# SAT Math : How to find an angle in an acute / obtuse isosceles triangle

## Example Questions

### Example Question #1 : Isosceles Triangles

Triangle ABC has angle measures as follows:   What is ?

44

79

57

19

90

57

Explanation:

The sum of the measures of the angles of a triangle is 180.

Thus we set up the equation After combining like terms and cancelling, we have Thus ### Example Question #4 : Acute / Obtuse Isosceles Triangles

The base angle of an isosceles triangle is five more than twice the vertex angle.  What is the base angle?      Explanation:

Every triangle has 180 degrees.  An isosceles triangle has one vertex angle and two congruent base angles.

Let = the vertex angle and = the base angle

So the equation to solve becomes Thus the vertex angle is 34 and the base angles are 73.

### Example Question #5 : Acute / Obtuse Isosceles Triangles

The base angle of an isosceles triangle is 15 less than three times the vertex angle.  What is the vertex angle?      Explanation:

Every triangle contains 180 degrees.  An isosceles triangle has one vertex angle and two congruent base angles.

Let = vertex angle and = base angle

So the equation to solve becomes .

### Example Question #1 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

The base angle of an isosceles triangle is ten less than twice the vertex angle.  What is the vertex angle?      Explanation:

Every triangle has 180 degrees.  An isosceles triangle has one vertex angle and two congruent base angles.

Let = vertex angle and = base angle

So the equation to solve becomes So the vertex angle is 40 and the base angles is 70

### Example Question #2 : Isosceles Triangles

The base angle of an isosceles triangle is 10 more than twice the vertex angle.  What is the vertex angle?      Explanation:

Every triangle has 180 degrees.  An isosceles triangle has one vertex angle and two congruent base angles.

Let = the vertex angle and = the base angle

So the equation to solve becomes The vertex angle is 32 degrees and the base angle is 74 degrees

### Example Question #11 : Isosceles Triangles

In an isosceles triangle, the vertex angle is 15 less than the base angle.  What is the base angle?      Explanation:

Every triangle has 180 degrees.  An isosceles triangle has one vertex angle and two congruent base angles.

Let = base angle and = vertex angle

So the equation to solve becomes Thus, 65 is the base angle and 50 is the vertex angle.

### Example Question #12 : Isosceles Triangles

In an isosceles triangle the vertex angle is half the base angle.  What is the vertex angle?      Explanation:

Every triangle has 180 degrees.  An isosceles triangle has one vertex angle and two congruent base angles.

Let = base angle and = vertex angle

So the equation to solve becomes , thus is the base angle and is the vertex angle.

### Example Question #13 : Isosceles Triangles

If the average (arithmetic mean) of two noncongruent angles of an isosceles triangle is , which of the following is the measure of one of the angles of the triangle?      Explanation:

Since the triangle is isosceles, we know that 2 of the angles (that sum up to 180) must be equal. The question states that the noncongruent angles average 55°, thus providing us with a system of two equations:  Solving for x and y by substitution, we get x = 70° and y = 40° (which average out to 55°).

70 + 70 + 40 equals 180 also checks out.

Since 70° is not an answer choice for us, we know that the 40° must be one of the angles.

### Example Question #14 : Isosceles Triangles

The base angle of an isosceles triangle is .  What is the vertex angle?      Explanation:

Every triangle has 180 degrees.  An isosceles triangle has one vertex angle and two congruent base angles.

Solve the equation for x to find the measure of the vertex angle.

x = 180 - 27 - 27

x = 126

Therefore the measure of the vertex angle is . 