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Isosceles Triangles

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What if you were presented with an isoceles triangle and told that its base angles measure x^\circ and y^\circ ? What could you conclude about x and y ? After completing this Concept, you'll be able to apply important properties about isoceles triangles to help you solve problems like this one.

Watch This

CK-12

Watch the first part of this video.

James Sousa: How To Construct An Isosceles Triangle

Then watch this video.

James Sousa: Proof of the Isosceles Triangle Theorem

Finally, watch this video.

James Sousa: Using the Properties of Isosceles Triangles to Determine Values

Guidance

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 . One of the important properties of isosceles triangles is that their base angles are always congruent. This is called the Base Angles Theorem.

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

Another important property of isosceles triangles is that the angle bisector of the vertex angle is also the perpendicular bisector of the base. This is called the Isosceles Triangle Theorem . ( Note this is ONLY true of the vertex angle. ) The converses of the Base Angles Theorem and the Isosceles Triangle Theorem are both true as well.

Base Angles Theorem Converse: If two angles in a triangle are congruent, then the sides opposite those angles are also congruent. So for \triangle DEF , if \angle D \cong \angle F , then \overline{DE} \cong \overline{EF} .

Isosceles Triangle Theorem Converse: The perpendicular bisector of the base of an isosceles triangle is also the angle bisector of the vertex angle. So for isosceles \triangle DEF , if \overline{EG} \perp \overline{DF} and \overline{DG} \cong \overline{GF} , then \angle DEG \cong \angle FEG .

Example A

Which two angles are congruent?

This is an isosceles triangle. The congruent angles are opposite the congruent sides. From the arrows we see that \angle S \cong \angle U .

Example B

If an isosceles triangle has base angles with measures of 47^\circ , what is the measure of the vertex angle?

Draw a picture and set up an equation to solve for the vertex angle, v . Remember that the three angles in a triangle always add up to 180^\circ .

47^\circ + 47^\circ + v & = 180^\circ\\v & = 180^\circ - 47^\circ - 47^\circ\\v & = 86^\circ

Example C

If an isosceles triangle has a vertex angle with a measure of 116^\circ , what is the measure of each base angle?

Draw a picture and set up and equation to solve for the base angles, b .

116^\circ + b + b & = 180^\circ\\2b & = 64^\circ\\b & = 32^\circ

CK-12 Isosceles Triangles

Guided Practice

1. Find the value of x and the measure of each angle.

2. Find the measure of x .

3. True or false: Base angles of an isosceles triangle can be right angles.

Answers:

1. The two angles are equal, so set them equal to each other and solve for x .

(4x+12)^\circ & = (5x-3)^\circ\\15 = x

Substitute x = 15 ; the base angles are [4(15) +12]^\circ , or 72^\circ . The vertex angle is 180^\circ -72^\circ-72^\circ =36^\circ .

2. The two sides are equal, so set them equal to each other and solve for x .

2x-9 & = x+5\\x & = 14

3. This statement is false. Because the base angles of an isosceles triangle are congruent, if one base angle is a right angle then both base angles must be right angles. It is impossible to have a triangle with two right ( 90^\circ ) angles. The Triangle Sum Theorem states that the sum of the three angles in a triangle is 180^\circ . If two of the angles in a triangle are right angles, then the third angle must be 0^\circ and the shape is no longer a triangle.

Practice

Find the measures of x and/or y .

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 are acute.

Fill in the proofs below.

  1. Given : Isosceles \triangle CIS , with base angles \angle C and \angle S \overline{IO} is the angle bisector of \angle CIS Prove : \overline{IO} is the perpendicular bisector of \overline{CS}
Statement Reason
1. 1. Given
2. 2. Base Angles Theorem
3. \angle CIO \cong \angle SIO 3.
4. 4. Reflexive PoC
5. \triangle CIO \cong \triangle SIO 5.
6. \overline{CO} \cong \overline{OS} 6.
7. 7. CPCTC
8. \angle IOC and \angle IOS are supplementary 8.
9. 9. Congruent Supplements Theorem
10. \overline{IO} is the perpendicular bisector of \overline{CS} 10.
  1. Given : Isosceles \triangle ICS with \angle C and \angle S \overline{IO} is the perpendicular bisector of \overline{CS} Prove : \overline{IO} is the angle bisector of \angle CIS
Statement Reason
1. 1.
2. \angle C \cong \angle S 2.
3. \overline{CO} \cong \overline{OS} 3.
4. m\angle IOC = m\angle IOS = 90^\circ 4.
5. 5.
6. 6. CPCTC
7. \overline{IO} is the angle bisector of \angle CIS 7.

On the x-y 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)

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