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

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Question

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?

Answer

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:

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

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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Question

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 $\dpi{100} \small m\angle BAC$ ?

Answer

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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Question

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

Answer

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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Question

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

Answer

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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Question

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

Answer

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 .

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Question

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

Answer

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

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Question

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

Answer

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

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Question

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

Answer

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.

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Question

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

Answer

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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Question

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?

Answer

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:

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

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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Question

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 $\dpi{100} \small m\angle BAC$ ?

Answer

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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Question

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

Answer

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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Question

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

Answer

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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Question

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

Answer

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 .

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Question

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

Answer

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

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Question

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

Answer

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

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Question

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

Answer

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.

Compare your answer with the correct one above

Question

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

Answer

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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Question

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?

Answer

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:

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

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.

Compare your answer with the correct one above

Question

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 $\dpi{100} \small m\angle BAC$ ?

Answer

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

Compare your answer with the correct one above

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