Triangles - GED Math

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Question

Which of the following can be the measures of the three angles of an acute isosceles triangle?

Answer

For the triangle to be acute, all three angles must measure less than $90 ^{\circ }$ . We can eliminate $32 ^{\circ }, 32 ^{\circ }, 116^{\circ }$ and $45 ^{\circ }, 45 ^{\circ }, 90^{\circ }$ for this reason.

In an isosceles triangle, at least two angles are congruent, so we can eliminate $50 ^{\circ }, 60 ^{\circ }, 70 ^{\circ }$ .

The degree measures of the three angles of a triangle must total 180, so, since $80 ^{\circ }+ 80 ^{\circ }+ 40 ^{\circ } = 200^{\circ }$ , we can eliminate $80 ^{\circ }, 80 ^{\circ }, 40 ^{\circ }$ .

$36^{\circ }, 72 ^{\circ }, 72 ^{\circ }$ is correct.

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Question

Which of the following describes a triangle with sides of length 9 feet, 3 yards, and 90 inches?

Answer

One yard is equal to three feet, and one foot is equal to twelve inches. Therefore, 9 feet is equal to $9 \times 12 = 108$ inches, and 3 yards is equal to $3 \times 36 = 108$ inches. The triangle has sides of measure 90 inches, 108 inches, and 108 inches. Exactly two sides are of equal measure, so it is isosceles but not equilateral.

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Question

Note: Figure NOT drawn to scale.

Refer to the above triangle. Evaluate .

Answer

The degree measures of a triangle total $180^{\circ }$ , so

$4x \div 4 = 156 \div 4$

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Question

Note: Figure NOT drawn to scale.

Refer to the above figure. Evaluate .

Answer

The degree measures of the interior angles of a triangle total $180 ^{\circ }$ , so, if we let $x^{\circ }$ be the measure of the unmarked angle, then

Three angles with measures $67^{\circ}, y^{\circ}, y^{\circ}$ together form a straight angle, so

$2y \div 2 = 113 \div 2$

$y = 56 \frac{1}{2}$

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Question

Figure drawn to scale.

Refer to the above diagram.

Which of the following is a valid description of $\Delta EFC$ ?

Answer

One of the angles of $\Delta EFC$ - namely, $\angle EFC$ - can be seen to be an obtuse angle, as it is wider than a right angle. This makes $\Delta EFC$ , by definition, an obtuse triangle.

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Question

Refer to the above diagram.

Which of the following is a valid description of $\Delta CED$ ?

Answer

One of the angles of $\Delta CED$ - namely, $\angle CDE$ - is marked as a right angle. This makes $\Delta CED$ , by definition, a right triangle.

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Question

Which of the following follows from the fact that $\Delta MNO \cong \Delta PQR$ ?

Answer

A congruency statement about two triangles implies nothing about the relationship between two angles of one of the triangles, so $\angle M \cong \angle N$ is not correct.

Also, letters in the same position between the two triangles refer to corresponding - and subsequently, congruent - angles. Therefore, $\Delta MNO \cong \Delta PQR$ implies that:

$\angle M \cong \angle P$

$\angle N \cong \angle Q$

$\angle O \cong \angle R$

Of the given choices, only $\angle M \cong \angle P$ is a consequence. It is the correct response.

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Question

An exterior angle of an isosceles triangle measures $70 ^{\circ }$ . What is the least measure of any of the three angles of the triangle?

Answer

The triangle has an exterior angle of $70 ^{\circ }$ , so it has an interior angle of $(180-70)^{\circ } = 110 ^{\circ }$ .

Since this is an obtuse angle, its other two angles must be acute. By the Isosceles Triangle Theorem, an isosceles triangle must have two congruent angles - the acute angles are those. Since the sum of their measures is the same as their remote exterior angle - $70 ^{\circ }$ - each has measure $35 ^{\circ }$ .This is the correct response.

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Question

An exterior angle of an isosceles triangle measures $80 ^{\circ }$ . What is the greatest measure of any of the three angles of the triangle?

Answer

The triangle has an exterior angle of $80 ^{\circ }$ , so it has an interior angle of $(180-80)^{\circ } = 100 ^{\circ }$ .

