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# 4.5: Isosceles and Equilateral Triangles

Difficulty Level: At Grade Created by: CK-12

## 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 \begin{align*}x\end{align*} and/or \begin{align*}y\end{align*}.

1. 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 4-5: Isosceles Triangle Construction

Tools Needed: pencil, paper, compass, ruler, protractor

1. Refer back to Investigation 4-2. 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 \begin{align*}72.5^\circ\end{align*} and the vertex angle should be \begin{align*}35^\circ\end{align*}.

We can generalize this investigation for all isosceles triangles.

Base Angles Theorem: The base angles of an isosceles triangle are congruent.

For \begin{align*}\triangle DEF\end{align*}, if \begin{align*}\overline{DE} \cong \overline{EF}\end{align*}, then \begin{align*}\angle D \cong \angle F\end{align*}.

To prove the Base Angles Theorem, we need to draw the angle bisector (Investigation 1-5) of \begin{align*}\angle E\end{align*}.

Given: Isosceles triangle \begin{align*}\triangle DEF\end{align*} above, with \begin{align*}\overline{DE} \cong \overline{EF}\end{align*}.

Prove: \begin{align*}\angle D \cong \angle F\end{align*}

Statement Reason
1. Isosceles triangle \begin{align*}\triangle DEF\end{align*} with \begin{align*}\overline{DE} \cong \overline{EF}\end{align*} Given

2. Construct angle bisector \begin{align*}\overline{EG}\end{align*} of \begin{align*}\angle E\end{align*}

Every angle has one angle bisector
3. \begin{align*}\angle DEG \cong \angle FEG\end{align*} Definition of an angle bisector
4. \begin{align*}\overline{EG} \cong \overline{EG}\end{align*} Reflexive PoC
5. \begin{align*}\triangle DEG \cong \triangle FEG\end{align*} SAS
6. \begin{align*}\angle D \cong \angle F\end{align*} CPCTC

Let’s take a further look at the picture from step 2 of our proof.

Because \begin{align*}\triangle DEG \cong \triangle FEG\end{align*}, we know \begin{align*}\angle EGD \cong \angle EGF\end{align*} by CPCTC. These two angles are also a linear pair, so \begin{align*}90^\circ\end{align*} each and \begin{align*}\overline{EG} \perp \overline{DF}\end{align*}.

Additionally, \begin{align*}\overline{DG} \cong \overline{GF}\end{align*} by CPCTC, so \begin{align*}G\end{align*} is the midpoint of \begin{align*}\overline{DF}\end{align*}. This means that \begin{align*}\overline{EG}\end{align*} is the perpendicular bisector of \begin{align*}\overline{DF}\end{align*}.

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 \begin{align*}\angle S \cong \angle U\end{align*}.

Example 2: If an isosceles triangle has base angles with measures of \begin{align*}47^\circ\end{align*}, what is the measure of the vertex angle?

Solution: Draw a picture and set up an equation to solve for the vertex angle, \begin{align*}v\end{align*}.

\begin{align*}47^\circ + 47^\circ + v & = 180^\circ\\ v & = 180^\circ - 47^\circ - 47^\circ\\ v & = 86^\circ\end{align*}

Example 3: If an isosceles triangle has a vertex angle with a measure of \begin{align*}116^\circ\end{align*}, what is the measure of each base angle?

Solution: Draw a picture and set up and equation to solve for the base angles, \begin{align*}b\end{align*}.

\begin{align*}116^\circ + b + b & = 180^\circ\\ 2b & = 64^\circ\\ b & = 32^\circ\end{align*}

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 \begin{align*}\triangle DEF\end{align*}, if \begin{align*}\angle D \cong \angle F\end{align*}, then \begin{align*}\overline{DE} \cong \overline{EF}\end{align*}.

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 \begin{align*}\triangle DEF\end{align*}, if \begin{align*}\overline{EG} \perp \overline{DF}\end{align*} and \begin{align*}\overline{DG} \cong \overline{GF}\end{align*}, then \begin{align*}\angle DEG \cong \angle FEG\end{align*}.

Equilateral Triangles By definition, all sides in an equilateral triangle have the same length.

