar. They are all the same shape, no matter how they are turned or what their sizes.
Reading Check:
1. Fill in the blanks: Similar figures have the same ________________________ but different _____________________.
2. True/False: If two triangles are similar, then their corresponding angles are congruent.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
3. Can you think of some similar shapes in the real world?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
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\begin{align*}{\;}\end{align*}
Look at the quadrilaterals below:
- The quadrilaterals in the upper left are not similar because they are not the same shape.
- The quadrilaterals in the upper right are similar. They are all the same shape, no matter what their sizes.
- The quadrilaterals in the lower left are similar. They are all the same shape, no matter how they are turned or what their sizes.
Two polygons are similar if and only if:
- they have the same number of sides
- for each angle in either polygon there is a corresponding angle in the other polygon that is congruent
- the lengths of all corresponding sides in the polygons are proportional
Reminder: Just as we did with congruent figures, we name similar polygons according to corresponding parts. The symbol \begin{align*}\sim\end{align*} is used to represent “is similar to.”
Reading Check:
1. Fill in the blanks: Two polygons are similar if and only if:
They have the same number of ________________________.
Corresponding angles are ________________________________.
Corresponding side lengths are ________________________________.
2. What does the symbol \begin{align*}\sim\end{align*} mean?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
3. In the space below, draw two similar rectangles.
\begin{align*}{\;}\end{align*}
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\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
Example 2
Suppose \begin{align*}\Delta ABC \sim \Delta JKL\end{align*}. Based on this statement, which angles are congruent and which sides are proportional? Write true congruence statements and proportions.
In similar triangles, all matching angles are congruent.
First, list the angles that match up in the names of the triangles:
\begin{align*}\angle A \cong \angle J, \angle B \cong \angle K\end{align*}, and \begin{align*}\angle C \cong \angle L\end{align*}
Next, match up the side segments by the order of the letters in the similar triangles:
\begin{align*}\frac{AB}{JK} = \frac{BC}{KL} = \frac{AC}{JL}\end{align*}
Remember that there are many equivalent ways to write a proportion. The answer above is not the only set of true proportions you can create based on the given similarity statement.
Example 3
Given: \begin{align*}MNPQ \sim RSTU\end{align*}
What are the values of \begin{align*}x, y,\end{align*} and \begin{align*}z\end{align*} in the diagram above?
In similar figures, corresponding side lengths are proportional.
Side \begin{align*}x\end{align*} in \begin{align*}MNPQ\end{align*} corresponds to the side with length 25 in \begin{align*}RSTU\end{align*}.
Set up a proportion to solve for \begin{align*}x\end{align*}:
\begin{align*}&\frac{x}{25} = \frac{18}{30} \quad \text{you can reduce} \ \frac{18}{30} \ \text{to be} \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\\ &\frac{x}{25} = \frac{3}{5}\\ &\text{Cross multiply to get} \ 5x = 75 \quad \text{so} \quad x = 15\end{align*}
Side \begin{align*}y\end{align*} in \begin{align*}RSTU\end{align*} corresponds to the side with length 15 in \begin{align*}MNPQ\end{align*}.
Set up a proportion to solve for \begin{align*}y\end{align*}:
\begin{align*}&\frac{y}{15} = \frac{30}{18} \quad \text{you can reduce}\ \frac{30}{18} \ \text{to be} \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\\ &\frac{y}{15} = \frac{5}{3}\\ &\text{Cross multiply to get} \ 3y = 75 \quad \text{so} \quad y = 25\end{align*}
Finally, since \begin{align*}Z\end{align*} is an angle, we are looking for \begin{align*}m \angle R\end{align*}:
\begin{align*}Z = m \angle R = m \angle M = 115^\circ\end{align*}
Because corresponding angles are congruent in similar figures.
Example 4
\begin{align*}ABCD\end{align*} is a rectangle with length 12 and width 8.
\begin{align*}UVWX\end{align*} is a rectangle with length 24 and width 18.
A. Are corresponding angles in the rectangles congruent?
Yes. Since both are rectangles, all four angles in both figures are congruent right angles.
B. Are the lengths of the sides of the rectangles proportional?
No. The ratio of the lengths is \begin{align*}12 : 24 = 1 : 2\end{align*}.
The ratio of the widths is \begin{align*}8 : 18 = 4 : 9\end{align*} which \begin{align*}\neq 1 : 2\end{align*}.
Therefore, the lengths of the sides are not proportional.
C. Are the rectangles similar?
No. Although the corresponding angles are congruent, the lengths of corresponding sides are not proportional.
Scale Factors
If two polygons are similar, we know that the lengths of corresponding sides are proportional.
Similar polygons have ___________________________________ corresponding sides.
If \begin{align*}k\end{align*} is the length of a side in one polygon, and \begin{align*}m\end{align*} is the length of the corresponding side in the other polygon, then the ratio \begin{align*}\frac{k}{m}\end{align*} is called the scale factor relating the first polygon to the second.
Another way to say this is:
The length of every side of the first polygon is \begin{align*}\frac{k}{m}\end{align*} times the length of the corresponding side of the other polygon.
- \begin{align*}\frac{k}{m}\end{align*} represents the _________________ factor of similar shapes.
If two corresponding segments are shown below, where the segment on the left has a length of 3 and segment on the right has a length of 9, the scale factor relating them is \begin{align*}\frac{3}{9}\end{align*} which reduces to \begin{align*}\frac{1}{3}\end{align*}
Example 5
Look at the diagram below, where \begin{align*}ABCD\end{align*} and \begin{align*}AMNP\end{align*} are similar rectangles.
A. What is the scale factor?
Since \begin{align*}ABCD \sim AMNP\end{align*}, then \begin{align*}AM\end{align*} and \begin{align*}AB\end{align*} are corresponding sides. Since \begin{align*}ABCD\end{align*} is a rectangle, you know that \begin{align*}AB = DC = 45\end{align*}.
