Similar triangles

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Congruence and similarity

Congruence and similarity on the SAT test

SAT Subscore: Additional topics in math

Studying congruence and similarity

On the SAT test congruence and similarity topic is the third topic of additional topics in math that include 7 advanced topics (see the full topics list on the top menu). It is recommended to start learning additional topics in math with its first topic called complex numbers.

Congruence and similarity topic is divided into sections from easy to difficult (the list of the sections appears on the left menu). Each section includes detailed explanations of the required material with examples followed by a variety of self-practice questions with solutions.

Finish studying heart of algebra subscore topics before you study this topic or any other additional topic in math. (Heart of algebra subscore includes basic algebra topics which knowledge is required for understanding additional topics in math). 

Congruence and similarity- summary

Congruence and similarity questions include congruent angles and similar triangles topics.

Similar triangles have the same angle measures and their corresponding side lengths are related by a constant ratio.

Sum of angles in a triangle: The sum of the measures in degrees of the angles of a triangle is 180.

Angle measures in triangles: In an isosceles triangle the angles opposite the two equal sides are equal; the angles of an equilateral triangle are equal to 60°.

Congruent angles are angles that have the same angle measure.

Vertical Angles are the angles opposite each other when two lines cross, vertical angles are equal.

Supplementary angles are those angles that measure up to 180 degrees. Angles that lie on the same side of a straight line, are always supplementary.

Alternate angles are angles located in opposite positions relative to a transversal intersecting two parallel lines. Alternate angles are equal.

Corresponding angles are angles located in the same position relative to parallel lines intersected by a transversal. Corresponding angles are equal.

Two intersecting lines create vertical angles and supplementary angles.

Two parallel lines combined with two intersecting lines form 2 similar triangles and equal alternate angles.

A parallel line inside a triangle forms 2 similar triangles and equal corresponding angles.

Continue reading this page for detailed explanations and examples.

Angle relationships inside a triangle

Sum of angles in a triangle:
The sum of the measures in degrees of the angles of a triangle is 180. (This statement is provided at the beginning of each SAT math section).
If use the variables x°, y° and z° to represent angles then x°+y°+z°=180°.

Angle measures in an isosceles triangle:
An isosceles triangle is a triangle that has two equal sides, the angles opposite the two equal sides are equal.

Angle measures in an equilateral triangle:
An equilateral triangle is a triangle with all three sides of equal length, the angles of an equilateral triangle are equal to 60°.

The figure below presents a triangle and its angles.

Angle sum inside a triangle

Consider the following example:

In a triangle ABC given that the angle A is equal to 50° and AB=BC.

What is the value of the angles B and C?

The sum of the measures of the angles of a triangle is 180°, therefore the angles A+B+C=180°.
We are given that AB=BC, therefore the angle B is equal to the angle C.
We are given that A=50°.

Therefore, we can represent by x the value of angle B and C and solve an equation with 1 variable:
x+x+50°=180°
2x=180°-50°
2x=130°
x=65°

Checking the answer:
65°+65°+50°=180°
180°=180°

Consider the following example:

In a triangle ABC given that the angle A is equal to 60° and the angle B is twice bigger than the angle C.

What is the value of the angles B and C?

The sum of the measures of the angles of a triangle is 180°, therefore the angles A+B+C=180°.
We are given that A=60°.
We are given that the angle B is twice bigger than the angle C.

Therefore, we can represent by x the value of angle C and solve an equation with 1 variable:
2x+x+60°=180°
3x=120°
x=40° therefore C=40° and D=40°*2=80°

Checking the answer:
40°+80°+60°=180°
180°=180°

Angle relationships between intersecting lines- vertical and supplementary angles

Two intersecting lines create vertical angles and supplementary angles.

Vertical Angles are the angles opposite each other when two lines cross (vertical means that they share the same vertex). Vertical angles are equal.

Supplementary angles are those angles that measure up to 180 degrees. Angles that lie on the same side of a straight line, are always supplementary.

The figure below shows the angles created between two intersecting lines.

Angle relationships- vertical and supplementary angles

The angles 1 and 3 (180-α) and the angles 2 and 4 (α) are vertical angles. All pairs of vertical angles are equal.

 

The angles 1 and 2, the angles 2 and 3, the angles 3 and 4 and the angles 4 and 1 are supplementary angles. The sum of each pair of supplementary angles equals to 180 degrees (180-α+α=180°).

Consider the following example:

In the figure below are presented 4 lines that cross each other: AB, FI, AH and BG. The triangle ABC is an isosceles triangle, so that AS=BC. The angle ∠FEG is equal to 30° and the angle ∠FDH is equal to 120°.

What is the value of the angle BAC?

Angle relationships between intersecting lines

We are given that the angle ∠FEG is equal to 30°, therefore the angle ∠CED is also equal to 30° (the angles are vertical, therefore they are equal).

