CK-12 Geometry
Page 17
Draw segments connecting the points on the bisector to the original endpoints.
Knowing that the center point is the midpoint of both line segments and that all angles formed around point are right angles, you can prove that all four triangles created are congruent by the SAS rule.
Lesson Summary
In this lesson, we explored applications triangle congruence. Specifically, we have learned to:
Identify various triangles congruence postulates and theorems.
Use the fact that corresponding parts of congruent triangles are congruent.
Find distances using congruent triangles.
Use construction techniques to create congruent triangles.
These skills will help you understand issues of congruence involving triangles. Always look for triangles in diagrams, maps, and other mathematical representations.
Points to Consider
You now know all the different ways in which you can prove two triangles congruent. In the next chapter you’ll learn more about isosceles and equilateral triangles.
Review Questions
Use the following diagram for exercises 1-5
Find in the diagram above.
Find in the diagram above.
What is ? How do you know?
What postulate can you use to show
Use the distance formula to find . How can use this to find
6-8: For each pair of triangles, complete the triangle congruence statement, or write “no congruence statement possible.” Name the triangle congruence postulate you use, or write a sentence to explain why you can’t write a triangle congruence statement.
__________.
__________.
__________.
In the following diagram, Midtown is exactly halfway between Uptown and Downtown. What is the distance between Downtown and Lower East Side? How do you know? Write a few sentences to convince a reader your answer is correct.
Given: is the midpoint of and Prove:
Review Answers
Since is horizontal (parallel to the axis) and is vertical (parallel to the axis), we can conclude that they intersect at a right angle.
SAS (other answers are possible)
. Since the triangles are congruent, we can conclude
. SAS triangle congruence postulate
. SSS triangle congruence postulate
No congruence statement is possible; we don’t have enough information.
. Since Midtown is the midpoint of the line connecting Uptown and Downtown, we can use the vertical angle theorem for the angles made by the two lines that meet at Midtown, and then we can conclude that the triangles are congruent using AAS. If the triangles are congruent then all corresponding parts are also congruent
Statement Reason
1. is the midpoint of
1. Given
2.
2. Definition of midpoint
3.
3. Given
4.
4. Alternate interior angles theorem
5.
5. Alternate interior angles theorem
6.
6. AAS triangle congruence postulate
7. 7. Definition of congruent triangles (corresponding parts are congruent)
Isosceles and Equilateral Triangles
Learning Objectives
Prove and use the Base Angles Theorem.
Prove that an equilateral triangle must also be equiangular.
Use the converse of the Base Angles Theorem.
Prove that an equiangular triangle must also be equilateral.
Introduction
As you can imagine, there is more to triangles than proving them congruent. There are many different ways to analyze the angles and sides within a triangle to understand it better. This chapter addresses some of the ways you can find information about two special triangles.
Base Angles Theorem
An isosceles triangle is defined as a triangle that has at least two congruent sides. In this lesson you will prove that an isosceles triangle also has two congruent angles opposite the two congruent sides. The congruent sides of the isosceles triangle are called the legs of the triangle. The other side is called the base and the angles between the base and the congruent sides are called base angles. The angle made by the two legs of the isosceles triangle is called the vertex angle.
The Base Angles Theorem states that if two sides of a triangle are congruent, then their opposite angles are also congruent. In other words, the base angles of an isosceles triangle are congruent. Note, this theorem does not tell us about the vertex angle.
Example 1
Which two angles must be congruent in the diagram below?
The triangle in the diagram is an isosceles triangle. To find the congruent angles, you need to find the angles that are opposite the congruent sides.
This diagram shows the congruent angles. The congruent angles in the triangle are and .
So, how do we prove the base angles theorem? Using congruent triangles.
Given: Isosceles with
Prove
Statement Reason
1. is isosceles with .
1. Given
2. Construct Angle Bisector
2. Angle Bisector Postulate
3.
3. Definition of Angle Bisector
4.
4. Reflexive Property
5.
5. SAS Postulate
6.
6. Definition of congruent triangles (all pairs of corresponding angles are congruent)
Equilateral Triangles
The base angles theorem also applies to equilateral triangles. By definition, all sides in an equilateral triangle have exactly the same length.
