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Intermediate Geometry : How to find an angle in an acute / obtuse triangle

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Example Question #21 : How To Find An Angle In An Acute / Obtuse Triangle

Two of the exterior angles of a triangle, taken at different vertices, measure $145^{\circ }$ and $125^{\circ }$ . Is the triangle acute, right, or obtuse?

Possible Answers:

Right

Acute

Obtuse

Correct answer:

Right

Explanation:

At a given vertex, an exterior angle and an interior angle of a triangle form a linear pair, making them supplementary - that is, their measures total $180^{\circ }$ . The measures of two interior angles can be calculated by subtracting the exterior angle measures from $180^{\circ }$ :

$180^{\circ } - 145^{\circ } = 35^{\circ }$

$180^{\circ } - 125^{\circ } = 55^{\circ }$

The triangle has two interior angles of measures $35^{\circ }$ and $55^{\circ }$ . The sum of these measures is $35^{\circ }+ 55 ^{\circ } = 90 ^{\circ }$ , thereby making them complementary. A triangle with two complementary acute angles is a right triangle.

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Example Question #51 : Acute / Obtuse Triangles

True or false: It is possible for a triangle to have angles of measure $50^{\circ }$ , $60^{\circ }$ , and $70^{\circ }$ .

Possible Answers:

False

True

Correct answer:

True

Explanation:

The sum of the measures of the angles of a triangle is $180^{\circ }$ . The sum of the three given angle measures is

$50^{\circ }+ 60 ^{\circ }+ 70^{\circ } = 180^{\circ }$ .

This makes the triangle possible.

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Example Question #23 : How To Find An Angle In An Acute / Obtuse Triangle

True or false: It is possible for a triangle to have three interior angles, each of whose measures are $50^{\circ }$ .

Possible Answers:

True

False

Correct answer:

False

Explanation:

A triangle with three congruent angles is an equiangular - and equilateral - triangle; such an angle must have three angles that measure $60^{\circ }$ .

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Example Question #24 : How To Find An Angle In An Acute / Obtuse Triangle

Given: $\bigtriangleup ABC$ with perimeter 40;

AB = 12 , BC = 14

True or false: $\angle A \cong \angle B$

Possible Answers:

False

True

Correct answer:

True

Explanation:

The perimeter of $\bigtriangleup ABC$ is the sum of the lengths of its sides - that is,

AB + BC + AC = P

The perimeter is 40, so set AB = 12 , BC = 14, P = 40 , and solve for :

AB + BC + AC = P

12+14 + AC = 40

26 + AC = 40

Subtract 26 from both sides:

26 + AC - 26 = 40 - 26

AC= 14

$\overline{BC}\cong \overline{AC}$ , so by the Isosceles Triangle Theorem, their opposite angles are congruent - that is,

$\angle A \cong \angle B$ .

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Example Question #25 : How To Find An Angle In An Acute / Obtuse Triangle

$\bigtriangleup ABC$ is an equilateral triangle; is the midpoint of $\overline{BC}$ ; the segment $\overline{AM }$ is constructed.

True or false: $m \angle BAM = 40^{\circ }$ .

Possible Answers:

False

True

Correct answer:

False

Explanation:

The referenced triangle is below:

Equilateral 2

In an equilateral triangle, the median from - the segment from to , the midpoint of the opposite side $\overline{BC}$ - is also the bisector of the angle $\angle BAC$ , so

$m \angle BAM = \frac{1}{2} \cdot \angle BAC$

Each interior angle of an equilateral triangle, including $\angle BAC$ , measures $60^{\circ }$ , so substitute and evaluate:

$m \angle BAM = \frac{1}{2} \cdot 60^{\circ } = 30 ^{\circ }$ .

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Example Question #26 : How To Find An Angle In An Acute / Obtuse Triangle

$\bigtriangleup ABC$ is an equilateral triangle. Locate a point along $\overline{BC}$ and construct $\overline{AD}$ . $m \angle ADB = 80^{\circ }$ .

Evaluate $m \angle DAC$ .

Possible Answers:

$20^{\circ }$

$30^{\circ }$

$\textup{Insufficient information is given to answer the problem}$

$25^{\circ }$

$15^{\circ }$

Correct answer:

$20^{\circ }$

Explanation:

The referenced figure is below. Note that $m \angle C = 60 ^{\circ }$ , as is the case with all of the interior angles of an equilateral triangle.

Equilateral

The interior angles of an equilateral triangle each measure $60^{\circ }$ . An exterior angle of a triangle has as its degree measure the sum of its remote interior angles; specifically,

$m \angle ADB = m\angle C + m \angle DAC$

Substitute the known angle measures, and solve:

$80^{\circ }= 60^{\circ } + m \angle DAC$

$80^{\circ }- 60^{\circ } = 60^{\circ } + m \angle DAC - 60^{\circ }$

$20^{\circ } = m \angle DA C$

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Intermediate Geometry : How to find an angle in an acute / obtuse triangle

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #21 : How To Find An Angle In An Acute / Obtuse Triangle

Example Question #51 : Acute / Obtuse Triangles

Example Question #23 : How To Find An Angle In An Acute / Obtuse Triangle

Example Question #24 : How To Find An Angle In An Acute / Obtuse Triangle

Example Question #25 : How To Find An Angle In An Acute / Obtuse Triangle

Example Question #26 : How To Find An Angle In An Acute / Obtuse Triangle

All Intermediate Geometry Resources

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