The scale factor is the ratio of the lengths of any two corresponding sides.
So the scale factor (relating \begin{align*}ABCD\end{align*} to \begin{align*}AMNP\end{align*}) is \begin{align*}\frac{45}{30} = \frac{3}{2}\end{align*} or 1.5.
We now know that the length of each side of \begin{align*}ABCD\end{align*} is 1.5 times the length of the corresponding side in \begin{align*}AMNP\end{align*}.
- Comment: We can turn this relationship around “backwards” and talk about the scale factor relating \begin{align*}AMNP\end{align*} to \begin{align*}ABCD\end{align*}. This scale factor is just \begin{align*}\frac{30}{45} = \frac{2}{3}\end{align*}, which is the reciprocal of the scale factor relating \begin{align*}ABCD\end{align*} to \begin{align*}AMNP\end{align*}.
B. What is the ratio of the perimeters of the rectangles?
Find each perimeter by adding up the lengths of the sides of the rectangle:
\begin{align*}ABCD\end{align*} is a 45 by 60 rectangle. Its perimeter is \begin{align*}45 + 60 + 45 + 60 = 210\end{align*}.
\begin{align*}AMNP\end{align*} is a 30 by 40 rectangle. Its perimeter is \begin{align*}30 + 40 + 30 + 40 = 140\end{align*}.
The ratio of the perimeters of \begin{align*}ABCD\end{align*} to \begin{align*}AMNP\end{align*} is \begin{align*}\frac{210}{140} = \frac{3}{2}\end{align*}.
- Comment: You see from this example that the ratio of the perimeters of the rectangles is the same as the scale factor. This relationship for the perimeters holds true in general for any similar polygons.
The scale factor of similar shapes is the ratio of corresponding sides and the ratio of the ______________________________ of the shapes.
The AA Rule for Similar Triangles
Suppose that the triangles \begin{align*}\Delta ABC\end{align*} and \begin{align*}\Delta MNP\end{align*} have two pairs of congruent angles, say \begin{align*}\angle A \cong \angle M\end{align*} and \begin{align*}\angle B \cong \angle N\end{align*}.
But we know that if triangles have two pairs of congruent angles, then the third pair of angles are also congruent (by the Triangle Sum Theorem).
The AAA rule for similar triangles reduces to the AA triangle similarity postulate.
The AA Triangle Similarity Postulate
If two pairs of corresponding angles in two triangles are congruent, then the triangles are similar.
- Two triangles are _____________________________ if two pairs of corresponding angles are congruent.
Example 6
Look at the diagram below:
A. Are the triangles similar? Explain your answer.
Yes. They both have congruent right angles, and they both have a \begin{align*}35^\circ\end{align*} angle.
These are two pairs of congruent corresponding angles so the triangles are similar by AA.
You need _________ pairs of corresponding congruent angles to prove similarity by AA.
B. Write a similarity statement for the triangles.
\begin{align*}\Delta ABC : \Delta TRS\end{align*} or equivalent
C. Name all pairs of congruent angles.
Remember, congruent angles match up within the names of the triangles (first letter first letter, and so on...)
\begin{align*}\angle A \cong \angle T, \angle B \cong \angle R,\end{align*} and \begin{align*}\angle C \cong \angle S\end{align*}
D. Write equations stating the proportional side lengths in the triangles.
\begin{align*}\frac{AB}{TR} = \frac{BC}{RS} = \frac{AC}{TS}\end{align*} or equivalent
Reading Check:
1. True/False: Scale factor is a relationship that is expressed as a ratio or fraction.
2. True/False: The ratio of the perimeters of two polygons is the same as the scale factor of the polygons.
Example 7
Flo wants to measure the height of a windmill. She held a 6 foot vertical pipe with its base touching the level ground, and the pipe’s shadow was 10 feet long. At the same time, the shadow of the windmill tower was 85 feet long. How tall is the windmill?
Draw a diagram:
Note: It is safe to assume that the sun’s rays hit the ground at the same angle. It is also proper to assume that the windmill tower is vertical (perpendicular to the ground).
The diagram shows two similar right triangles. They are similar because each has a right angle, and the angle where the sun’s rays hit the ground is the same for both objects. Because they are similar triangles, their corresponding side lengths are proportional.
We can write a proportion with only one unknown, \begin{align*}x\end{align*}, the height of the windmill tower:
\begin{align*}\frac{x}{85} &= \frac{6}{10}\\ \text{Cross multiply to get}: 10x &= 85 \cdot 6\\ 10x &= 510\\ x &= 51\end{align*}
Thus, the tower is 51 feet tall.
Note: This method is considered indirect measurement because it would be difficult to directly measure the height of a tall windmill tower. Imagine how difficult it would be to hold a tape measure up to something that is 51 feet tall!
SSS for Similar Triangles
The SSS Triangle Similarity Postulate
If the lengths of the sides of two triangles are proportional, then the triangles are similar.
- Two triangles with ___________________________________ sides are similar.
Example 8
Imagine a diagram with two triangles. (You may want to draw these in the margin). One triangle has sides 6-8-10. The other has sides 9-12-15.
Are the two triangles similar?
What do you notice? All three side lengths in the two triangles are proportional:
\begin{align*}\frac{6}{9} = \frac{8}{12} = \frac{10}{15} \left ( = \frac{2}{3} \right )\end{align*}
Yes, the two triangles are similar!
Reading Check:
1. The following statement is TRUE. Why? Explain:
All equilateral triangles are similar.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. The following statement is FALSE. Why? Explain in words (but you may also draw a picture to show an example):
All isosceles triangles are similar.
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\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
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