Because of the base angles theorem, we know that angles opposite congruent sides in an isosceles triangle are congruent. So, if all three sides of the triangle are congruent, then all of the angles are congruent as well.
A triangle that has all angles congruent is called an equiangular triangle. So, as a result of the base angles theorem, you can identify that all equilateral triangles are also equiangular triangles.
Converse of the Base Angles Theorem
As you know, some theorems have a converse that is also true. Recall that a converse identifies the “backwards,” or reverse statement of a theorem. For example, if I say, “If I turn a faucet on, then water comes out,” I have made a statement. The converse of that statement is, “If water comes out of a faucet, then I have turned the faucet on.” In this case the converse is not true. For example the faucet may have a drip. So, as you can see, converse statements are sometimes true, but not always.
The converse of the base angles theorem is always true. The base angles theorem states that if two sides of a triangle are congruent the angles opposite them are also congruent. The converse of this statement is that if two angles in a triangle are congruent, then the sides opposite them will also be congruent. You can use this information to identify isosceles triangles in many different circumstances.
Example 2
Which two sides must be congruent in the diagram below?
has two congruent angles. By the converse of the base angles theorem, it is an isosceles triangle. To find the congruent sides, you need to find the sides that are opposite the congruent angles.
This diagram shows arrows pointing to the congruent sides. The congruent sides in this triangle are and .
The proof of the converse of the base angles theorem will depend on a few more properties of isosceles triangles that we will prove later, so for now we will omit that proof.
Equiangular Triangles
Earlier in this lesson, you extrapolated that all equilateral triangles were also equiangular triangles and proved it using the base angles theorem. Now that you understand that the converse of the base angles theorem is also true, the converse of the equilateral/equiangular relationship will also be true.
If a triangle has three congruent angles, it is be equiangular. Since congruent angles hav
e congruent sides opposite them, all sides in an equiangular triangle will also be congruent. Therefore, every equiangular triangle is also equilateral.
Lesson Summary
In this lesson, we explored isosceles, equilateral, and equiangular triangles. Specifically, we have learned to:
Prove and use the Base Angles Theorem.
Prove that an equilateral triangle must also be equiangular.
Use the converse of the Base Angles Theorem.
Prove that an equiangular triangle must also be equilateral.
These skills will help you understand issues of analyzing triangles. Always look for triangles in diagrams, maps, and other mathematical representations.
Review Questions
Sketch and label an isosceles with legs and that has a vertex angle measuring
What is the measure of each base angle in from 1?
Find the measure of each angle in the triangle below:
below is equilateral. If bisects , find:
Which of the following statements must be true about the base angles of an isosceles triangle ? The base angles are congruent.
The base angles are complementary.
The base angles are acute.
The base angles can be right angles.
One of the statements in 5 is possible (i.e., sometimes true), but not necessarily always true. Which one is it? For the statement that is always false draw a sketch to show why.
7-13: In the diagram below, . Use the given angle measure and the geometric markings to find each of the following angles.
_____
_____
_____
_____
_____
_____
_____
Review Answers
Each base angle in measures
and
,
a. and c. only.
b. is possible if the base angles are When this happens, the vertex angle is d. is impossible because if the base angles are right angles, then the “sides” will be parallel and you won’t have a triangle.
Congruence Transformations
Learning Objectives
Identify and verify congruence transformations.
Identify coordinate notation for translations.
Identify coordinate notation for reflections over the axes.
Identify coordinate notation for rotations about the origin.
Introduction
Transformations are ways to move and manipulate geometric figures. Some transformations result in congruent shapes, and some don’t. This lesson helps you explore the effect of transformations on congruence and find location of the resulting figures. In this section we will work with figures in the coordinate grid.
Congruence Transformations
Congruent shapes have exactly the same size and shape. Many types of transformations will keep shapes congruent, but not all. A quick review of transformations follows.
Transformation Diagram Congruent or Not?
Translation (Slide) Congruent
Reflection (Flip) Congruent
Rotation (Turn) Congruent
Dilation (Enlarge or Shrink) Not Congruent
As you can see, the only transformation in this list that interferes with the congruence of the shapes is dilation. Dilated figures (whether larger or smaller) have the same shape, but not the same size. So, these shapes will be similar, but not congruent.