Investigation 4-6: 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, \begin{align*}60^\circ\end{align*} each.

Equilateral Triangle Theorem: All equilateral triangles are also equiangular. Also, all equiangular triangles are also equilateral.

If \begin{align*}\overline{AB} \cong \overline{BC} \cong \overline{AC}\end{align*}, then \begin{align*}\angle A \cong \angle B \cong \angle C\end{align*}.

If \begin{align*}\angle A \cong \angle B \cong \angle C\end{align*}, then \begin{align*}\overline{AB} \cong \overline{BC} \cong \overline{AC}\end{align*}.

Example 4: Algebra Connection Find the value of \begin{align*}x\end{align*}.

Solution: Because this is an equilateral triangle \begin{align*}3x-1=11\end{align*}. Solve for \begin{align*}x\end{align*}.

\begin{align*}3x-1 & = 11\\ 3x & = 12\\ x & = 4\end{align*}

Example 5: Algebra Connection Find the value of \begin{align*}x\end{align*} 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 \begin{align*}x\end{align*}.

\begin{align*}(4x+12)^\circ & = (5x-3)^\circ\\ 15^\circ & = x\end{align*}

Substitute \begin{align*}x = 15^\circ\end{align*}; the base angles are \begin{align*}4(15^\circ) +12\end{align*}, or \begin{align*}72^\circ\end{align*}. The vertex angle is \begin{align*}180^\circ -72^\circ-72^\circ =36^\circ\end{align*}.

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 \begin{align*}60^\circ\end{align*} angles. This makes our equilateral hexagon also equiangular, with each angle measuring \begin{align*}120^\circ\end{align*}. Because there are 6 angles, the sum of the angles in a hexagon are \begin{align*}6 \cdot 120^\circ\end{align*} or \begin{align*}720^\circ\end{align*}.

## Review Questions

• Questions 1-5 are similar to Investigations 4-5 and 4-6.
• Questions 6-14 are similar to Examples 2-5.
• Question 15 uses the definition of an equilateral triangle.
• Questions 16-20 use the definition of an isosceles triangle.
• Question 21 is similar to Examples 2 and 3.
• Questions 22-25 are proofs and use definitions and theorems learned in this section.
• Questions 26-30 use the distance formula.

Constructions For questions 1-5, use your compass and ruler to:

1. Draw an isosceles triangle with sides 3.5 in, 3.5 in, and 6 in.
2. Draw an isosceles triangle that has a vertex angle of \begin{align*}100^\circ\end{align*} and legs with length of 4 cm. (you will also need your protractor for this one)
3. Draw an equilateral triangle with sides of length 7 cm.
4. Using what you know about constructing an equilateral triangle, construct (without a protractor) a \begin{align*}60^\circ\end{align*} angle.
5. Draw an isosceles right triangle. What is the measure of the base angles?

For questions 6-14, find the measure of \begin{align*}x\end{align*} and/or \begin{align*}y\end{align*}.

1. \begin{align*}\triangle EQG\end{align*} is an equilateral triangle. If \begin{align*}\overline{EU}\end{align*} bisects \begin{align*}\angle LEQ\end{align*}, find:
1. \begin{align*}m\angle EUL\end{align*}
2. \begin{align*}m\angle UEL\end{align*}
3. \begin{align*}m\angle ELQ\end{align*}
4. If \begin{align*}EQ = 4\end{align*}, find \begin{align*}LU\end{align*}.

Determine if the following statements are true or false.

1. Base angles of an isosceles triangle are congruent.
2. Base angles of an isosceles triangle are complementary.
3. Base angles of an isosceles triangle can be equal to the vertex angle.
4. Base angles of an isosceles triangle can be right angles.
5. Base angles of an isosceles triangle are acute.
6. In the diagram below, \begin{align*}l_1||l_2\end{align*}. Find all of the lettered angles.

Fill in the blanks in the proofs below.

