GED Math - Angles and Triangles

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

$36^{\circ }, 72 ^{\circ }, 72 ^{\circ }$

$32 ^{\circ }, 32 ^{\circ }, 116^{\circ }$

$45 ^{\circ }, 45 ^{\circ }, 90^{\circ }$

$50 ^{\circ }, 60 ^{\circ }, 70 ^{\circ }$

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

Explanation

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.

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

The triangle is isosceles but not equilateral.

The triangle is scalene.

The triangle is equilateral.

Insufficient information is given to answer this question.

Explanation

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.

Triangle

Note: Figure NOT drawn to scale.

Refer to the above triangle. Evaluate .

Explanation

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

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

Triangle

Note: Figure NOT drawn to scale.

Refer to the above figure. Evaluate .

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

$y = 20 \frac{1}{2}$

Explanation

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

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

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

Thingy

Figure drawn to scale.

Refer to the above diagram.

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

Obtuse

Acute

Right

Equiangular

Explanation

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.

Thingy

Refer to the above diagram.

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

Right

Equiangular

Acute

Obtuse

Explanation

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.

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

$\angle M \cong \angle P$

$\angle N \cong \angle R$

$\angle O \cong \angle Q$

$\angle M \cong \angle N$

Explanation

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.

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

$35 ^{\circ }$

Insufficient information is given to answer this question.

$70^{\circ }$

$40^{\circ }$

Explanation

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.

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

$100 ^{\circ }$

$40^{\circ }$

$80^{\circ }$

Insufficient information is given to answer this question.

Explanation

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.

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

Insufficient information is given to answer this question.

$50^{\circ }$

$65 ^{\circ }$

$80 ^{\circ }$

Explanation

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: