Acute / Obtuse Isosceles Triangles - 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

A triangle with two equal angles is called a(n) __________.

Answer

An isoceles triangle is a triangle that has at least two congruent sides (and therefore, at least two congruent angles as well).

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Question

An isosceles triangle has an area of 12. If the ratio of the base to the height is 3:2, what is the length of the two equal sides?

Answer

Area of a triangle is ½ x base x height. Since base:height = 3:2, base = 1.5 height. Area = 12 = ½ x 1.5 height x height or 24/1.5 = height2. Height = 4. Base = 1.5 height = 6. Half the base and the height form the legs of a right triangle, with an equal leg of the isosceles triangle as the hypotenuse. This is a 3-4-5 right triangle.

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Question

Two sides of a triangle each have length 6. All of the following could be the length of the third side EXCEPT

Answer

This question is about the Triangle Inequality, which states that in a triangle with two sides A and B, the third side must be greater than the absolute value of the difference between A and B and smaller than the sum of A and B.

Applying the Triangle Inequality to this problem, we see that the third side must be greater than the absolute value of the difference between the other two sides, which is |6-6|=0, and smaller than the sum of the two other sides, which is 6+6=12. The only answer choice that does not satisfy this range of possible values is 12 since the third side must be LESS than 12.

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Question

Two sides of an isosceles triangle are 20 and 30. What is the difference of the largest and the smallest possible perimeters?

Answer

The trick here is that we don't know which is the repeated side. Our possible triangles are therefore 20 + 20 + 30 = 70 or 30 + 30 + 20 = 80. The difference is therefore 80 – 70 or 10.

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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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