Since this is an obtuse angle, its other two angles must be acute. Therefore, this angle is the one of greatest measure.

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Question

An exterior angle of an isosceles triangle measures $130 ^{\circ }$ . What is the greatest measure of any of the three angles of the triangle?

Answer

The triangle has an exterior angle of $130 ^{\circ }$ , so it has an interior angle of $(180-130)^{\circ } = 50^{\circ }$ . By the Isosceles Triangle Theorem, an isosceles triangle must have two congruent angles; there are two possible scenarios that fit this criterion:

Case 1: Two angles have measure $50^{\circ }$ . The third angle will have measure

$\left [180 - (50 + 50 ) \right ] ^{\circ } = 80 ^{\circ }$ .

$80 ^{\circ }$ will be the greatest of the angle measures.

Case 2: One angle has measure $50^{\circ }$ and the others are congruent. Their common measure will be

$\frac{1}{2} (180^{\circ }- 50^{\circ } ) = 65 ^{\circ }$ .

$65 ^{\circ }$ will be the greatest of the angle measures.

The given information is therefore inconclusive.

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Question

An exterior angle of an isosceles triangle measures $130 ^{\circ }$ . What is the least measure of any of the three angles of the triangle?

Answer

The triangle has an exterior angle of $130 ^{\circ }$ , so it has an interior angle of $(180-130)^{\circ } = 50^{\circ }$ . By the Isosceles Triangle Theorem, an isosceles triangle must have two congruent angles; there are two possible scenarios that fit this criterion:

Case 1: Two angles have measure $50^{\circ }$ . The third angle will have measure

$\left [180 - (50 + 50 ) \right ] ^{\circ } = 80 ^{\circ }$ .

$50^{\circ }$ will be the least of the angle measures.

Case 2: One angle has measure $50^{\circ }$ and the others are congruent. Their common measure will be

$\frac{1}{2} (180^{\circ }- 50^{\circ } ) = 65 ^{\circ }$ .

$50^{\circ }$ will be the least of the angle measures.

In both cases, the least of the degree measures of the angles will be $50^{\circ }$ .

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Question

Which of the following could give the measures of the three angles of a triangle?

Answer

The degree measures of a triangle must total $180 ^{\circ }$ , so we find the sum of the degree measures in each choice.

$75^{\circ }, 75^{\circ }, 65^{\circ }$ :

$25^{\circ }, 65^{\circ }, 100^{\circ }$ :

$35^{\circ },40^{\circ }, 95^{\circ }$ :

$50 ^{\circ }, 55^{\circ }, 75^{\circ }$ :

This is the correct choice.

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Question

Which of the following cannot be true of equilateral triangle $\Delta ABC$ ?

Answer

Every equilateral triangle has three congruent angles - all of which have measure $60 ^{\circ }$ - as a result of the Isosceles Triangle Theorem. Therefore, the statements:

$m \angle C = 60 ^{\circ }$

$m \angle B = m \angle C$

$\Delta ABC$ is an acute triangle

immediately follow.

However, if $\angle A$ and $\angle B$ are complementary, then the sum of their measures is $90 ^{\circ }$ ; since each angle measures $60 ^{\circ }$ , then their measures add up to $120^{\circ }$ . This is the correct choice.

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Question

Refer to the above diagram. $\Delta ABC$ is equilateral; $\Delta ABX \cong \Delta ACX$ . How many of the following statements must be true?

I) $\overline{AX}$ bisects $\angle BAC$

II) $\overline{AX}$ bisects $\overline{BC}$

III) $\overline{AX} \perp \overline{BC}$

Answer

Since corresponding parts of congruent triangles are congruent, it follows that $\angle BAX \cong \angle CAX$ . Therefore, $\overline{AX}$ , by definition, bisects $\angle BAC$ , and the first statement is true. A bisector of an angle of an equilateral triangle is also the perpendicular bisector of the opposite side, so the other two statements are immediate consequences.

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Question

Which of the following could be the measures of two angles of a scalene triangle?

Answer

By the Isosceles Triangle Theorem, angles opposite congruent sides of a triangle are congruent. Therefore, a scalene triangle, having three noncongrent sides, must have three noncongruent angles.

$50 ^{\circ } , 50 ^{\circ }$ can be eliminated immediately.

For the other three choices, we need to find the measure of the third angle using the fact that the degree measures of the angles of a triangle must total $180 ^{\circ }$ .

$140^{\circ } , 20^{\circ }$ :

The third angle measure is $180^{\circ } - \left (140^{\circ } + 20^{\circ } \right ) = 180^{\circ } - 160^{\circ } = 20^{\circ }$ .

This triangle has two $20^{\circ }$ angles and can be eliminated.

$70 ^{\circ } , 40 ^{\circ }$ :

The third angle measure is $180^{\circ } - \left (70^{\circ } + 40^{\circ } \right ) = 180^{\circ } - 110^{\circ } = 70^{\circ }$ .

This triangle has two $70^{\circ }$ angles and can be eliminated.

$80^{\circ } , 40^{\circ }$ :

The third angle measure is $180^{\circ } - \left (80^{\circ } + 40^{\circ } \right ) = 180^{\circ } - 120^{\circ } = 60^{\circ }$ .

This triangle has three angles of different measure, making it scalene. This is the correct choice.

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Question

For a triangle, suppose a given interior angle is degrees and the other two angles are . What is the value of ?

Answer

The sum of the three interior angles of a triangle is 180 degrees.

Set up an equation such that all three angles are equal to 180 degrees.

Combine like-terms.

Subtract 34 on both sides.

Divide by 6 on both sides.

$\frac{146}{6} =\frac{73}{3}$

The answer is: $\frac{73}{3}$

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Question

If a triangle is isosceles, and the vertex angle is 100 degrees, what must be another angle?

Answer

The isosceles triangle has two congruent sides and two congruent angles.

The interior angles will sum to 180 degrees.

Let the unknown angles be . These angles are congruent to each other because the triangle is isosceles.

Divide by 2 on both sides.

$\frac{2x }{2}= \frac{80}{2}$

The answer is:

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Question

Figure NOT drawn to scale.

Refer to the above diagram. Evaluate $m \angle 1$

Answer

$\triangle ABC$ is indicated to be isosceles, with $\overline{AB} \cong \overline{AC}$ , so, by the isosceles triangle theorem,

$\angle B \cong \angle ACB$ .

Since the measures of the interior angles of a triangle total $180^{\circ }$ ,

$m \angle A + m\angle B + m \angle ACB = 180^{\circ }$

Substitute $107^{\circ }$ for $\angle A$ and $m \angle B$ for $m \angle ACB$ , then solve for $m \angle B$ :

$107^{\circ }+ m\angle B + m \angle B = 180^{\circ }$

$107^{\circ }+ 2 m \angle B = 180^{\circ }$

Subtract $107^{\circ }$ from both sides:

$107^{\circ }+ 2 m \angle B - 107^{\circ } = 180^{\circ } - 107^{\circ }$

$2 m \angle B = 73^{\circ }$

Multiply both sides by $\frac{1}{2}$ :

$\frac{1}{2} \cdot 2 m \angle B =\frac{1}{2} \cdot 73^{\circ }$

$m \angle B =36 \frac{1}{2} ^{\circ }$

$\angle 1$ is an exterior angle of $\triangle ABC$ ; its measure is the sum of those of its remote interior angle, $\angle A$ and $\angle B$ , so

$m \angle 1 = m \angle A + m \angle B$

Substitute and add:

$m \angle 1 =107 ^{\circ } + 36 \frac{1}{2} ^{\circ }$

$m \angle 1 =143 \frac{1}{2} ^{\circ }$ .

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Question

The figure below is what type of triangle?

Answer

Start by figuring out the degree measurements of each angle.

Since we know that the angles of a triangle must add up to , we can write the following equation:

Solve for .

Now, we know the measure of the angles:

Since each angle of the triangle is different, this means the the legs of the triangle must also be different. Thus, this is a scalene triangle.

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Question

For an isosceles triangle, if the vertex angle is measured 30 degrees, what must a base angle equal?

Answer

An isosceles has two equivalent base angles. We can set up an equation such that both angles with the vertex angle add up to 180 degrees.

Solve for , which is a base angle. Subtract 30 from both sides.

Divide by 2 on both sides.

$\frac{2x }{2}= \frac{150}{2}$

The answer is:

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