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

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Example Questions

Example Question #4 : Isosceles Triangles

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:

Correct answer:

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

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Example Question #5 : Isosceles Triangles

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

Possible Answers:

$62$

$34$

$47$

$55$

$73$

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.

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Example Question #6 : Isosceles Triangles

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 3x-15 = base angle

So the equation to solve becomes x+(3x-15)+(3x-15)=180 .

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Example Question #4 : Isosceles Triangles

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

Possible Answers:

$70^{\circ}$

$35^{\circ}$

$20^{\circ}$

$40^{\circ}$

$65^{\circ}$

Correct answer:

$40^{\circ}$

Explanation:

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

Let = vertex angle and 2x - 10 = base angle

So the equation to solve becomes x + (2x - 10) + (2x - 10) = 180

So the vertex angle is 40 and the base angles is 70

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Example Question #5 : 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:

$60^{\circ}$

$50^{\circ}$

$32^{\circ}$

$74^{\circ}$

$45^{\circ}$

Correct answer:

$32^{\circ}$

Explanation:

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

Let = the vertex angle and 2x + 10 = the base angle

So the equation to solve becomes x + (2x +10) + (2x +10) = 180

The vertex angle is 32 degrees and the base angle is 74 degrees

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Example Question #6 : How To Find An Angle In An Acute / Obtuse Triangle

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 x - 15 = vertex angle

So the equation to solve becomes (x - 15) + x + x = 180

Thus, 65 is the base angle and 50 is the vertex angle.

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Example Question #11 : Isosceles Triangles

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

Possible Answers:

$108$

$54$

$45$

$36$

$72$

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.

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Example Question #1 : Acute / Obtuse Isosceles Triangles

If the average (arithmetic mean) of two noncongruent angles of an isosceles triangle is 55^o , which of the following is the measure of one of the angles of the triangle?

Possible Answers:

50^o

90^o

45^o

40^o

30^o

Correct answer:

40^o

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:

$\frac{x+y}{2}=55^o$

x+x+y=180^o

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.

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Example Question #2 : Acute / Obtuse Isosceles Triangles

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

Possible Answers:

$75^{\circ}$

$135^{\circ}$

$108^{\circ}$

$126^{\circ}$

$149^{\circ}$

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

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #4 : Isosceles Triangles

Example Question #5 : Isosceles Triangles

Example Question #6 : Isosceles Triangles

Example Question #4 : Isosceles Triangles

Example Question #5 : How To Find An Angle In An Acute / Obtuse Triangle

Example Question #6 : How To Find An Angle In An Acute / Obtuse Triangle

Example Question #11 : Isosceles Triangles

Example Question #1 : Acute / Obtuse Isosceles Triangles

Example Question #2 : Acute / Obtuse Isosceles Triangles

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