What if your parents want to redo the bathroom? Below is the tile they would like to place in the shower. The blue and green triangles are all equilateral. What type of polygon is dark blue outlined figure? Can you determine how many degrees are in each of these figures? Can you determine how many degrees are around a point? After completing this Concept, you'll be able to apply important properties about equilateral triangles to help you solve problems like this one.
By definition, all sides in an equilateral triangle have exactly the same length.
Investigation: Constructing an Equilateral Triangle
Tools Needed: pencil, paper, compass, ruler, protractor
1. Because all the sides of an equilateral triangle are equal, pick a length to be all the sides of the triangle. Measure this length and draw it horizontally on your 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.
3. 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 or
Equilateral Triangles Theorem: All equilateral triangles are also equiangular. Also, all equiangular triangles are also equilateral.
Find the value of
Because this is an equilateral triangle
Find the values of
Let’s start with
Two sides of an equilateral triangle are
The two given sides must be equal because this is an equilateral triangle. Write and solve the equation for
To figure out how long each side is, plug in
Watch this video for help with the Examples above.
Concept Problem Revisited
Let’s focus on one tile. First, these triangles are all equilateral, so this is an equilateral hexagon (6 sided polygon). Second, we now know that every equilateral triangle is also equiangular, so every triangle within this tile has
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 and are always congruent by the Base Angles Theorem. The angle made by the two legs is called the vertex angle. An equilateral triangle is a triangle with three congruent sides. Equiangular means all angles are congruent. All equilateral triangles are equiangular.
1. Find the measure of
2. Fill in the proof:
|2.||2. Base Angles Theorem|
|3.||3. Base Angles Theorem|
|4.||4. Transitive PoC|
3. True or false: All equilateral triangles are isosceles triangles.
1. The markings show that all angles are congruent. Since all three angles must add up to
||2. Base Angles Theorem|
||3. Base Angles Theorem|
||4. Transitive PoC|
||5. Definition of equiangular.|
3. This statement is true. The definition of an isosceles triangle is a triangle with at least two congruent sides. Since all equilateral triangles have three congruent sides, they fit the definition of an isosceles triangle.
The following triangles are equilateral triangles. Solve for the unknown variables.
- Find the measures of