We are given that the angle ∠FDH is equal to 120°, therefore the angle ∠CDI is equal to 180°-120°=60° (the angles are supplementary, therefore their sum is 180°).

The sum of angles in a triangle is equal to 180°, therefore the sum of angles in the triangle CDE is 180° and the angle ∠DCE=180°-60°-30°=90°.

We are given that the triangle ABC is an isosceles triangle, therefore the angles ∠ABS and ∠BAC are equal.
The sum of angles in a triangle is equal to 180°, therefore the sum of angles in the triangle ABC is 180°.
We calculated that the angle ∠DCE is equal to 90° the angles ∠ABS and ∠BAC are equal.
From these 3 statements we can conclude that the angle ∠BAC=∠ABC=(180°-90°)/2=90°/2=45°.

Angle relationships between intersecting and parallel lines- alternate angles

Two parallel lines combined with two intersecting lines form 2 similar triangles and equal alternate angles.

Similar triangles have the same shape, but not the same size (they have the same angle measures, but not the same side lengths).
Note that the corresponding side lengths of similar triangles are related by a constant ratio, which is called k. See further details below.

Alternate angles are angles located in opposite positions relative to a transversal intersecting two parallel lines. Alternate angles are equal.

In the figure below two parallel lines AB and DE (marked in blue) combined with two intersecting lines AE and BD form 2 similar triangles ABC and CDE.

The angles x°, y° and z° are equal.

Angles between intersecting lines and parallel lines

The equal angles x°, y° and z°:

The equal angles z° are vertical angles that are formed by the lines AE and BD.

The equal angles x° are called alternate angles, they are formed by the parallel lines AB and DE and the line AE that crosses them.

The equal angles y° are called alternate angles, they are formed by the parallel lines AB and DE and the line BD that crosses them.

Since the angles x°, y° and z° are equal between the triangle ABC and CDE, the triangles are similar.  

Consider the following example:

In the figure below are presented 4 lines that cross each other: HI, DE, EF and BG. The lines HI and DE are parallel. The angle ∠FAH is equal to 40° and the angle ∠EDG is equal to α. In addition, AC=BC.

What is the value of α?

Angle relationships between intersecting lines and parallel lines question

We are given that the angle ∠EDG is equal to α.

Angles that lie on the same side of a straight line, are always supplementary and their sum is equal to 180°. Therefore, the angles ∠EDG and ∠CDE are supplementary and ∠EDG+ ∠CDE=180°.

The angle ∠CDE=180°-α.

Two parallel lines combined with two intersecting lines form 2 similar triangles. We are given that the lines HI and DE are parallel, therefore the triangles ABC and CDE are similar and the angles ∠CDE and ∠ABC are equal and their value is 180°-α (we found that ∠CDE=180°-α).

Note that instead of using similar triangles we can use the statement that two parallel lines combined with two intersecting lines form 2 equal alternate angles. We are given that the lines HI and DE are parallel, therefore the angles ∠CDE and ∠ABC are equal (these angles are alternate) and their value is 180°-α (we found that ∠CDE=180°-α).

We are given that the angle ∠FAH is equal to 40°, therefore the angle ∠BAC is also equal to 40° (the angles are vertical, therefore they are equal).

We are given that AC=BC, therefore the angle ∠BAC=∠ABC.

We found that the angle ∠BAC=∠ABC, the angle ∠BAC=40° and the angle ∠ABC=∠CDE=180°-α. Therefore, the angle ∠ABC=∠CDE=∠BAC=∠FAH=180°-α=40°

We can solve the equation 180°-α=40°, getting α=180°-40°=140°

Angle relationships between parallel lines inside a triangle- corresponding angles

A parallel line inside a triangle (the line is parallel to one of the sides of the triangle) forms 2 similar triangles (a small triangle and a large triangle) and equal corresponding angles.

Corresponding angles are angles located in the same position relative to parallel lines intersected by a transversal. Corresponding angles are equal.

In the figure below a parallel line EB inside a triangle ACD (EB is parallel to CD) forms 2 similar triangles (a small triangle ABE and a large triangle ACD).

 

Angle relationships between parallel lines inside a triangle- corresponding angles

The equal angles x° and y°:

The equal angles x° are called corresponding angles, they are formed by the parallel lines BE and CD and the line AD.

The equal angles y° are called corresponding angles, they are formed by the parallel lines BE and CD and the line AC.

Since the angles x°, y° and z° are equal between the triangle ABE and ACD, the triangles are similar.  

Consider the following example:

In the figure below are presented 4 lines that cross each other: AD,AC,BF and CE. The lines BF and CE are parallel. The angle ∠BAF is equal to 20° and the angle ADE is equal to α. In addition, AF=AB.

What is the value of α?

Angle relationships between parallel lines inside a triangle- corresponding angles

Angles that lie on the same side of a straight line, are always supplementary and their sum is equal to 180°. Therefore, the angles ADE and ADC are supplementary and ADE+ ADC=180°.

