4.5: Isosceles and Equilateral Triangles
Learning Objectives
 Understand the properties of isosceles and equilateral triangles.
 Use the Base Angles Theorem and its converse.
 Understand that an equilateral triangle is also equiangular.
Review Queue
Find the value of and/or .
 If a triangle is equiangular, what is the measure of each angle?
Know What? Your parents now want to redo the bathroom. To the right are 3 of the tiles they would like to place in the shower. Each blue and green triangle is an equilateral triangle. What shape is each dark blue polygon? Find the number of degrees in each of these figures?
Isosceles Triangle Properties
An isosceles triangle is a triangle that has at least two congruent sides. The congruent sides of the isosceles triangle are called the legs. The other side is called the base. The angles between the base and the legs are called base angles. The angle made by the two legs is called the vertex angle.
Investigation 45: Isosceles Triangle Construction
Tools Needed: pencil, paper, compass, ruler, protractor
1. Refer back to Investigation 42. Using your compass and ruler, draw an isosceles triangle with sides of 3 in, 5 in and 5 in. Draw the 3 in side (the base) horizontally at least 6 inches down the page.
2. Now that you have an isosceles triangle, use your protractor to measure the base angles and the vertex angle.
The base angles should each be and the vertex angle should be .
We can generalize this investigation for all isosceles triangles.
Base Angles Theorem: The base angles of an isosceles triangle are congruent.
For , if , then .
To prove the Base Angles Theorem, we need to draw the angle bisector (Investigation 15) of .
Given: Isosceles triangle above, with .
Prove:
Statement  Reason 

1. Isosceles triangle with  Given 
2. Construct angle bisector of

Every angle has one angle bisector 
3.  Definition of an angle bisector 
4.  Reflexive PoC 
5.  SAS 
6.  CPCTC 
Let’s take a further look at the picture from step 2 of our proof.
Because , we know by CPCTC. These two angles are also a linear pair, so each and .
Additionally, by CPCTC, so is the midpoint of . This means that is the perpendicular bisector of .
Isosceles Triangle Theorem: The angle bisector of the vertex angle in an isosceles triangle is also the perpendicular bisector of the base.
Note this is ONLY true of the vertex angle. We will prove this theorem in the review questions.
Example 1: Which two angles are congruent?
Solution: This is an isosceles triangle. The congruent angles, are opposite the congruent sides. From the arrows we see that .
Example 2: If an isosceles triangle has base angles with measures of , what is the measure of the vertex angle?
Solution: Draw a picture and set up an equation to solve for the vertex angle, .
Example 3: If an isosceles triangle has a vertex angle with a measure of , what is the measure of each base angle?
Solution: Draw a picture and set up and equation to solve for the base angles, .
The converses of the Base Angles Theorem and the Isosceles Triangle Theorem are both true.
Base Angles Theorem Converse: If two angles in a triangle are congruent, then the opposite sides are also congruent.
For , if , then .
Isosceles Triangle Theorem Converse: The perpendicular bisector of the base of an isosceles triangle is also the angle bisector of the vertex angle.
For isosceles , if and , then .
Equilateral Triangles By definition, all sides in an equilateral triangle have the same length.
Investigation 46: Constructing an Equilateral Triangle
Tools Needed: pencil, paper, compass, ruler, protractor
1. Because all the sides of an equilateral triangle are equal, pick one length to be all the sides of the triangle. Measure this length and draw it horizontally on you paper.
2. Put the pointer of your compass on the left endpoint of the line you drew in Step 1. Open the compass to be the same width as this line. Make an arc above the line. Repeat Step 2 on the right endpoint.
4. Connect each endpoint with the arc intersections to make the equilateral triangle.
Use the protractor to measure each angle of your constructed equilateral triangle. What do you notice?
From the Base Angles Theorem, the 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, each.
Equilateral Triangle Theorem: All equilateral triangles are also equiangular. Also, all equiangular triangles are also equilateral.
If , then .
If , then .
Example 4: Algebra Connection Find the value of .
Solution: Because this is an equilateral triangle . Solve for .
Example 5: Algebra Connection Find the value of and the measure of each angle.
Solution: Similar to Example 4, the two angles are equal, so set them equal to each other and solve for .
Substitute ; the base angles are , or . The vertex angle is .
Know What? Revisited Let’s focus on one tile. First, these triangles are all equilateral, so this is an equilateral hexagon (6 sides). Second, we now know that every equilateral triangle is also equiangular, so every triangle within this tile has 3 angles. This makes our equilateral hexagon also equiangular, with each angle measuring . Because there are 6 angles, the sum of the angles in a hexagon are or .
Review Questions
 Questions 15 are similar to Investigations 45 and 46.
 Questions 614 are similar to Examples 25.
 Question 15 uses the definition of an equilateral triangle.
 Questions 1620 use the definition of an isosceles triangle.
 Question 21 is similar to Examples 2 and 3.
 Questions 2225 are proofs and use definitions and theorems learned in this section.
 Questions 2630 use the distance formula.
Constructions For questions 15, use your compass and ruler to:
 Draw an isosceles triangle with sides 3.5 in, 3.5 in, and 6 in.
 Draw an isosceles triangle that has a vertex angle of and legs with length of 4 cm. (you will also need your protractor for this one)
 Draw an equilateral triangle with sides of length 7 cm.
 Using what you know about constructing an equilateral triangle, construct (without a protractor) a angle.
 Draw an isosceles right triangle. What is the measure of the base angles?
For questions 614, find the measure of and/or .

is an equilateral triangle. If bisects , find:
 If , find .
Determine if the following statements are true or false.
 Base angles of an isosceles triangle are congruent.
 Base angles of an isosceles triangle are complementary.
 Base angles of an isosceles triangle can be equal to the vertex angle.
 Base angles of an isosceles triangle can be right angles.
 Base angles of an isosceles triangle are acute.
 In the diagram below, . Find all of the lettered angles.
Fill in the blanks in the proofs below.
 Given: Isosceles , with base angles and is the angle bisector of Prove: is the perpendicular bisector of
Statement  Reason 

1.  Given 
2.  Base Angles Theorem 
3.  
4.  Reflexive PoC 
5.  
6.  
7.  CPCTC 
8. and are supplementary  
9.  Congruent Supplements Theorem 
10. is the perpendicular bisector of 
 Given: Equilateral with Prove: is equiangular
Statement  Reason 

1.  Given 
2.  Base Angles Theorem 
3.  Base Angles Theorem 
4.  Transitive PoC 
5. is equiangular 
 Given: Isosceles with and is the perpendicular bisector of Prove: is the angle bisector of
Statement  Reason 

1.  
2.  
3.  
4.  
5.  
6.  CPCTC 
7. is the angle bisector of 
 Given: Isosceles with base angles and Isosceles with base angles and Prove:
Statement  Reason 

1.  
2.  
3.  
4. 
Coordinate Plane Geometry On the plane, plot the coordinates and determine if the given three points make a scalene or isosceles triangle.
 (2, 1), (1, 2), (5, 2)
 (2, 5), (2, 4), (0, 1)
 (6, 9), (12, 3), (3, 6)
 (10, 5), (8, 5), (2, 3)
 (1, 2), (7, 2), (3, 9)
Review Queue Answers
 Each angle is , or