1. Given: Isosceles \begin{align*}\triangle CIS\end{align*}, with base angles \begin{align*}\angle C\end{align*} and \begin{align*}\angle S\end{align*} \begin{align*}\overline{IO}\end{align*} is the angle bisector of \begin{align*}\angle CIS\end{align*} Prove: \begin{align*}\overline{IO}\end{align*} is the perpendicular bisector of \begin{align*}\overline{CS}\end{align*}
Statement Reason
1. Given
2. Base Angles Theorem
3. \begin{align*}\angle CIO \cong \angle SIO\end{align*}
4. Reflexive PoC
5. \begin{align*}\triangle CIO \cong \triangle SIO\end{align*}
6. \begin{align*}\overline{CO} \cong \overline{OS}\end{align*}
7. CPCTC
8. \begin{align*}\angle IOC\end{align*} and \begin{align*}\angle IOS\end{align*} are supplementary
9. Congruent Supplements Theorem
10. \begin{align*}\overline{IO}\end{align*} is the perpendicular bisector of \begin{align*}\overline{CS}\end{align*}
1. Given: Equilateral \begin{align*}\triangle RST\end{align*} with \begin{align*}\overline{RT} \cong \overline{ST} \cong \overline{RS}\end{align*} Prove: \begin{align*}\triangle RST\end{align*} is equiangular
Statement Reason
1. Given
2. Base Angles Theorem
3. Base Angles Theorem
4. Transitive PoC
5. \begin{align*}\triangle RST\end{align*} is equiangular
1. Given: Isosceles \begin{align*}\triangle ICS\end{align*} with \begin{align*}\angle C\end{align*} and \begin{align*}\angle S\end{align*} \begin{align*}\overline{IO}\end{align*} is the perpendicular bisector of \begin{align*}\overline{CS}\end{align*} Prove: \begin{align*}\overline{IO}\end{align*} is the angle bisector of \begin{align*}\angle CIS\end{align*}
Statement Reason
1.
2. \begin{align*}\angle C \cong \angle S\end{align*}
3. \begin{align*}\overline{CO} \cong \overline{OS}\end{align*}
4. \begin{align*}m\angle IOC = m\angle IOS = 90^\circ\end{align*}
5.
6. CPCTC
7. \begin{align*}\overline{IO}\end{align*} is the angle bisector of \begin{align*}\angle CIS\end{align*}
1. Given: Isosceles \begin{align*}\triangle ABC\end{align*} with base angles \begin{align*}\angle B\end{align*} and \begin{align*}\angle C\end{align*} Isosceles \begin{align*}\triangle XYZ\end{align*} with base angles \begin{align*}\angle Y\end{align*} and \begin{align*}\angle Z\end{align*} \begin{align*}\angle C \cong \angle Z, \ \overline{BC} \cong \overline{YZ}\end{align*} Prove: \begin{align*}\triangle ABC \cong \triangle XYZ\end{align*}
Statement Reason
1.
2. \begin{align*}\angle B \cong \angle C, \ \angle Y \cong \angle Z\end{align*}
3. \begin{align*}\angle B \cong \angle Y\end{align*}
4. \begin{align*}\triangle ABC \cong \triangle XYZ\end{align*}

Coordinate Plane Geometry On the \begin{align*}x-y\end{align*} plane, plot the coordinates and determine if the given three points make a scalene or isosceles triangle.

1. (-2, 1), (1, -2), (-5, -2)
2. (-2, 5), (2, 4), (0, -1)
3. (6, 9), (12, 3), (3, -6)
4. (-10, -5), (-8, 5), (2, 3)
5. (-1, 2), (7, 2), (3, 9)

1. \begin{align*}(5x - 1)^\circ + (8x + 5)^\circ + (4x + 6)^\circ = 180^\circ\!\\ {\;} \qquad \qquad \qquad \qquad \qquad \ 17x + 10 = 180^\circ\!\\ {\;} \qquad \qquad \qquad \qquad \qquad \ \qquad 17x = 170^\circ\!\\ {\;} \qquad \qquad \qquad \qquad \qquad \ \qquad \quad x = 10^\circ\end{align*}
2. \begin{align*}x = 40^\circ, y = 70^\circ\end{align*}
3. \begin{align*}x - 3 = 8\!\\ {\;} \quad x = 5\end{align*}
4. Each angle is \begin{align*}\frac{180^\circ}{3}\end{align*}, or \begin{align*}60^\circ\end{align*}

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