When in doubt, check the length of each side of a triangle in the coordinate grid by using the distance formula. Remember that if triangles have three pairs of congruent sides, the triangles are congruent by the SSS triangle congruence postulate.
Example 1
Use the distance formula to prove that the reflected image below is congruent to the original triangle .
Begin with triangle . First write the coordinates.
is
is
is
Now use the coordinates and the distance formula to find the lengths of each segment in the triangle.
The lengths are as follows.
, , and
Next find the lengths in triangle . First write down the coordinates.
is
is
is
Now use the coordinates to find the lengths of each segment in the triangle.
The lengths are as follows.
, , and
Using the distance formula, we demonstrated that the corresponding sides of the two triangles have the same lengths. Therefore, by the SSS congruence postulate, these triangles are congruent. This example shows that reflected figures are congruent.
Translations
The transformation you saw above is called a reflection. Translations are another type of transformation. You translate a figure by moving it right or left and up or down. It is important to know how a transformation of a figure affects the coordinates of its vertices. You’ll now have the opportunity to practice translating images and changing the coordinates.
For each unit a figure is translated to the right, add to each coordinate in the vertices. For each unit a figure is translated to the left, subtract from the coordinates. Always remember that moving a figure left and right only affects the coordinates.
If a figure is translated up or down, it affects the coordinate. So, if you move a figure up by , then add to each of the coordinates in the vertices. Similarly, if you translate a figure down by , subtract from the coordinates.
Example 2
is shown on the coordinate grid below. What would be the coordinates of if it has been translated to the left and up?
Analyze the change and think about how that will affect the coordinates of the vertices. The translation moves the figure to the left. That means you will subtract from each of the coordinates. It also says you will move the figure up , which means that you will add to each of the coordinates. So, the coordinate change can be expressed as follows.
Carefully adjust each coordinate using the formula above.
This gives us the new coordinates , , and .
Finally, draw the translated triangle to verify that your answer is correct.
Reflections
Reflections are another form of transformation that also result in congruent figures. When you “flip” a figure over the axis or axis, you don’t actually change the shape at all. To find the coordinates of a reflected figure, use the opposite of one of the coordinates.
If you reflect an image over the axis, the new coordinates will be the opposite of the old coordinates. The coordinates remain the same.
If you reflect an image over the axis, take the opposite of the coordinates. The coordinates remain the same.
Example 3
Triangle is shown on the coordinate grid below. What would be the coordinates of if it has been reflected over the axis?
Since you are finding the reflection of the image over the axis, you will find the opposite of the coordinates. The coordinates will remain the same. So, the coordinate change can be expressed as follows.
Carefully adjust each coordinate using the formula above.
This gives new coordinates , and .
Draw the translated triangle to verify that your answer is correct.
Rotations
The most complicated of the congruence transformations is rotations. To simplify rotations, we will only be concerned with rotations of or about the origin . The rules describe how coordinates change under rotations.
rotations: Take the opposite of both coordinates.
becomes
clockwise rotations: Find the opposite of the coordinate, and reverse the coordinates.
becomes
counterclockwise rotations: Find the opposite of the. coordinate, and reverse the coordinates.
becomes
Example 4
Triangle is shown on the following coordinate grid. What would be the coordinates of if it has been rotated counterclockwise about the origin?
Since you are finding the rotation of the image c
ounterclockwise about the origin, you will find the opposite of the coordinates and then reverse the order. So, the coordinate change can be expressed as follows.
Carefully adjust each coordinate using the formula above.
This gives new coordinates , , and .
Finally, we draw the rotated triangle to verify that your answer is correct.
Lesson Summary
In this lesson, we explored transformations with triangles. Specifically, we have learned to:
Identify and verify congruence transformations.
Identify coordinate notation for translations.
Identify coordinate notation for reflections over the axes.
Identify coordinate notation for rotations about the origin.
These skills will help you understand many different situations involving coordinate grids. Always look for triangles in diagrams, maps, and other mathematical representations.
Review Questions