We are given that the angle ADE= α, therefore α+ADC=180° and the angle ADC=180°-α.

A parallel line inside a triangle forms 2 similar triangles. We are given that the lines BF and CE are parallel, therefore the triangles ABF and ACD are similar and the angles AFB and ADC are equal and their value is 180°-α (we found that ADC=180°-α).

Note that instead of using similar triangles we can use the statement that a parallel line inside a triangle forms 2 equal corresponding angles. We are given that the lines BF and CE are parallel, therefore the angles AFB and ADC are equal (these angles are corresponding) and their value is 180°-α (we found that ADC=180°-α).

We are given that AB=AF, therefore the angle ABF=AFB. We are also given that the angle BAF=20°.

The sum of angles in a triangle is equal to 180°, therefore the sum of angles in the triangle ABF is 180° and the angle ABF=AFB=(180°-20)/2=160°/2=80°.

We have an equation that we can solve: ABF=AFB=ADC=180°-α=80°.

180°-α=80°

α=180°-80°

α=100°

Note that the angle ACD=ABF=80°.

Calculating side lengths with triangles similarity

Similar triangles have the same shape, but not the same size (they have the same angle measures, but not the same side lengths).

The corresponding side lengths of similar triangles are related by a constant ratio, which is called k.

Note that:
Similar triangles have the same angle measures.
If 2 lines are parallel, the triangles that are formed by them are similar.

The figure below shows two similar triangles: ABC and DEF. The triangle ABC is similar to the triangle DEF.

Finding side lengths with triangles similarity

Similar triangles have the same angle measures, therefore the angle ∠A is equal to the angle ∠D; the angle ∠B is equal to the angle ∠E and the angle ∠C is equal to the angle ∠F.

The corresponding side lengths of similar triangles are related by a constant ratio, which is called k, therefore the following ratios between pairs of the sides exist:
DE=k*AB (marked in orange in the figure above)
DF=k*AC (marked in green in the figure above)
EF=k*BC (marked in black in the figure above)

We can also present all the ratios as being equal to k:

k =  DE   =   DF   =   EF
       ____     ____      ____
       AB        AC        BC

Note that we could write the ratio in an opposite way by switching between the numerator and the denominator (so that the numerator will include the sides of the small triangle and the denominator will include the sides of the big triangle). The parameter k will be smaller than 1 and equal to 1 divided by the k parameter above.
For example: if the ratio k between the sides of the big triangle divided by the sides of the small triangle is 2 (like 4 divided by 2), then the ratio k between the sided of the small triangle divided by the sides of the big triangle will be equal to 1/2 (like 2 divided by 4).

Consider the following example:

In the figure below is given that AB=5 centimeters; CD=27 centimeters; BE is parallel to CD; AC is 3 times bigger than AB.

What is the value of BE?

An example- Calculating side lengths with triangles similarity

We are given that AC is 3 times bigger than AB, therefore we can write an equation AC=AB*3.

We are given that AB=5 centimeters, therefore AC=AB*3=5*3=15. Meaning that AC/AB=3.

A parallel line inside a triangle forms 2 similar triangles. We are given that BE is parallel to CD, therefore the triangles ABE and ACD are similar.

The corresponding side lengths of similar triangles are related by a constant ratio, which is called k.

We found AC/AB=3, therefore k=3.

K is also equal to BE/CD, therefore we can write an equation CD/BE=3.

We are given that CD=27 centimeters, therefore we can write an equation 27/BE=3.

27/BE=3

BE=27/3

BE=9 centimeters

Calculating side lengths with triangles similarity when one side is common for both triangles

In these questions one side is common for both triangles, therefore we need the length of two sides instead of three sides to calculate the length of the fourth side using the similar triangles ratio.  

Consider the following example:

The figure below presents two similar triangles, ABC and BDC.

The length of AB is 20 centimeters and the length of BC is 15 centimeters.

What is the length of DC?

Example -calculating side lengths with triangles similarity when one side is common for both triangles

In this example the side BC is common for both the triangles. Therefore we can you it twice in the similar triangles ratio.

The similar triangles ratio is

k   =   AB     =    BC
          ____        ____
           BC           DC

k   =   20   =  15
         ____     ____
          15        DC

4  =  15
__      __
3      DC

4DC=45

DC=45/4=11 1/4

The length of the side DC is 11 1/4 centimeters.

Note that we could write the ratio in an opposite way by switching between the numerator and the denominator and get the same answer:

k    =   BC     =    DC
           ____         ____
           AB            BC

K   =   15   =    DC
         _____      _____
           20          15

 3   =   DC
___      ___
 4         15

4DC=45

DC=45/4=11 1/4

The ratio k in this case will be 15/20=3/4, which is 1 divided by the previous ratio k=4/3.

You just finished studying congruence and similarity topic, the third topic of additional topics in math!

Continue studying the next additional topic in math- right triangle trigonometry and word problems.