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

Study concepts, example questions & explanations for SAT Math

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Example Questions

Example Question #3 : How To Find An Angle In An Acute / Obtuse Triangle

Triangle ABC has angle measures as follows:

\dpi{100} \small m\angle ABC=4x+3 

\dpi{100} \small m\angle ACB=2x+6

\dpi{100} \small m\angle BAC=3x

What is \dpi{100} \small m\angle BAC?

Possible Answers:

57

90

44

19

79

Correct answer:

57

Explanation:

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

Thus we set up the equation \dpi{100} \small 4x+3+2x+6+3x=180

After combining like terms and cancelling, we have \dpi{100} \small 9x=171\rightarrow x=19

Thus \dpi{100} \small m\angle BAC=3x=57

Example Question #4 : How To Find An Angle In An Acute / Obtuse Triangle

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

Possible Answers:

34

47

62

55

73

Correct answer:

73

Explanation:

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

Let x = the vertex angle and 2x+5 = the base angle

So the equation to solve becomes  x+(2x+5)+(2x+5)=180

Thus the vertex angle is 34 and the base angles are 73.

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

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

Possible Answers:

Correct answer:

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?

Possible Answers:

Correct answer:

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 #7 : How To Find An Angle In An Acute / Obtuse Triangle

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

Possible Answers:

Correct answer:

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?

Possible Answers:

Correct answer:

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 #1 : Triangles

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

Possible Answers:

36

72

45

108

54

Correct answer:

36

Explanation:

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

Let x = base angle and 0.5x = vertex angle

So the equation to solve becomes x+x+0.5x=180, thus x=72 is the base angle and 0.5x=36 is the vertex angle.

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

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?

Possible Answers:

Correct answer:

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 #3 : Triangles

The base angle of an isosceles triangle is 27^{\circ}.  What is the vertex angle?

Possible Answers:

149^{\circ}

75^{\circ}

135^{\circ}

108^{\circ}

126^{\circ}

Correct answer:

126^{\circ}

Explanation:

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

Solve the equation 27+27+x=180 for x to find the measure of the vertex angle. 

x = 180 - 27 - 27

x = 126

Therefore the measure of the vertex angle is 126^{\circ}.

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